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The development of analysis of variance techniques for angular data.
HARRISON, David.
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THE DEVELOPMENT OF ANALYSIS OF VARIANCE TECHNIQUES
FOR ANGULAR DATA
DAVID HARRISON
A thesis submitted to the Council for National Academic Awards in partial fulfilment of the requirements for the degree of Doctor of Philosophy
Sponsoring Establishment Department of Applied Statistics and Operational Research
Sheffield City Polytechnic
September 1987
^ v \
TH E DEVELOPMENT OF ANALYSIS OF VARIANCE TECHNIQUES FOR
ANGULAR DA TA
D HARRISON
ABSTRACT
In many areas of research, such as within medical statistics, biology and geostatistics, problems arise requiring the analysis of angular (or directional) data. Many possess experimental design problems and require analysis of variance techniques for suitable analysis of the angular data. These techniques have been developed for very limited cases and the sensitivity of such techniques to the violation of assumptions made, and their possible extension to larger experimental models, has yet to be investigated.
The general aim of this project is therefore to develop suitable experimental design models and analysis of variance type techniques for the analysis of directional data.
Initially a generalised linear modelling approach is used to derive parameter estimates for one-way classification designs leading to maximum likelihood methods. This approach however, when applied to larger experimental designs is shown to be intractable due to optimization problems.
The limited analysis of variance techniques presently available for angular data are reviewed and extended to take account of the possible addition of further factors within an experimental design. These are shown to breakdown under varying conditions and question basic underlying assumptions regarding the components within the original approach.
A new analysis of variance approach is developed which possesses many desirable properties held in standard 'linear' statistical analysis of variance.
Finally several data sets are analysed to support the validity of the new techniques.
ACKNOWLEDGEMENTS
I would like to express my gratitude to Dr G K Kanji, acting Head of Department,
and Dr R J Gadsden, Principal Lecturer, for introducing me to this field and under
whose supervision and constant encouragement this work has been carried out. I
would also like to thank the members of the Departments of Mathematics and
Statistics for their assistance on specific points.
My indebtedness to the work of Professor K V Mardia will be evident; I should
here like to thank him, in addition, for his valuable comments and suggestions.
I wish to thank Ms Ann Moon for the excellent typing of this thesis produced from
the demanding material with which she had to work.
Finally, and most of all, I would like to express my love and gratitude to my wife
Carole, for her patience and understanding.
CONTENTS
PageCHAPTER 1
INTRODUCTION
1.1 Directional Data 1
1.2 A General Thesis Review 3
1.3 Probability Distributions on the Circle 6
1.4 Statistics of a Circular Distribution 10
CHAPTER 2
TH E VON MISES DISTRIBUTION
2.1 Derivation 15
2.2 Properties of the von Mises Distribution and its
Parameters 16
2.3 The Distribution of R for the Uniform and von Mises
Distributions 21
2.3.1 Preliminary Results 21
2.3.2 Distribution of R for the Uniform Distribution 22
2.3.3 Distribution of R for the von Mises Distribution 24
2.4 Combining von Mises Distributions 24
2.5 A Summary of Exact and Approximate Moments of R 27
CHAPTER 3
LIKELIHOOD RATIO TESTS AND APPROXIMATIONS TO TH E
M AXIM UM LIKELIHOOD ESTIMATES
3.1 The Method of Maximum Likelihood and Likelihood Ratio
Tests 30
- iii -
3.2 Approximations for the von Mises Concentration Statistic, k,
when k is small 33
3.3 Approximations for the von Mises Concentration Statistic, k ,
when k is large 42
3.4 The Best Approximation for the von Mises Concentration
Statistic, k, in the range 1 < k < 2.5 52
3.5 Summary of Approximations 54
3.6 Expansion of [ I0(A )/I0(B)]N 56
CHAPTER 4
THE DEVELOPMENT OF CIRCULAR ANALYSIS OF VARIANCE
TECHNIQUES
4.1 Introduction 58
4.2 Exact and Approximate Tests of Significance from Watson
and Williams 59
4.2.1 An Exact Test 59
4.2.2 Approximate Tests 59
4.3 Hypothesis Testing Concerning the Mean Direction 61
4.4 Multi-sample Tests Concerning the Mean Direction 65
4.4.1 For Small Concentration Parameter, k 66
4.4.2 For Large Concentration Parameter, k 66
4.5 Multi-sample Tests for the Equality of Concentration
Parameters, fcj 69
4.5.1 Rj/Nj < 0.45 70
4.5.2 0.45 < Rj/Nj 0.70 71
4.5.3 Rj/Nj > 0.70 71
- iv -
CHAPTER 5
PARAMETER ESTIMATION LEADING TO M A X IM U M LIKELIHOOD
METHODS FOR LARGER EXPERIMENTAL DESIGNS
5.1 Introduction 73
5.2 Constraints for Circular Models 75
Example 5.2.1
5.3 One-way Classification 77
Example 5.3.1
5.4 Randomised Complete Block and Larger Experimental Designs 84
Example 5.4.1
5.5 Summary 90
CHAPTER 6
EXTENDING TO LARGER EXPERIMENTAL DESIGNS
6.1 Introduction 91
6.2 Nested or Hierarchical Design 92
6.3 The Randomised Complete Block Design and Two-way Classification
with Interaction Design 94
6.4 Accuracy of the Associated x 2 Distributions for the
Randomised Block Design and the Two-way Classification
and their Corresponding F Statistics 98
6.5 Summary 114
- v -
CHAPTER 7
THE ROBUSTNESS AND POSSIBLE COLLAPSE OF TH E EXTENDED
TECHNIQUES
7.1 Introduction 115
7.2 Robustness of Assumptions 115
Example 7.2.1
7.3 A Further Approach to Watson and Williams Design Using a
Regression Model 119
7.3.1 Total Measure of Variation 122
7.3.2 Residual or Within Measure of Variation 123
7.3.3 Between Measure of Variation 124
7.3.3.1 Combining Mean Angular Directions and Resultant
Lengths 125
7.4 Cross Product Terms and the Analysis of
Cross-Classification 130
7.5 Summary 134
CHAPTER 8
DEVELOPMENT OF A NEW ANALYSIS OF VARIANCE PROCEDURE
8.1 Introduction 135
8.2 Minimising Chord Distances to Build New Model Components 135
8.2.1 Total Measure of Variation 136
8.2.2 Residual Measure of Variation 138
8.2.3 Between Measure of Variation 139
8.3 Cross Product Terms 141
8.4 Analysis of Cross-Classification and Interaction 143
8.4.1 The Interpretation of Interaction on the Circle 143
8.4.2 The Calculation of Interaction on the Circle 147
8.5 Other Design Models 151
8.5.1 The Randomised Complete Block Design 151
8.5.2 Latin Square Design 152
8.5.3 Nested or Hierarchical Design 154
8.5.4 Three-way Classification 155
8.6 Summary 156
CHAPTER 9
TH E DISTRIBUTION OF THE NEW COMPONENTS A ND TEST
STATISTICS
9.1 Introduction 157
9.2 Small Concentration Parameter, k 158
9.3 Large Concentration Parameter, k 160
9.3.1 Distribution of fc(N-(C2/N)) 161
9.3.2 Distribution of fc((R2/N )-(C 2/N)) 162
9.3.3 Distribution of fc(N-(R2/N )) 162
9.4 The Variance of the Component Chi-Squared Approximations 163
9.4.1 Variance of (N -(R 2/N )) 163
9.4.2 Variance of (N-^(R2j/N 2j)) 166
9.4.3 Variance of ® R 2j/N j)-(R 2/N)) 167
9.5 The Adequacy of the New Procedure, for Large k 168
9.6 Comparing the Power of the Tests, for Large k 184
9.7 The Adequacy of the New Procedure, for Small k 187
9.8 Comparing the Power of the Tests, for Small k 191
9.9 Summary 194
- vii -
CHAPTER 10
TH E RANDOMISED COMPLETE BLOCK AND T W O -W A Y DESIGNS
V IA TH E NEW APPROACH
10.1 Introduction 196
10.2 Randomised Complete Block and Two-way Classification with
Interaction Designs, for Large k 196
10.3 Accuracy of the Associated \ 2 Approximations for the
Randomised Complete Block and the Two-way Classification
Designs with their Corresponding F Statistics, for
Large k 200
10.3.1 The Randomised Complete Block Design 200
10.3.2 Two-way Classification with Interaction 211
10.4 Randomised Complete Block and Two-way Classification with
Interaction Designs, for Small k 218
10.5 Accuracy of the Associated Chi-squared Approximations for
Small k 219
10.6 Summary 227
CHAPTER 11
ANALYSIS OF VARIANCE EXAMPLES FOR CIRCULAR STATISTICS
11.1 Introduction 228
11.2 One-way Analysis 228
Example 11.2.1 for Large k
Example 11.2.2 for Small k
11.3 Randomised Complete Block Design 234
Example 11.3.1 for Large k
Example 11.3.2
11.4 Two-way Classification with Interaction
Example 11.4.1 for Large k
Example 11.4.2 for Small k
11.5 The Latin or Graeco-Latin Square
Example 11.5.1
11.6 Split Plot Design
Example 11.6.1
11.7 Summary
CHAPTER 12
SYNOPSIS OF RESULTS AND CONCLUSIONS
Appendix A
Index of Notation
Appendix B
The Simulation and Accuracy of Numerical Results
References
CHAPTER 1
INTRODUCTION
1.1 Directional Data
In many scientific fields the experimenter is interested primarily in the direction of a
measured variable. These observations will be bearings from some central point, or
origin, ending on a sphere or circumference of a circle, and may be regarded as
vectors. The radius can be represented as a unit vector, while the length or
magnitude of the vector is not important. Directions may be thought of in any
number of dimensions but in practice they are invariably collected in two or three
dimensional space. The tests and new results presented in this thesis are solely
concerned with directions in two-dimensions. Directions are measured by angles
ranging from 0* to 360*, or, equivalently, from 0 to 2x radians. Circular or
directional data is the name given to data which arise when the observations are
angles.
There are many examples of circular data originating from various disciplines. For
example, geologists study the orientation of fractures in deformed rocks to interpret
structural changes, and the orientation of cross-bedding or particles in undisturbed
sediments to the direction of depositing currents of wind and water. (Pincus (1953),
Curray (1956), Sengupta and Rao (1966), and Sanderson (1976)). The classic example
of directional data is from the study of bird orientation in homing or migration
which involves observing the birds vanishing angles from their release point.
Zoologists use such data to investigate consistency of bird migration under certain
conditions. Many examples from this field of study are well cited and illustrated by
Batschelet (1965, 1981).
- 1 -
Directional data is not confined to observations directly measured in degrees or
radians, but may also occur in the area of biological rhythms. A period of 24 hours
corresponds to a full turn of 360 degrees. Similarly, a month, a year or any other
period of a cyclic event may be represented by a rotation of 360 degrees. The
number of deaths due to a disease or the number of onsets of a disease in each
month over years fall in this category and can be treated as directional or circular
data. Other examples of this type can be seen in Gumbel (1954).
It is tempting to use the conventional measures of location and spread used in linear
analysis to analyse directional data. For example, suppose our data are the four
values 5, 14, 351 and 10, a simple arithmetic mean would give a value of 95. For
linear analysis this is understandable as the value of 351 has a large influence and
draws the mean away from the other data points. If these values are now regarded
as angles the spread of the whole sample is reduced, since in angular terms the
sample value of 351* is now situated close to the other data points. Similarly the
point of central location will have changed considerably and can now be seen to be
around zero degrees. Then the simple arithmetic mean would not, in general, give a
meaningful mean direction of the sample, similarly, the standard deviation would not
give a good measure of dispersion. If, however, the zero direction was taken at a
different position on the circle such as the y-axis in place of the x-axis then the
linear measure may give a sensible result. For example, if the above sample values
were rotated by 90*, to become 95*, 104*, 81* and 100*, the arithmetic mean would
give a sensible result of 95*. It is therefore not possible to define an arithmetic
mean or standard deviation in such a way that it is invariant under a rotation of the
circle. This heavy dependence on the zero direction shows the inappropriate use of
basic linear methods for circular statistics. Simple examples of such problems are
given by Batschelet (1965, 1981), Mardia (1972) and Watson (1983). Distribution
functions, characteristic functions and moments all suffer from the same draw-back
and must in some way take account of the natural periodicity of the circle.
- 2 -
1.2 A General Thesis Review
Having introduced the study of directional data, this section gives a brief review of
the work discussed within each of the following chapters, whilst indicating the
structure and progression of the thesis as a whole. The following two sections within
this chapter give further background to the subject of directional data. The first
discusses different types of probability distributions that may exist on the circle, whilst
the second states the elementary statistics of angular data required for further use in
later chapters.
In Chapter 2 the von Mises distribution is discussed from estimation and distributional
view points. The maximum likelihood estimates of the von Mises parameters are
seen to be asymptotically independent so that construction of simple large-sample
tests, for differing hypotheses regarding the parameters, may be carried out. Interest
is focused on the distributional form of the resultant length on the random (Uniform)
distribution, k - 0, and the von Mises distribution. The results from the special
case of the random distribution are required since terms in its solution arise again in
the general distribution theory. A summary of exact and approximate moments of
the resultant length, R, are given in preparation for the derivation of further circular
statistical tests.
Chapter 3 is a review of the maximum likelihood results and tests for the von Mises
distribution. Extensive investigation of the many approximations for the von Mises
concentration parameter, k, for both small and large k, has been undertaken. Seven
approximations for the maximum likelihood estimate of small k, k, which have been
cited by various authors are reviewed. Their corresponding values, residuals and
relative residuals are plotted to enable comparison and evaluation of their accuracy.A
Similar work is carried out for nine approximations to large k, k. A summary of
the 'best' and 'best simple1 approximations is given in Section 3.5 against varying
- 3 -
ranges of the concentration parameter.
Chapter 4 gives an historical review of the development of analysis of variance
techniques. The work covers the exact and approximate tests for differing mean
directions derived by Watson (1956), Watson and Williams (1956) and Stephens
(1962a, b and c), to multi-sample tests for the equality of concentration parameters.
The homogeneity tests for varying ranges of concentration parameter are cited for
later use with new design tests.
Chapter 5 discusses the use of the generalised linear modelling approach for circular
statistics to derive parameter estimates leading to maximum likelihood methods. For
circular statistics it is shown to be desirable to choose the constraint on the angles
specifying the factor parameters so that their sines sum to zero. Section 5.3 shows
how parameter estimates for the one-way classification design may be found and
therefore assist in further understanding of the underlying structure under
investigation. The approach, however, when applied to larger experimental designs is
seen, at this time, to be intractable since the optimization procedures cannot be
solved due to the numerous local maxima found within the constrained equations.
Chapters 6 and 7 examine the possibility of extending the original procedure for the
one-way analysis, derived by Watson and Williams (1956), to larger designs, for large
k. Chapter 6 shows the construction of the nested or hierarchical design, the
randomised complete block and two-way classification design with interaction together
with a comparison of their accuracy to the chi-squared and F distributions. Chapter
7, however, shows the possible collapse of these test statistics under particular
circumstances. The problems associated with the combining of circular mean
directions are shown to be influential in this collapse whilst the cross-product terms
are seen to be non-zero and requiring a correction factor to eliminate them.
- 4 -
Chapter 8 develops a new analysis of variance approach by taking account of the
resultant lengths together with their corresponding mean directions to eliminate the
possible collapse discussed in Chapter 7. The method is still based on maximum
likelihood techniques but requires the user to test for equality of concentration
parameters prior to testing for any difference between mean directions. The
cross-product terms are examined and found to equal their desired combined value of
zero. An investigation of the interpretation and representation of interaction on the
circle is given in Section 8.4 prior to its calculation via the new approach. For the
two-way design the cross-product terms are again shown to equal zero. Further
designs are then constructed in the same manner.
Following the development of the procedures in Chapter 8, Chapter 9 examines the
statistical theory and distributions behind the new design components and test
statistics. The exact theoretical distributions are seen to be intractable, and therefore
distribution approximations are used to examine the theory whilst simulation
techniques reproduce the distributions of the test statistics for comparison with their
assumed expected distributions. The comparisons are carried out for both large and
small k and test statistic improvements are made using the component moments.
The power of the new tests are also compared with existing tests for the
multi-sample case and are seen to compare favourably for both large and small k.
Chapter 10 reproduces the components within the new procedure for the randomised
complete block and two-way designs together with their improvement factor derived
in Chapter 9. The component statistics and test statistics are compared to their
respective exact chi-squared and F distributions. These two designs are used to
illustrate the validity of the approach for larger more complex design situations.
- 5 -
Chapter 11 gives several examples where the new approach is applied to real data
sets with varying sizes of concentration parameter. The examples vary from the
one-way design to the Graeco-Latin square and split plot designs.
Finally Chapter 12 summarises the development of the new analysis of variance
techniques. The adequacy of the new procedures, produced for both large and small
concentration parameter, are discussed together with their respective components and
test statistics.
Appendix A gives a list of notations used throughout the thesis together with the
design notations set out in tabular form. Appendix B reviews the techniques used to
simulate the von Mises distribution and the required experimental designs. The size
and accuracy of the numerical results are also discussed.
1.3 Probability Distributions on the Circle
There is no single distribution on the circle which has all the desirable properties
which the Normal distribution possesses on the line. Most of the distributions on the
circle have been derived either from transformations of the standard univariate (or
bivariate) distributions or as circular analogues of important univariate characteristics.
Linear distributions may have a finite range, range to infinity, or may even extend
over the whole straight line. Circular distributions, however, are always finite
ranging from 0* to 360’ (or, equivalently, 0 to 2ir), or are fractions within this
range.
In general, circular distributions are continuous over the circumference of the circle
and may be specified by a probability density function f(0), which is a periodic
function satisfying
- 6 -
" 27T
f ( 0 ) d0 - 1 ( 1 . 3 . 1 )
. o
Although no circular distribution holds all the desirable properties seen in the Normal
distribution, the von Mises distribution (originally referred to as the Circular Normal
distribution) is the most generally used distribution in statistical inference on the
circle. The importance of the von Mises distribution on the circle is often compared
to that of the Normal distribution on the line.
The distribution has probability density function
f ( 0 ) = 2ffT (E) exp^ cos( 6 ~ Mo)l 0 < 6 < 2x ( 1 . 3 . 2 )
where I 0(fc) is a modified Bessel function, and A: is a parameter of concentration of
the data about a mean direction nQ. A complete discussion of the von Mises and
its properties can be seen in Chapter 2.
There are two limiting cases of circular distributions. The first case occurs when all
the angles are the same i.e. concentrated at one single point 8 = fiQ. The 'point'
distribution has little if no practical or theoretical interest here, but has been used
for the analysis of Brownian movement and the paths of beta rays.
The second limiting case is the Uniform distribution, where every angle on the unit
circle has an equal chance of occurring or no sector is preferred to any other sector.
The probability density of 6 is constant over the whole circumference and is defined
by
f ( 0 ) = 0 < 8 < 2tt ( 1 . 3 . 3 )
As there is no concentration of points about any given direction on the circle, then
no mean direction exists. Polya (1935) used an analogue of this when he investigated
whether the stars are distributed at random over the celestial sphere.
- 7 -
As in the linear case an infinite number of circular distributions exist. Among these,
a few, possessing some desirable properties, have received attention. After the von
Mises, probably the most important is the wrapped Normal distribution. This
distribution is a natural conversion of the Normal distribution and is obtained, as its
name suggests, by wrapping the Normal distribution around the circumference of the
circle and adding those probabilities that fall into the same sector of the circle. The
addition of the overlapping tails leads to a rather complicated density function,
though when cr is small the distribution will be approximately Gaussian in (0 ,2k ).
Like the Circular Uniform distribution the wrapped Normal distribution possess the
additive property i.e. the sum of two or more wrapped Normal distributions produces
another wrapped Normal distribution with related parameters. The probability density
of a random variable 0, with mean angle / i0 = 0* , from the wrapped Normal
distribution is
f ( 0 ) ----------- — T exp[~ (<> t A m ) 2 ] o < e < 2x ( 1 . 3 . 4 )O' V 2 7 1 L 1 2<r 1n=-oo
where the mean vector length is
- c r 2exp
As p tends to zero, the wrapped Normal distribution approaches the Uniform
distribution, and as p tends to one it is concentrated at a single point. The
distribution has applications in the study of diffusion processes, and amongst others
has been examined by Stephens (1963) and Bingham (1971).
Another distribution which has been wrapped around the circle in a similar manner
to the Normal is the Cauchy distribution. The result is again a unimodal and
symmetric circular distribution possessing the additive property. With mean angle fiQ
and mean vector length p the probability density function for the wrapped Cauchy is
- 8 -
introduced by Levy (1939) and studied by Wintner (1947).
Other circular distributions of less importance include the cosine distribution, also
called the sine wave distribution, with mean vector length p and mean angle p 0, and
has the density function
f ( 0 ) — 7s— + — cos(0 - p 0) ( 1 . 3 . 6 )Z tt tt
and the cardioid distribution with the density function
f ( 0 ) - [1 + 2p cos(0 - (iQ) ] ( 1 . 3 . 7 )
introduced by Jeffreys (1948).
All the circular distributions discussed so far have been unimodal distributions, with a
single preferred direction of p Q. There are, however, circular distributions with
multimodal directions. (1.3.8) gives the density function of a multimodal von Mises
distribution where v denotes the number of modes.
f (0) = 2^1 ( k ) exP ^ cos v ( 0 " ^o)3 ( 1 . 3 . 8 )
This was suggested by Breitenberger (1963) and further investigated by Stephens
(1965).
Many examples of bimodal data can be found, particularly in scientific fields where
orientation is measurable but direction is not. Batschelet (1965, 1981) gives many
examples of bimodal data from animal orientation and navigation. A similar situation
occurs if we observe the position of undirected straight lines or undirected axes.
Gadsden and Kanji (1983) collected this type of data on clay particles following the
removal of their electric charge. These particles do not have a 'head' or 'tail' and
so the observations lie in the range 0 to x radians.
Other unimodal circular distributions were discussed by Mardia (1972 p.48-61).
Batschelet (1981 p.275-90) also reviews skewed, flat-topped and sharply peaked
circular distributions.
The most widely used circular distribution, however, is the von Mises distribution and
it is on this distribution that this thesis is based. The following chapters investigate
and extend the theory and uses of this distribution.
1.4 Statistics of a Circular Distribution
As we noted in Section 1.1 the simple arithmetic mean would not, in general, give a
meaningful mean direction of a sample of angles 0 1> e 2, ...., %• The mean
direction in circular statistics is determined by applying trigonometric functions.
Let Pj be one of the N observed angles 0j, i = 1, 2,...,N , with origin 0. Let q
and S[ be the rectangular components of Pj. Then by definition of sine and cosine,
c j = cos 0j s j = s in ( 1 . 4 . 1 )
where
N N
( 1 . 4 . 2 )i = l
Therefore, if R is the length of the resultant vector with components
mean direction, with components C and S, then
r = (C 2 + S2) 2 ( 1 . 4 . 3 )
N N
R - Nr ( 1 . 4 . 4 )
i= l i= l - 10 -
Applying basic trigonometry to calculate 6, the mean direction of the sample
arc tan
180 + arc tan
i f C > 0
i f C < 0
with the exceptional cases of
90* i f C = 0 and S > 0
270* i f C = 0 and S < 0
undetermined i f C - 0 and S = 0
Figure 1.4.1 illustrates these circular measures:
( 1 . 4 . 5 )
( 1 . 4 . 6 )
uni
= Sin 0£.
Cc = coS Oc
Figure 1.4.1 The Circular Statistics
- 11 -
It is clear from Figure 1.4.1 that the length of the resultant cannot exceed N and
similarly, from (1.4.4), that r cannot exceed 1.
Hence,
In the extreme case when all sample points fall onto the same point, the length of
the mean vector, r, equals 1. When points are close together, concentrated over a
small arc, the centre of mass is still very close to the circumference of the unit
circle, and r is close to 1. Less concentration leads to smaller values of r. At the
lower end r = 0, with no concentration around a single direction. Hence, in
unimodal samples, the mean vector length, r, serves as a measure of concentration.
From the above results we may now see that 6 has some desirable properties as a
measure of location. One such property is that the mean vector does not depend on
the zero direction of the sample. If a rotation of \p is applied to each angle, then
the sample values 6\ turn into 6[ = 6\ - \p. Similarly the new mean angle is
6" = 6 - \p, but the mean vector length, r, remains invariant.
Examining the sine of the difference between the mean angular direction and the
sample angles, it is easily shown that
( 1 . 4 . 7 )
and
( 1 . 4 . 8 )
N
d . 4 . 9 )
i = l
which is analogous to
N
( 1 . 4 . 1 0 )
in linear statistical analysis.- 12 -
Similarly, for the cosine of the difference
N
J cos(6{ - 6) = R ( 1 . 4 . 1 1 )
i = l
and therefore
N
i T 2[1 - cos( 9 j - ? ) ] - 2(1 - r ) ( 1 . 4 . 1 2 )
1=1
For small deviations, 6\ - 6
2[1 - cos( 0 j - 0) ] « ( e i - I ) 2 ( 1 . 4 . 1 3 )
hence,
N
1 J (« i - 9 ) 2 * 2(1 - r ) ( 1 . 4 . 1 4 )
i= l
which is analogous to
N
i ^ (X | - x ) 2 = s 2 ( 1 . 4 . 1 5 )
i = l
in linear statistics.
Equation (1.4.12) may be defined as the angular variance and, from (1.4.14), is
asymptotically equivalent to the variance in linear statistics.
Taking the square root of (1.4.12) gives a measure of dispersion, equivalent to the
standard deviation
s = [2(1 - r ) ] i ( 1 . 4 . 1 6 )
called the mean angular deviation.
- 13 -
The basic results of this section were adapted from Batschelet (1981) where further
discussion of the properties are given. The analogies have been reiterated here so
further use may be made of them in later chapters.
- 14 -
CHAPTER 2
TH E VON MISES DISTRIBUTION
2.1 Derivation
Gauss showed that the Normal distribution can be derived by the method of
maximum likelihood with the single assumption that the mean is the most probable
value. Von Mises (1918) applied this to a circular variate, and for this reason
Gumbel, Greenwood and Durand (1953) referred to the distribution as the Circular
Normal Distribution. Von Mises procedure was for a distribution f(0| - n 0), such
that the direction f iQ upon N observations 0 lt 0 2, ...... 0^ is a maximum given by
the constraints
where f(0) is the required distribution and f'(0 ) is the first derivative of f(0) with
respect to n Q. Since the equations (2.1.1) and (2.1.2) are identical for each 0j,
therefore
N
(2 . 1 . 1)i= l
and
(2 . 1 .2 )
f ' ( * - Mo)-------------------------s in (0 - /t0)f (0 - H0)
( 2 . 1 . 3 )
The equation has the solution
f (0 - j i0) - U exp[/c cos(0 - / i0) ] ( 2 . 1 . 4 )
where the two variables U and k are linked by the condition (1.3.1).
- 15 -
Hence
U -1
exp[k cos (6 - n Q) ] d(0 - /tQ)o
1 ( 2 . 1 . 5 )2x10(k)
where 10(k) is the modified Bessel function of the first kind and order zero. A
proof of (2.1.5) can be seen in Mardia (1972, p.58).
The von Mises distribution then, denoted as M (/*0,fc), is given by
The von Mises distribution is unimodal and symmetric with its mode at f iQ and
anti-mode at n Q + x. For k=0, the von Mises degenerates into the Uniform
distribution, and for large k the distribution concentrates around the mean direction.
Therefore, k is called the parameter of concentration. The concentration parameter
k is analogous to the inverse of the variance parameter cr2 of the Normal
distribution in its effect on the shape of the distribution. For sufficiently large k we
may approximate the von Mises by the Normal distribution. Using an approximation
quoted by Bickley (1957), for large k
2x10(k) exP ^ cos(0 “ *o>] (2 . 1 .6)
2.2 Properties of the von Mises Distribution and its Parameters
1 exp (k)10c « - — -----------
(2 r k ) i ( c ( k ) )
where
2 1(2 .2 . 1)
- 16 -
Extending the limits of this approximation would be reasonable since, when k is
large, the additional area is negligible. Replacing this into (2.2.2), a von Mises with
mean at zero;
f (» )1
2x 1n(k) exp(/c cos 0) ( -X < 0 < x) (2 .2 .2)
gives
f (0 ) * c(/c) k_2x exp{k((cos 0) - 1)} ( —go < 0 < °°) ( 2 . 2 . 3 )
Gumbel, Greenwood and Durand (1953) simplified (2.2.3) to obtain
M(0, k) » N ( 0 , / H )
alternatively
0 V F « N( 0 ,1) ( 2 . 2 . 4 )
Upton (1974) considered more accurate approximations by taking further terms in the
power series of ((cos 0) - 1), and produced two new approximations
v / F 0 » N ( 0 ,1) ( 2 . 2 . 5 )
and more accurately,
J T 1 - 8k124 1 + 4k ' )■
03 « N ( 0 , 1) (2 .2 .6)
A fourth approximation was considered by Upton (1974) given by Mardia (1972,
p.64), without proof;
i0 « N ( 0 , 1) ( 2 . 2 . 7 )
Upton tested the power of all the approximations, finding that all four consistently
overestimated the upper tail probability. Approximation (2.2.6) was found to be the
best, and this was later confirmed from further work by Hill (1978).
Stephens- 17 -
(1962c) gave another approach to the equality of the two distributions using the
moment generating function.
Estimates of the parameters k and /x0 may be obtained by means of the maximum
likelihood method.
A sample of N angles •••% are collected from a von Mises distribution with
unknown population parameters k and pL0 which we wish to estimate. The
probability density for these angles is
CN exp[/c cos(0 1 - /*0) ] exp[/c cos(0 2 - /*0) ] ............
- CN exp k[cos(6^ - / i0) + cos(0 2 - fiQ) + (2 .2 . 8)
where
12x10(k)
The log likelihood function is
N
log L(f i0 ,k) - -N log 10 (k ) + k ^ cos(0 j - / i0) + const
i= l
( 2 . 2 . 9 )
For the maximum likelihood estimate of the parameter (i0
“ k [ s i n ( 6 , - n 0) + s in (0 2 - fLQ) +
N
(2 .2 . 10)
aand vanishes for the particular value (nQ) with
N
- 18 -
From (1.4.yj we Know mat mis equauon is sausriea ior me sampie mean angie a
which we calculated in (1.4.5). Hence, the maximum likelihood estimate of the
parameter p Q of a von Mises distribution is the sample mean angle 0. Bingham and
Mardia (1975) showed that there exists only one circular distribution for which the
sample mean angle is the maximum likelihood estimate of the population mean angle,
namely the von Mises distribution.
For the maximum likelihood estimate of the parameter k
Ndlog L 1, (10------------------N ---------
d(/c) I 0(k)+ ^ c o s ( 0 { - H0) ( 2 . 2 . 1 1 )
Therefore dlog Ud{k) is zero if
N
- £ cos( 0 j - IX0) ( 2 . 2 . 1 2 )I 00 0 N ^
The right hand side is the mean vector length of the sample, r, as indicated by
equation (1.4.11). Hence, the maximum likelihood estimate of k is the solution of
I , < £ ) RA ( k ) ------------------------ r ( 2 . 2 . 1 3 )
I 0<£) N
If the mean direction is known to be p Qt then the maximum likelihood estimate of
k, k, is no longer given by equation (2.2.13), but instead by equation (2.2.14)
I , ( * ) X----------------- ( 2 . 2 . 1 4 )I 0W N
where X is the component of R on 6, when /x0 is known.
The solution of (2.2.14) is obtained numerically. Tables are not provided here since
adequate tables have been produced by Upton (1970, Appendix G), Mardia (1972,
- 19 -
Appendix 2.2) and Batschelet (1981, Table B). For the extreme cases, there are
approximate solutions to (2.2.13) which will be discussed in Chapter 3.
Figure 2.2.1, taken from Upton (1970), illustrates the measures R and X with their
relationship to C and S from (1.4.2). 5 is the angle between R and X.
2encjfh O Z = R
OT = Xv z = c = 2cos QO V = < 5 = 2 Sin ©£
VO
Figure 2.2.1 Statistics R, X, C and S
Clearly from Figure 2.2.1
C = R cos(0 - 6)
S = R sin(0 - 6)
R2 = C 2 + S2
The estimates of /<0 and k by the method of moments are the solutions of
C — A(/c)cos fiQ S = A(/c)sin (iQ ( 2 . 2 . 1 5 )
- 20 -
which give the same results as the maximum likelihood estimates. Hogg and Craig
(1965) have shown that 8 and R are jointly complete sufficient statistics for p 0 and
k. However, if k is known then Rcos 8 and Rsin 8 are minimal sufficient statistics
for p 0 which implies that 8 itself does not contain all the information about fiQ.
This underlines the difficulty in constructing an optimal criterion for estimating the
circular mean direction. Yet if n 0 is known then C is a complete sufficient statistic
of k, and C an unbiased estimate of A (k).
2.3 The Distribution of R for the Uniform and von Mises Distributions
The distribution of R for the Uniform and von Mises distributions were derived and
discussed by Stephens (1962a, 62b), Upton (1970) and Mardia (1972) and therefore
will not be fully reiterated here. A brief summary of results, however, will be given
in order that they may be utilised in later chapters.
2.3.1 Preliminary Results
Using the notation of Mardia (1972)
(a) Let i/{p,$) be the characteristic function of a continuous two-dimensional random
variable (x,y) where
x — r cos 8 y — r s in 8 ( 2 . 3 . 1 )
^(p,3>) - E [exp { ip r cos(0 - $ ) } ] ( 2 . 3 . 2 )
The joint density of r and 8 is given by
00 f 27T
e x p [ - ip r c o s (0 - <i>) ] p ^ (p ,<!>) dpd$ ( 2 . 3 . 3 )
- 21 -
Integrating over 0 gives the density of r
P(r) <2t )
2ir( 2 . 3 . 4 )
where Jm(x) is the standard Bessel function of order m and real argument.
Equation (2.3.4) is described as an inversion formula for the distribution of r.
Let 0 j,j= l,2 ...,N be distributed independently with probability density function fj(0),
j= l,2 ,...,N and of unit length and
N N
C - ^ cos 0j S - J s in 0 j
J - l J - l
The joint characteristic function of (C,S) is given by
Nn imp.*>
J -l
where ^j(p,$) is the joint characteristic function of (cos0j,sin0j).
( 2 . 3 . 5 )
Hence, from equation (2.3.5) the probability density function of R is given by
P(R) <2x)
2ir NJ 0<PR) (II l h < P . * ) ) P dpd4»
j “ l( 2 . 3 . 6 )
2.3.2 Distribution of R for the Uniform Distribution
To enable the construction of the distribution of R when the observations are taken
from the von Mises we shall initially consider the special case of the von Mises when
k - 0 and the distribution is Uniform.
- 22 -
The problem of finding the density of R is analogous to the problem of random
walk. Pearson (1905) required the probability that after N steps a man is at a
distance between R and R+5R from his starting point, 0. Here his steps, /, are
regarded as of unit length.
Using (1.3.3), the Uniform distribution, in (2.3.2), the characteristic function of
(cos0j,sin0j) is given by
* & ■ * > - < hexp[ ip cos(0 - <£)] d0 ( 2 . 3 . 7 )
o
From (2.3.5) the c.f. of (C,S) is given by
J„N(P) ' ( 2 . 3 . 8 )
Then substituting (2.3.8) into the inversion formula (2.3.6) for R, the probability
density function of R for the Uniform distribution is
pu (R) - R uJ0 (Ru) J N(u) du ( 2 . 3 . 9 )
where
pu (R) = 0 fo r R > N
The integral (2.3.9) is often referred to as Kluyver’s integral (1906). The asymptotic
solution of Pearson had been obtained already by Lord Rayleigh (1880). Pearson
(1906) gave another proof of Kluyvers result. Rayleigh (1919) used Kluyvers
technique to obtain the solution to the problem in three dimensions, or random
flights. Tables of pu(R) for differing N are given by Greenwood and Durand (1955)
and updated and extended by Durand and Greenwood (1957). Asymptotic
approximations will be discussed in Chapter 4.
- 23 -
2.3.3 Distribution of R for the von Mises Distribution
The joint probability density function of C and S is given by
1 Pu<R>g(C,S) - 7J exp[k#C + /cvS] --------- ( 2 . 3 . 1 0 )
2*1 ” (lc) R
where
H — cos fi0 v - s in p0
On transforming C and S to 0 and R by C = R cos 0 and S = R sin 8 in (2.3.10)
the joint p.d.f. of 8 and R is seen to be
1g ( 0 , R ) -rr exp[RR co s (0 - p 0) ] pu (R) ( 2 . 3 . 1 1 )
2x I q( /c)
0 < 0 < 2 * 0 < R < N
Integrating with respect to 0, the p.d.f. of R for the von Mises is given by
1pv ( R ) -------------------10(RR) PU(R) 0 < R < N ( 2 . 3 . 1 2 )
( i „ W ] N
where pu(R) denotes the p.d.f. for the Uniform distribution given by (2.3.9).
Equation (2.3.12) is due to Greenwood and Durand (1955). Asymptotic
approximations will be discussed in Chapter 4.
2.4 Combining von Mises Distributions
Let 0 j , 0 2 be independently distributed as von Mises M M ( v Qtk 2) respectively.
The probability distribution function of 0 = 8 y + 0 2, using the convolution formula
is given by
f 2 7rexp[r cos($- /3) ] d£ ( 2 . 4 . 1 )
o47r21 0 (k, ) I 0 (/c2)
- 24 -
where
r cos (3 - k., + k2cos(0 - a )
r sin /3 - k2sin(0 - a )
a “ Mo + v0
Proof of Equation (2.4.1), Outlined by Mardia (1972)
The convolution formula is given by
2*f 1( 0 ) f 2( r - 0) d0
2 *
2x 1 ^ ; - exP[fc.cos<« - /*»)] 2tIq(Ic2) exP[*2cos(f - 0
4 x * I 0 ( k , ) l 0 ( k 2)
2 *
exp[/c1cos(0 - f i0) + k 2cos(T - (0 - v,
Taking the exponential term of the integral
k ,c o s (0 - p 0) + k 2cos( f - (0 - v Q) )
Let £ = 0 - p 0
Then (2.4.4) becomes
k,cos £ + k 2c o s ( f - ( £ . + p0 - v Q) )
Expanding and using the sine and cosine rules produces
[k , + k 2cos( f - p 0 - v 0)]cos £ + [ k 2s i n ( f - p 0 - v 0) ] s i n £
Using the equalities (2.4.2)
- [ r cos /3]cos £ + [ r s in /3]sin £
r cos(£ - 0)
(2.4.2)
(2.4.3)
- v 0) ) ]
) ) ] d0
(2.4.4)
(2.4.5)
(2.4.6)
(2.4.7)
(2.4.8)
- 25 -
Replacing (2.4.7) into (2.4.4) gives equation (2.4.1)
As2x
exp[/c cos 0 ] d0 - 2 x l 0 (/c) ( 2 . 4 . 9 )o
(2.4.1) may be reduced i.e.
2 Xexp[/c cos 0] d0
o
2Xexp[r cos(£ - 0 ) ] d£
o
where k=r
r » [ + k 2 cos(0 - a ) ] 2 + [ k 2 s in (0 - a ) ] 2 ]^
= k 2 + k 2 + 2k.k? cos (0 - a) ( 2 . 4 . 1 0 )1 2 1
From (2.4.9), (2.4.4) now becomes
2xl„(fe)4 x M 0( k , ) I 0(/c2) ‘•“ ‘ 0
~ a , i B( O l B(ic2) l o K * ? + * i + 2 M 2 o o . ( * - a ) ) i ] ( 2 . 4 . 1 1 )
I f k^=k2=k, and using cos20 = 2cos20 - l , (2.4.11) becomes
I 0 [2k COS j ( e - Of)] ( 2 . 4 . 1 2 )0
The expression (2.4.12) is not the density of a von Mises distribution, i.e. the
convolution of von Mises distributions is not a von Mises distribution. However,
expression (2.4.12) may be approximated by a von Mises distribution. Without loss
of generality, let fiQ and v0 equal zero, the distributions M (0 ,k t) and M(0,fc2) may
now be approximated by the wrapped Normal distribution. The wrapped Normal, as
discussed in Section 1.3, holds the additive property, therefore two wrapped Normals
gives another wrapped Normal with parameter <r| = c f + a \ . This distribution can
then be approximated by M(0,fc3) where k 3 is the solution of
A(fc3) - A(/c1 )A(/c2) ( 2 . 4 . 1 3 )
- 26 -
Hence
( 0 1 + 6 2) mod 2x is approximately M (/i0 + v 0 , k 3)
and ( 2 . 4 . 1 4 )
( 0 1 - 6 2) mod 2 t is approximately M(f iQ - v Q, k 3)
Stephens (1963) has shown numerically that this approximation is satisfactory,
although reducing in accuracy as k decreases.
Full details of the exact and approximate moments of R may be found in Upton
(1970) and Mardia (1972). Here only those necessary for the improvement and
examination of tests discussed in later chapters will be given.
The distribution (2.3.12) cannot be used directly to obtain the expected values of R,
however, for large or small values of k (2.3.12) may be replaced by approximation
expressions from which the expectation of R for differing k may be calculated.
Stephens (1969) was the first to suggest and undertake the method of repeated
differentiation of the probability density function to obtain the exact even moments of
R. Upton (1970) having defined S and C by
N N
utilised the moment generating functions of S and C to obtain their exact
expectations and in turn produce the expectation of R 2 as did Stephens, as
2.5 A Summary of Exact and Approximate Moments of R
i -1 i-1
where R2 - S2 + C2 ( 2 . 5 . 1 )
E(R2) - N + N(N - l ) p 2 ( 2 . 5 . 2 )where
As k-*0, then p-»0 and E (R 2)-*N, a result expected for the Uniform distribution. As
the distribution becomes more concentrated about its angular mean direction (&-*»)
then p-»1 and E (R 2)—»N2.
Using both these approaches the exact expectation of R 4 has been calculated and
given as
Upton (1970) gives several approximations to the expectation of R by equating
distribution approximations given by Watson and Williams (1956) and Stephens (1969)
to their associated expected chi-squared values for large and small values of k. The
expectation of R for the von Mises distribution, as N— is given by
( 2 . 5 . 3 )
where
1 - 2 —(Q.k
E(R) - Np + i - ( l - p ’ ) + 0 -4 p1
IN J
( 2 . 5 . 4 )
From Watson and Williams approximation, for large k
( 2 . 5 . 5 )
From Stephens approximation, for large k
( 2 . 5 . 6 )
- 28 -
I f k and N are large, by substituting the Bessel functions, discussed later in Chapter
3;
E(R) - N - ^ ( 2 . 5 . 7 )
which is in close agreement with the result (2.5.5).
I f k is small, without being too small for the approximation to the ratio of Bessel
function to be invalid.
CHAPTER 3
LIKELIHOOD RATIO TESTS AND APPROXIMATIONS TO TH E M A XIM U M
LIKELIHOOD ESTIMATES
In the early 1920's, R A Fisher proposed a general method of estimation, called the
method of maximum likelihood. Fisher demonstrated the advantage of this method
by showing that (1) it yields sufficient estimates whenever they exist, and (2) it yields
estimates which are asymptotically (when N—*») minimum variance unbiased
estimators. In principle, the method of maximum likelihood consists of selecting that
value of the parameter 0 under consideration for which f(x 1,x2,...,xn;^), the
probability of obtaining the sample values, is a maximum.
The joint likelihood of the N observations 0 1, 0 2, . . . ,0 N from a von Mises
distribution with parameters k and fi0 is
as was given and used in Chapter 2.2.
Likelihood ratio tests utilise maximum likelihood estimates to test whether a particular
set of data is consistent with some hypothesis about its underlying distribution. The
likelihood ratio test is a uniformly most powerful test. A detailed discussion of these
tests originally formulated by Neyman and Pearson, can be seen in Kendall and
Stuart (1967).
3.1 The Method of Maximum Likelihood and Likelihood Ratio Tests
N
( 3 . 1 . 1 )
i= l
- 30 -
The likelihood ratio test provides a means by which a null hypothesis can be tested
against an alternative hypothesis. A null hypothesis k = k Q, fiQ = j tQ may be tested
against an alternative hypothesis A: = & ,, /*0 = ? 0 parameters of the population given
by f(0 ; k, fiQ). Let L 0 and L , denote the likelihoods of k Q, / *0 and k , given
the population with its parameters k and /t0. Symbolically,
N N
L 0 - II f ( 0 iJ k 0 , ?0) and L 1 — II f (0 j ; k , , £ , ) ( 3 . 1 . 2 )i - 1 i - 1
These quantities are both values of random variables, they depend on the observed
sample values 0 , , 0 2,... 0N, and their ratio.
max {L 0} under null hypothesisX -------------------: ------------------------------------------------ ( 3 . 1 . 3 )
max { L , } under alternative hypothesis
which is referred to as a value of the likelihood ratio statistic X.
Since max L 0 is apt to be small compared to max L , when the null hypothesis is
false, then the null hypothesis should be rejected when X is small.
Usually the natural logarithm of the ratio (3.1.3) is taken since, for large N, the
distribution of -21og X approaches, under very general conditions, the chi-squared
distribution with its degrees of freedom given by the number of parameters which are
constrained by the null hypothesis. Let, under the null hypothesis, the best estimatesa a
of k and /i0 be k Q and /*0 respectively, where these are either given values specified
by the hypothesis or the maximum likelihood estimates of the parameters under the
null hypothesis.
- 31 -
Similarly, let k y and be the corresponding estimates under the alternative
hypothesis. Then, using (3.1.3) the likelihood ratio statistic is
( 3 . 1 . 4 )X -
For observations from the von Mises distribution
X -
It is, therefore, not necessary to rely on being able to derive the distribution of our
test statistic theoretically since a good approximation to the distribution, using
-21og X, is available.
Since this, and as we will see later, several other successive approximations to the
likelihood ratio test statistics are used, invariably the tests are biased. However, by
equating the expectation of the test statistic to its associated chi-square expectation
(given by its degrees of freedom), we may attempt to remove this bias, and hence
obtain a more effective test.
Upton (1973, 76) utilizes this method extensively to improve his statistics for
single-sample and multi-sample tests of the von Mises distribution. (These tests will
be discussed and summarised in Chapter 4.) Many of the test statistics resulting
from the likelihood ratio method could be used without simplication. By using the
various approximations, test statistics which are simpler, both in form and use, may
be derived.
- 32 -
3.2 Approximations for the von Mises Concentration Statistic, k, when k is
Small
In Chapter 2, we have seen that if 0N are a random sample from M ( f iQik)
then the maximum likelihood estimate of k, k, is the solution of
M * ) RA ( k ) ---------------------- ( c 2 + s 2) *
I „ ( k ) N( 3 . 2 . 1 )
thus
k = A" 1 ( 3 . 2 . 2 )
where the ratio of the Bessel functions I ,(k)/10(k) will be denoted as A (k).
Limited tables of A -1 are given by Mardia (1972, p 298) and Batschelet (1980,
Tables B, C and D) based on those in Gumbel, Greenwood and Durand (1953).
Mardia and Zemroch (1975) gave a computer algorithm for calculating k and other
circular statistics by an iterative process.
In this and section 3.3 several approximations to A ” 1, which do not need tables or
large computing equipment, are given. From these functions the statistic k can be
obtained fairly accurately. The approximations stated here are taken from Dobson
(1978) and Upton (1970). In a similar manner to Upton we shall denote R/N and
X/N by x. This causes no problems since the choice is determined by whether or
not the mean direction is known or not.
Dobson initially states four approximations to A -1 which are global approximations
for all values of k. The first uses Amos (1974) equation that
and hence that A" 1 (x) is approximately
A 7 ' (x ) [ l - x 2l f l *'
: f +C (1 - x 2) + - h ( 3 . 2 . 4 )
where c = 1.46 to minimize the maximum relative error, and so k can be estimated
using k =
The other three global approximations use a noticed feature that A(x) behaves like
(2/x)tan~1x and so A_ 1(x) is like tan(xx/2). From this Dobson states the
approximation
a ; 1( x ) i + x 2| - 4X | 4 XXXtan 2 ( 3 . 2 . 5 )
Improvements are found by replacing terms in (3.2.5) by minimax values, giving
1A ; 1( x ) 1.32 + X‘ 1.32 - 1.32 tan xx
2~ ( 3 . 2 . 6 )
and
A"1(x) = (1 .28 - 0 .5 3 x 2) t a n xx ( 3 . 2 . 7 )
Compared to these global approximations, approximations to A - 1 (x) for particular
parts of its range were studied by Upton. The power series for the Bessel functions
I 0(x) and I , (x ) for small x gives
A(/c) " 2 ^ ( 3 . 2 . 8 )
On inverting the series and taking the first three terms an approximation of A *00
is given by
a ; 1(x ) - x^2 + x 2 + ] ( 3 . 2 . 9 )
- 34 -
Using the first two terms
A ^ C x ) - x [2 + x 2 ] - k 6 ( 3 . 2 . 1 0 )
Finally if k is expected to be very small, then we may simply use the first term
approximation of the power series, which gives
A7 1(x ) - 2x - k 7 ( 3 . 2 . 1 1 )
To examine these approximations further Table 3.1 lists the true value of k against
the approximations k , to k 7 between 0 and 1. For quicker and easier appreciation
of the accuracy of the approximations Figures 3.1(a) to 3.1(g) plot the residuals,
k-k'v against the true k. Figures 3.2(a) to 3.2(g) plot the relative percentage errors,
\ k-k{ \ lk, for each of the approximations.
A A A A
From the global approximations, k , , . . . , for values of k less than 1, k 2 and k 4
are clearly the best, with maximum relative errors, illustrated in Figures 3.2(b) and
3.2(d), of 0.71% and 0.84%, respectively at k=l .
A
By far the best approximation obtained from the power series is k 5, for all values of
k < 1, with maximum relative error of 0.35%. For very small values of k (lessA
than 0.2 or R/N less than 0.1), however, k 7 is by far the simplest and quickest
method of estimating. From Figures 3.1(g) and 3.2(g) we can see that, outside the
range of 0 < /: < 0 .2 , k 7 deteriorates rapidly.
A
A disappointing function approximation is k , with maximum relative error of 9.61%
for k=0.05 (the minimum value of k tested). For values of x (ie R/N) near toA A
zero, the global approximations k A are not as good as those obtained from the
power series.
- 35 -
T A B L E 3 . 1
A P P R O X I M A T I O N S T O S M A L L K
T r u e k X N
A
K 2 * 3 * 4 * 5 * 6 *70 0 0 0 0 0 0 0 0
0 . 0 5 0 . 0 2 5 0 0 . 0 4 5 2 0 . 0 5 0 0 0 . 0 5 1 8 0 . 0 5 0 3 0 . 0 5 0 0 0 . 0 5 0 0 0 . 0 5 0 0
0 . 1 0 0 . 0 4 9 9 0 . 0 9 0 4 0 . 1 0 0 0 0 . 1 0 3 6 0 . 1 0 0 5 0 . 1 0 0 0 0 . 1 0 0 0 0 . 0 9 9 9
0 . 1 5 0 . 0 7 4 8 0 . 1 3 5 7 0 . 1 4 9 9 0 . 1 5 5 4 0 . 1 5 0 7 0 . 1 5 0 0 0 . 1 5 0 0 0 . 1 4 9 6
0 . 2 0 0 . 0 9 9 5 0 . 1 8 1 1 0 . 1 9 9 9 0 . 2 0 7 1 0 . 2 0 0 9 0 . 2 0 0 0 0 . 2 0 0 0 0 . 1 9 9 0
0 . 2 5 0 . 1 2 4 0 0 . 2 2 6 6 0 . 2 4 9 8 0 . 2 5 8 8 0 . 2 5 1 0 0 . 2 5 0 0 0 . 2 5 0 0 0 . 2 4 8 1
0 . 3 0 0 . 1 4 8 3 0 . 2 7 2 3 0 . 2 9 9 6 0 . 3 1 0 3 0 . 3 0 1 0 0 . 3 0 0 0 0 . 2 9 9 9 0 . 2 9 6 7
0 . 3 5 0 . 1 7 2 4 0 . 3 1 8 2 0 . 3 4 9 4 0 . 3 6 1 8 0 . 3 5 0 9 0 . 3 5 0 0 0 . 3 4 9 9 0 . 3 4 4 7
0 . 4 0 0 . 1 9 6 1 0 . 3 6 4 3 0 . 3 9 9 1 0 . 4 1 3 1 0 . 4 0 0 8 0 . 4 0 0 0 0 . 3 9 9 7 0 . 3 9 2 2
0 . 4 5 0 . 2 1 9 5 0 . 4 1 0 6 0 . 4 4 8 8 0 . 4 6 4 3 0 . 4 5 0 5 0 . 4 5 0 0 0 . 4 4 9 6 0 . 4 3 9 0
0 . 5 0 0 . 2 4 2 5 0 . 4 5 7 2 0 . 4 9 8 4 0 . 5 1 5 4 0 . 5 0 0 1 0 . 5 0 0 0 0 . 4 9 9 3 0 . 4 8 5 0
0 . 5 5 0 . 2 6 5 1 0 . 5 0 4 1 ' 0 . 5 4 8 0 0 . 5 6 6 3 0 . 5 4 9 6 0 . 5 4 9 9 0 . 5 4 8 8 0 . 5 3 0 2
0 . 6 0 0 . 2 8 7 3 0 . 5 5 1 3 0 . 5 9 7 5 0 . 6 1 7 1 0 . 5 9 9 1 0 . 5 9 9 9 0 . 5 9 8 2 0 . 5 7 4 2
0 . 6 5 0 . 3 0 9 0 0 . 5 9 8 9 0 . 6 4 6 9 0 . 6 6 7 8 0 . 6 4 8 4 0 . 6 4 9 8 0 . 6 4 7 4 0 . 6 1 7 9
0 . 7 0 0 . 3 3 0 2 0 . 6 4 6 8 0 . 6 9 6 4 0 . 7 1 8 4 0 . 6 9 7 6 0 . 6 9 9 6 0 . 6 9 6 3 0 . 6 6 0 4
0 . 7 5 0 . 3 5 0 9 0 . 6 9 5 1 0 . 7 4 5 8 0 . 7 6 8 9 0 . 7 4 6 7 0 . 7 4 9 4 0 . 7 4 5 0 0 . 7 0 1 8
0 . 8 0 0 . 3 7 1 1 0 . 7 4 3 8 0 . 7 9 5 2 0 . 8 1 9 2 0 . 7 9 5 8 0 . 7 9 9 1 0 . 7 9 3 2 0 . 7 4 2 1
0 . 8 5 0 . 3 9 0 7 0 . 7 9 2 9 0 . 8 4 4 6 0 . 8 6 9 5 0 . 8 4 4 8 0 . 8 4 8 7 0 . 8 4 1 1 0 . 7 8 1 5
0 . 9 0 0 . 4 0 9 8 0 . 8 4 2 4 0 . 8 9 4 0 0 . 9 1 9 7 0 . 8 9 3 7 0 . 8 9 8 1 0 . 8 8 8 5 0 . 8 1 9 7
0 . 9 5 0 . 4 2 8 4 0 . 8 9 2 4 0 . 9 4 3 4 0 . 9 6 9 8 0 . 9 4 2 7 0 . 9 4 7 4 0 . 9 3 5 4 0 . 8 5 6 8
1 . 0 0 0 . 4 4 6 4 0 . 9 4 2 8 0 . 9 9 2 9 1 . 0 1 9 9 0 . 9 9 1 6 0 . 9 9 6 5 0 . 9 8 1 7 0 . 8 9 2 8
- 36 "
approximations .for small k against k.
0.03
0.01
0 .40 .2 0. 8 1.0-o.oi
F i g u r e 3 . l C a )
0.05 •
0 .03
0.01
-0.010 .2 0 .4 0-5 o.s 1.0
-0 .0 3 '
-0 .0 5 '
F i g u r e 3 . 1 ( b )
0.05
0.03
0 . 01
"r1.32 tan1.32
-0 .0 5
Figure 3.1(c)
- 37 -
0 .05
0 .03
0.01-i<=3a
1 .00.801IL Itr -0.01
0 .40.2
0 .03
1.28 - 0 .5 3 r a ) tan
F i g u r e 3 . 1 ( d )
0 .05
0.03
J 0 .0 1
1.00 .80.60 .4£ - 0.01
■0.03'
-0 .0 5 '
F i g u r e 3 . l C e )
0 .05
< 0.01 CM ___cnliic - 0 . 0 1
1 .00 . 80.60 .4
•0.03
-0.05-
Figure 3.lCf)- 38 -
0.05
£ - 0.01
-0 .03
-0 .0 5
F i g u r e 3 . 1 C g )
F i g u r e s 3 . 2 ( a ) t o C g ) . P l o t s o f t h e r e l a t i v e p e r c e n t a g e e r r o r o f
t h e a p p r o x i m a t i o n s f o r s m a l l k a g a i n s t k .
*3*
7 .0
S .C -
5.0'
0 .4 0.5 0 .3
F i g u r e 3 . 2 ( a )
2.0
tan
0 .2
Figure 3.2Cb)_ 39 _
4 .0
1.32 tan
3 .0
(Eatr<rUJ
tu>t-it-<UJtr
00 .4 0.S 0.5 1 .0
F i g u r e 3 t. 2 C c ]
2 . 0'
(1 .25 - O.SSr1 ) tan
00 .80.50 .40 . 2
F i g u r e 3 . 2 C d 3
o0 .2 0 .4 o . s o . a
Figure 3.2Ce)
- 40 -
2.0
0 .4 0.6 0 .6 1.0
F i g u r e 3 . 2 C f )
3.0-
Ui
0.60.60 .40 .2
F i g u r e 3 . 2 C g )
- 41 -
3.3 Approximations for the von Mises Concentration Statistic, k, when k is
Large
For large values of k the power series expansion of Bessel functions does not provide
a simple approximation to (3.2.1), however, the asymptotic expansion quoted by
Abramowitz and Stegun (1965) may be used
exp[lc]I PW *
( 21c) i
m - 1 (m - l ) ( m - 9) (m - l ) ( m - 9 ) (m - 25) 1 + --------------------------------------------------------------------------
81c 2! (81c)2 3! (81c) 3
( 3 . 3 . 1 )
where m = 4p2. Using this relation
1 1 1A(k) * 1 ------------------------------+ 0 ( k “ 4) ( 3 . 3 . 2 )
2k 8 k 2 8 k 3
Denoting R/N and X/N by x, using Maclaurins theorem the solution of (3.3.2) is
1- = 2(1 - x ) - (1 - x ) 2 - (1 - x ) 3 ( 3 . 3 . 3 )k
On inverting we obtain, as the first three terms
^ - 2 ( r - h o + ? + § (1 + x) ( 3 -3 -4)
9 — 8x — 3x2 aA; 1(*> - 8 ( r - x) = ( 3 - 3 -5)
Using the first two terms of (3.3.3) only
- 4(1 I x) - (3-3-6)If k is expected to be very large, then we may simply use the first term
approximation of the power series, giving
A-1 (x ) - 75-7-=—"— r “ K o ( 3 . 3 . 7 )i o 2 ( l - x ) 10
Upton (1970) gives his solution of (3.3.2) as
- 42 -
1 3- = 2(1 - x) - (1 - x ) 2 - - ( 1 - x ) 3 ( 3 . 3 . 8 )k 2
On inverting
k = * + — + ^ (3 3 9)2(1 - x) 4 2
5 — 5v + 2v2 a
A71<*> <3 - 3 - 10)
Upton does not explain how he derives equation (3.3.8) and therefore does not state
why the third term is different to that of equation (3.3.3), however, as we shall seeA /V
later in the chapter, the approximation is better than k Q.
Upton (1973) states an approximation suggested by M A Stephens where the variance
of the maximum likelihood estimator is considered, giving
AT2( x ) “ T - ^ - k , 2 ( 3 . 3 . 1 1 )
A A A A
To examine the approximations k 8, . . . .k12 and the global approximations k 1 k A,
given in Section 3.2 for large values of k, Table 3.2 lists the true values of k
between 1 and 20 against these approximations.
A
As with small values of k , the residuals, k-k'v have been plotted for each
approximation for easier appreciation of their accuracy, given in Figures 3.3(a) to
3.3(i). Similarly Figures 3.4(a) to 3.4(i) plot the relative percentage errors for each
of the approximations.
Figures 3.3(a) to 3.3(d) show the residuals for the global approximations, k 1, . . . .k4.
Due to the large residuals shown within these approximations the graph scales, for
analysis of large k t have been increased compared to those of small k (Figures 3.1
(a) to 3.1(g)). Although these scales have increased, k , equation (3.2.4), has such
large residuals and relative errors, compared to the other approximations, that the
- 43 -
plots 3.3(a) and 3.4(a) still leave the graph. A residual of -0.8183 would be seen
at k=20, and a maximum relative error of 9.6 at approximately k - 4.5. Function
approximation, is unexpectedly poor for both small and large k especially
considering its complex form.
From the remaining global approximations, k 3 has the smallest maximum actual
residual, in the range k - 1 to 20, of 0.2634 at k - 20, in comparison k A has maximum
actual residual of 0.4944 at k=20. However, k A may be seen as the best global
function approximation since it has a maximum relative error of 2.47% at k=20,
while k 3 has a maximum relative error of 3.42 at k=3.85.
O f the five power series functions, from examination of Figures 3.3(e) to 3.3(i), k 8
and k 1 n are the better two approximations, with maximum residuals of 0.048 at
k=3.1, and 0.025 at k - 3.2, respectively, for values of k\ > 2.5. On examination of
the relative percentage errors k , 1 is the best approximation with maximum relative
error of 0.79% at k - 3.1, compared to 1.61% at k - 2.85 for k Q, for values of
k^> 2.5.
A
It is interesting to note the adequacy of k 10, equation (3.3.7), as this is often
quoted as a good approximation for large k. Form Figure 3.3(g) we can see that a
far greater improvement would be seen if 0.25 was added to the approximation,
producing k g.
A A
For almost all k , ......... k , 2 approximations the actual and relative errors fluctuate
most in the range 1 < k < 2.5.
- 44 -
T A B L E 3 . 2
A P P R O X I M A T I O N S T O L A R G E K
T r u e k XS k 3 *4 "a
CD<
K _ 1 0 k 1 1 K 1 2
1 . 0 0 . 4 4 6 4 0 . 9 4 2 8 0 . 9 9 2 9 1 . 0 1 9 9 0 . 9 9 1 6 1 . 3 6 0 8 1 . 1 5 3 2 0 . 9 0 3 2 1 . 4 3 0 0 1 . 2 4 8 9
1 . 5 0 . 5 9 6 1 1 . 4 7 1 7 1 . 4 9 4 7 1 . 5 2 2 2 1 . 4 8 3 5 1 . 6 3 9 5 1 . 4 8 8 0 1 . 2 3 8 0 1 . 6 9 0 0 1 . 5 5 1 3
2 . 0 0 . 6 9 7 8 2 . 0 3 9 2 2 . 0 1 5 3 2 . 0 3 5 6 1 . 9 8 8 5 2 . 0 1 7 7 1 . 9 0 4 4 1 . 6 5 4 4 2 . 0 5 5 5 1 . 9 4 8 9
2 . 5 0 . 7 6 5 0 2 . 6 2 8 1 2 . 5 5 3 1 2 . 5 6 1 2 2 . 5 0 6 8 2 . 4 6 5 8 2 . 3 7 7 6 2 . 1 2 7 6 2 . 4 9 5 1 2 . 4 1 0 9
3 . 0 0 . 8 1 0 0 3 . 2 2 0 0 3 . 0 9 8 1 3 . 0 9 1 0 3 . 0 3 0 2 2 . 9 5 2 6 2 . 8 8 1 4 2 . 6 3 1 4 2 . 9 7 6 4 2 . 9 0 7 6
3 . 5 0 . 8 4 1 1 3 . 8 0 3 4 3 . 6 4 0 9 3 . 6 1 7 4 3 . 5 5 0 5 3 . 4 5 6 3 3 . 3 9 6 7 3 , 1 4 6 7 3 . 4 7 6 2 3 , 4 1 8 3
4 . 0 0 . 8 6 3 5 4 , 3 7 4 0 4 . 1 7 7 2 4 . 1 3 6 5 1 . 0 6 3 8 3 . 9 6 4 8 3 . 9 1 3 6 3 . 6 6 3 6 3 . 9 8 1 8 3 . 9 3 1 9
4 . 5 0 . 8 8 0 3 4 . 9 3 1 9 4 . 7 0 5 9 4 . 6 4 7 9 4 . 5 6 9 7 4 . 4 7 3 1 4 . 4 2 8 2 4 . 1 7 8 2 4 . 4 8 8 0 4 . 4 4 4 1
5 . 0 0 . 8 9 3 4 5 . 4 7 9 1 5 . 2 2 8 2 5 . 1 5 2 7 5 . 0 6 9 2 4 . 9 7 9 6 4 . 9 3 9 6 4 . 6 8 9 7 4 . 9 9 3 0 4 . 9 5 3 7
5 . 5 0 . 9 0 3 8 6 . 0 1 8 0 5 . 7 4 5 5 5 . 6 5 2 5 5 . 5 6 3 7 5 . 4 8 4 5 5 . 4 4 8 4 5 . 1 9 8 4 5 . 4 9 6 5 5 , 4 6 1 0
6 . 0 0 . 9 1 2 4 6 . 5 5 0 6 6 . 2 5 9 2 6 . 1 4 8 6 6 . 0 5 4 7 5 . 9 8 8 0 5 . 9 5 5 1 5 . 7 0 5 1 5 . 9 9 8 9 5 . 9 6 6 6
6 . 5 0 . 9 1 9 5 7 . 0 7 8 1 6 . 7 7 0 0 6 . 6 4 1 8 6 . 5 4 2 8 6 . 4 9 0 4 6 . 4 9 0 2 6 . 2 1 0 2 6 . 5 0 0 5 6 . 4 7 0 7
7 . 0 0 . 9 2 5 5 7 . 6 0 2 0 7 . 2 7 8 9 7 . 1 3 3 2 7 . 0 2 9 1 6 . 9 9 2 3 6 . 9 6 4 2 6 . 7 1 4 3 7 . 0 0 1 6 6 . 9 7 4 0
7 . 5 0 . 9 3 0 7 8 . 1 2 2 7 7 . 7 8 6 3 7 . 6 2 2 9 7 . 5 1 3 9 7 . 4 9 2 5 7 . 4 6 7 5 7 . 2 1 7 5 7 . 5 0 2 2 7 . 4 7 6 5
8 . 0 0 . 9 3 5 2 8 . 6 4 1 0 8 . 2 9 2 4 8 . 1 1 1 5 7 . 9 9 7 5 7 . 9 9 4 5 7 . 9 7 0 2 7 . 7 2 0 2 8 . 0 0 2 6 7 . 9 7 8 6
8 . 5 0 . 9 3 9 2 9 . 1 5 7 2 8 . 7 9 7 8 8 . 5 9 9 2 8 . 4 8 0 2 8 . 4 9 5 2 8 . 4 7 2 4 8 . 2 2 2 4 8 . 5 0 2 8 8 . 4 8 0 3
9 . 0 0 . 9 4 2 7 9 . 6 7 2 0 9 . 3 0 2 5 0 . 0 8 6 3 8 . 9 6 2 4 8 . 9 9 6 0 8 . 9 7 4 5 8 . 7 2 4 5 9 . 0 0 3 1 8 . 9 8 1 8
9 . 5 0 . 9 4 5 8 1 0 . 1 8 5 2 9 . 8 0 6 6 9 . 5 7 2 7 9 . 4 4 3 9 9 . 4 9 6 5 9 . 4 7 6 2 9 . 2 2 6 2 9 . 5 0 3 3 9 . 4 8 3 1
1 0 . 0 0 . 9 4 8 6 1 0 . 6 9 7 2 1 0 . 3 1 0 2 1 0 . 0 5 8 7 9 . 9 2 4 9 9 . 9 9 6 8 9 . 9 7 7 6 9 . 7 2 7 6 1 0 . 0 0 3 3 9 . 9 8 4 1
1 1 . 0 0 . 9 5 3 4 1 1 . 7 1 8 0 1 1 . 3 1 6 1 1 1 . 0 2 9 3 1 0 . 8 8 5 7 1 0 . 9 9 7 3 1 0 . 9 7 9 8 1 0 . 7 2 9 8 1 1 . 0 0 3 1 1 0 . 9 8 5 9
1 2 . 0 0 . 9 5 7 4 1 2 . 7 3 5 9 1 2 . 3 2 1 2 1 1 . 9 9 9 0 1 1 . 8 4 5 7 1 1 . 9 9 7 8 1 1 . 9 8 1 8 1 1 . 7 3 1 8 1 2 . 0 0 3 1 1 1 . 9 8 7 3
1 3 . 0 0 . 9 6 0 7 1 3 . 7 5 1 4 1 3 . 3 2 5 4 1 2 . 9 6 7 9 1 2 . 8 0 4 8 1 2 . 9 9 8 3 1 2 . 9 8 3 6 1 2 . 7 3 3 6 1 3 . 0 0 3 2 1 2 . 9 8 8 6
1 4 . 0 0 . 9 6 3 6 1 4 . 7 6 4 7 1 4 . 3 2 8 9 1 3 . 9 3 6 0 1 3 . 7 6 3 1 1 3 . 9 9 8 5 1 3 . 9 8 4 9 1 3 . 7 3 4 9 1 4 . 0 0 3 1 1 3 . 9 8 9 5
1 5 . 0 0 . 9 6 6 1 1 5 . 7 7 6 6 1 5 . 3 3 1 9 1 4 . 9 0 3 7 1 4 . 7 2 1 1 1 4 . 9 9 8 9 1 4 . 9 8 6 2 1 4 . 7 3 6 2 1 5 . 0 0 3 2 1 4 . 9 9 0 5
1 6 . 0 0 . 9 6 8 2 1 6 . 7 8 6 7 1 6 . 3 3 4 3 1 5 . 8 7 0 7 1 5 . 6 7 8 4 1 5 . 9 9 8 8 1 5 . 9 8 6 9 1 5 . 7 3 6 9 1 6 . 0 0 2 8 1 5 . 9 9 0 9
1 7 . 0 0 . 9 7 0 1 1 7 . 7 9 5 9 1 7 . 3 3 6 4 1 6 . 8 3 7 4 1 6 . 6 3 5 5 1 6 . 9 9 8 9 1 6 . 9 8 7 7 1 6 . 7 3 7 7 1 7 . 0 0 2 7 1 6 . 9 9 1 5
1 8 . 0 0 . 9 7 1 8 1 8 . 8 0 4 4 1 8 . 3 3 8 8 1 7 . 8 0 4 4 1 7 . 5 9 2 9 1 7 . 9 9 9 3 1 7 . 9 8 8 7 1 7 . 7 3 8 7 1 8 . 0 0 2 8 1 7 . 9 9 2 2
1 9 . 0 0 . 9 7 3 3 1 9 . 8 1 1 4 1 9 . 3 4 0 1 1 8 . 7 7 0 3 1 8 . 5 4 9 0 1 8 . 9 9 9 0 1 8 . 9 8 9 0 1 8 . 7 3 9 0 1 9 . 0 0 2 4 1 8 . 9 9 2 4
2 0 . 0 0 . 9 7 4 7 2 0 . 8 1 8 2 2 0 . 3 4 1 8 1 9 . 7 3 6 6 1 9 . 5 0 5 6 1 9 . 9 9 9 0 1 9 . 9 8 9 5 1 9 . 7 3 9 5 2 0 . 0 0 2 1 1 9 . 9 9 2 7
- 45 -
0.05
15
-0.05oiUJ<r
- 0.10
-0 .1 5
- 0.20
-0 .2 5
- 0 . 2 0
F i g u r e 3 . 3 ( a )
0.05
13 14 15
-0 .0 5
« - 0.10
-0 .3 5
- 0 . 2 0
-0 .2 5
F i g u r e 3 . 3 C b 3
0.15
0.10
0.C5
0
-0 .0 5
1.32 1 . 2 2 :an(—-
-Q.1S
Figure 3.3Cc)- 46 -
0.20
0-25
0 .20
(1 -2 8 - O .S jr1 ] tan (-4r
0 .15
ce 0.10
0.05
-0 .0 5
F i g u r e 3 . 3 C d 3
0.05
«c= 3ah*UiUicr
-0 .0 5
a c 1 - r J
F i g u r e 3 . 3 C e )
0.15-
0.10
0.05
0
-Q.05-
- 0 . 10'
-0 .1 5 '
Figure 3.3(f]- 47 -
0.40
0.35
0.30
0 .2 5
0 . 20-
0 .1 5
0.10
0.05
1 2 3 4 5 6 7 6 9 10 11 12 13 14 15
F i g u r e 3 . 3 C g )
12
-0 .0 5
- 0 .1 0
- 0 .2 0
-0 .2 5
F i g u r e 3 . 3 ( h )
0.10
0.05
o12
-0 .0 5
12 1 - r- 0 . 1 0
-0 .1 5
- 0 . 2 0
-0 .2 5
Figure 3.3(i) 48 -
m e a p p r a x im c j uxu iib T u r i d r g t ; r\ a g a x i i& o c\.
g . n
7 .0
6 . 0
5 .0
CT§ 4 .0crtuUJ>HHt—<_]UJcr
1 -r -
2 .0
0
F i g u r e 3 . 4 C a )
5.0
4 .0
3 .0
2 .0tan
F i g u r e 3 . 4 C b )
4.0-
irr22 tancracrcrw 2.0
1.32
UJ>Mt—<- IUJcr
015
Figure 3.4Cc)- 49 -
2 .0a:ocrcrUJ
(1 .28 - 0 .5 3 ra J tan
ui> 1 .01-4»—<_JUJcr 0
F i g u r e 3 . 4 ( d )
3 .0ccacr£ 2.0tu><tucr
014
F i g u r e 3 . 4 C e )
6.0
s.o
~ 4 .0CCacrcr" 3 .0tu >HHt—<:
j 2 . 0UJcr
0
Figure 3.4Cf)
- 5 0 -
E.O
5 .0
4 .0
CCOCECCUI
3 .0
UJ>l-tt—<—1tuCC
2.0
01512 13
F i g u r e 3 . 4 C g )
3 . O'
ccaccccUJ
4 ( 1 - r J2 . 0-
tu>*-4>—<UJcc
Q14
F i g u r e 3 . 4 ( h )
4 .0
3 . 0cracccctu 1 - r
2 .0tu>t—<
010
Figure 3.4(i)- 51 -
3.4 The Best Approximation for the von Mises Concentration Statistic, k, in
the Range 1 < k < 2.5
From examining the results and graphs for the approximations for large and small k t
all the power series functions are very poor estimators in the range 1 < k < 2 .5 .
From the global approximations, Figures 3.3(a) to 3.3(d) and 3.4(a) to 3.4(d), k , is
a poor approximation and may be removed. Taking a closer examination of k 2, k 3
and k A in the range 1 < k < 2.5, the residuals for these are given in Figures 3.5(a)
to (c) respectively. Here we can see that all three are good approximations,
however, k A is the best with maximum residual of 0.0168 at k - 1.6. The maximum
percentage relative errors for k v k z and k 4 in this range are, 2.125%, 2.446% and
1 .12% respectively.
- 52 -
approximations to k in tne range
0 .0 5 .
0.03
0-01cS
2.1 2 .3 2 .5
•0.03
-0 .0 5
F i g u r e 3 . 5 ( a )
0 .05.
irrtan
0.01
02 .3 2 .5
- 0.01
-0 .0 3
-0.05
F i g u r e 3 . 5 ( b )
0.05
(1 .2 8 - 0 .5 3 ra ) tan0 .03
0.01
02.3
- 0 . 0 2
-0 .0 5
Figure 3.5Cc)- 53 -
3.5 Summary of Approximations
For the twelve approximations examined, Table 3.5.1 gives the best three
approximations of k with their respective ranges. Also shown are the best 'simple'
approximations of k.
TABLE 3.5.1
Range o f k
Best Function
Approx im ation Range o f k
Best 'S imple'
Funct ion
Approximat ion
0 < k < 1.25
or
0 < r < 0.528
5 r 4r [2+ r 2 + ^ i - ]
0 < k < 0 . 2
or
0 < r < 0 .1
2 r
0 .2 < k < 1.45
or
0.1 < r < 0.584
r [ 2+ r 2]1.25 < k « 2.45
or
0.528 < r < 0.759
[ 1 . 2 8 - 0 . 5 3 r 2] t a n ™
k > 1.45
or
r > 0.584
1
1 - r 2k > 2 .45
or
r > 0.759
5 -5 r+ 2 r 24 ( l - r )
Figures 3.6(a) and (b) plot the absolute residuals for the best and best 'simple'
approximations in their respective ranges as continuous functions.
- 54 -
ABSO
LUTE
R
ESIO
UA
t. AO
SOLU
TE
RES
IDU
AL
F i g u r e 3 . 6 ( a ) . P l o t o f t h e a b s o l u t e r e s i d u a l s f o r t h e b e s t
a p p r o x i m a t i o n s t o c o n c e n t r a t i o n s t a t i s t i c , k
0.05
0.04
0.03
[1 .2 8 - O .S S r^ tan i
0 .02
0.01
0 .5 1.0 2 .0 2 .5 3 .0 4 .0
F i g u r e 3 . 6 ( b ) . P l o t o f t h e a b s o l u t e r e s i d u a l s f o r t h e b e s t ' s i m p l e '
a p p r o x i m a t i o n t o c o n c e n t r a t i o n s t a t i s t i c , k
0 .09
o.oa
0.07
1-r-0 .06
0 .0 5
0 .04
0.03
0.02
02. 0 4 .0
- 55 -
Naturally we obtain better and better approximations using further terms in the power
series (3.3.4) and (3.3.9), however we wish to find and use fairly simple
approximations to (3.2.2) in order that tables or large computing equipment are not
required.
3.6 Expansion of [ I0(A )/IQ(B)]N
When calculating the likelihood ratio test for large or small values of k an expansion
and approximation of (3.6.1) is required.
N logIn (A)
I 0( B>( 3 . 6 . 1 )
If A and B are both small we may use the standard series expansion of Bessel
functions quoted by Bickley (1957) to obtain
i „<a >
I n ( B )
A2 A41 + — + +
4 64
1 + (A2 - B2) + 164
B2 B4 1 + — + — +
4 64
(A4 - 4A2B2 + 3B4) ( 3 . 6 . 2 )
For (3.6.1) the power series for log(l+x) is used to produce
N logI o (A) N
1 o ( B) 4(A 2 - B2) -
N
64(A4 - B4) ( 3 . 6 . 3 )
For very small values of A and B the second term may be neglected to give
N log' lo (A) ' N
1 o (B) 4(A2 - B2) ( 3 . 6 . 4 )
- 56 -
For large values of A and B the Bessel functions are expanded using the asymptotic
expansion (3.3.1) to obtain
M A> B i 1 1 1 1— exp[A - B] • 1 + - — ------- + --------
I 0<B> A 8 A B 128
9 2 7
A2 AB B2
For (3.6.1) we again expand the logorithm as a power series.
1 (A + B)’l 0(A) N BN log- , « - log -
I 0(B) 2 A+ N(A - B) 1 -
8AB 16A2B2
( 3 . 6 . 5 )
( 3 . 6 . 7 )
For very large A and B we may neglect the higher powers of 1/A and 1/B to give
’ l 0(A> N BN log . « - log -
I 0 (B) 2 A+ N(A - B) ( 3 . 6 . 7 )
- 57 -
CHAPTER 4
TH E DEVELOPMENT OF CIRCULAR ANALYSIS OF VARIANCE TECHNIQUES
4.1 Introduction
The analysis of data of a circular or spherical nature began when Lord Rayleigh
(1880) developed a one-sample test for uniformly distributed random vectors using
the resultant length. Investigation into circular distributions other than the uniform
distribution began in the early part of this century. The most important of which,
and the assumed circular distribution for much of this thesis, was the von Mises
distribution. True development of significance tests, however, did not appear until
Fisher (1953), whilst investigating the remanent magnetism of a sample of rock
specimens, considered the spherical analogue of the von Mises distribution where
observations are regarded as points on a sphere. Fisher derived the maximum
likelihood estimates of the concentration parameter and the mean direction and
provided the basic distribution theory in order to test a prescribed mean direction
when k is unknown. Watson (1956) gave a significance table for the test of k - 0
i.e. uniformity, and approximate tests for the equality of concentration parameters
and mean directions. As discussed in Chapter 2.3 Greenwood and Durand (1955)
utilised Fishers work to produce a similar distribution theory for the circular case.
In 1956 Watson and Williams derived tests for the direction and homogeneity in both
the two- and three-dimensional cases. Their exact test for the two-dimensional case
is summarised in the following section.
- 58 -
4.2 Exact and Approximate Tests of Significance from Watson and Williams
4.2.1 An Exact Test
Stephens (1972) produced an exact two-sample test for the null hypothesis that two
samples, size N 1 and N 2, have identical modal vectors, assuming they have the same
unknown value of k. For each sample the resultant lengths are found and the test
statistic Z calculated
R, + R2Z -------- ( 4 . 2 . 1 )
N
The resultant R of R , and R 2 is also found and W = R/N calculated.
The test consists of finding a critical value z satisfying Pr(Z > z/W ) = a, for
appropriate significance level a. Tables for the critical value z are presented by
Stephens (1972) and the null hypothesis is rejected at level a if Z = z. The exact
test is a conditional test based on the joint distribution of R 1t R 2 given R i.e.
P u ( R , ) P u ( R 2 )f ( R , , R 2 lR)--------------------------------- '■-------------1------------------------- ( 4 . 2 . 2 )
*Pu<R>{[(R, + R2) 2 - r2HR2 - (R, - R2>2]}*
where
0 < R 1 < n, 0 < R 2 < n 2 |R , - R 2 I < R < R , + R 2
and pu(R) is given by equation (2.3.9). Equation (4.2.2) was derived by Watson and
Williams (1956) for the circle following the derivation for the sphere by Fisher
(1953).
4.2.2 Approximate Tests
If 0 is an observation from the von Mises distribution with mode at zero then for
large k we have shown, (2.2.4), that 0 J~k is approximately N(0,1) and hence k 6 2
- 59 -
to be chi-squared distribution with 1 degree of freedom. Using this result we may
approximate for 0 2 to obtain the result that 2fc(l-cos0) is chi-squared distributed
with 1 degree of freedom.
Watson and Williams used this result and the additive properties of x 2 distributions
to produce the approximations
2 /c(l - cos 0 ) »
2/c(N - X) ( 4 . 2 . 3 )
2k(N - R)
for single sample tests.
In the two sample case, if 0 1 ( the mean direction for sample 1) equals 0 2 (the
mean direction for sample 2) then
R 1 + R 2 = R
However, if 0 1 * 0 2
then R 1 + R 2 > R
Therefore the greater the difference between 0 1 and 0 2 the greater the value of
R 1 + R 2 - R. Via equations (4.2.3), Watson and Williams showed
2/c(R1 + R2 - R) « x 2
Equations (4.2.4) are independently distributed. (Further proof of (4.2.4) and (4.2.3)
can be seen in Mardia (1972 p.114)).
- 60 -
2k(N - (R, + R2) ) ( 4 . 2 . 4 )
qwhere
J -l
Watson and Williams then stated that to test the equality of mean vectors for several
samples (for large k ), generalising Watson (1956), we should use
q
In 1962 two papers by M A Stephens produced exact and approximate tests for the
null hypothesis, concerning single sample tests, that the mean vector is a given
vector when k is unknown. For the exact test, given in section 4.2.1, Stephens
produced nomograms for differing significance levels to find R 0 given N and X,
where Pr(P > R 0 |X) = a . X being the component of R on 0, when /x 0 is known
(Figure 2.2.1).
Stephens' three approximate tests were also for the above null hypothesis, for
different ranges of N and X. For large concentration parameter the approximate test
was given by
calculating R 0 from (4.3.1). If R > R 0 , reject the null hypothesis.
Equation (4.3.1) was first suggested by Watson and Williams using the equations of
(4.2.3). Using these equations the chi-squared approximations may also be shown as;
( 4 . 2 . 5 )q
where q is the number of samples.
4.3 Hypothesis Testing Concerning the Mean Direction
<R0 - *>(N - 1) ( 4 . 3 . 1 )
2/c(N - X) = 2/c(R - X) + 2k(N - R) ( 4 . 3 . 2 )
- 61 -
For large k, this obeys the x 2 decomposition property parallel to
( 4 . 3 . 3 )
where x iy...,xN is a random sample from N(/t,cr) for linear statistics.
Equation (4.3.1) is therefore similar to the F-statistic for linear statistics, which is
the ratio square-root of the first and second terms on the right-hand side of
In 1969 Stephens stated that in practice the asympototic results of (4.2.3) are not
adequate for moderately large values of k. Illustrating this via Pearson curve
approximations and Monte Carlo studies, an improvement in (4.2.3) was found by
examining the expansion of A (k) for large k (equation (3.2.2)). This suggested that
the chi-squared form would be improved by replacing k by y in (4.2.3), where
y k 8k2
and that this should be used for tests when k > 2 .
In 1974 Upton considered further improvements of (4.2.3) by investigating the
distributions of 0, X and R. Taking the expectation of these distributions and
approximating the Bessel functions involved, Upton derived 0, to replace k in
(4.3.3).
1 1 3( 4 . 3 . 4 )
(4.2.3).
1 30 - k 1 ( 4 . 3 . 5 )
4k 16k2
Upton concluded that (1) both y and 0 approximations improve as k increases (2)
both 7 and 0 are considerable improvements over the original, and (3) there is little
to choose between the 0 and y approximations
Assuming equal concentration parameters k , and k 2 for the two sample case, we are
now interested in testing
Ho : Voi “ 02 “ ( 4 . 3 . 6 )
against
Hi : * 02 ( 4 . 3 . 7 )
where fi0 and k are unknown.
Using Stephens approximation improvement 7 , for large k , (4.2.5) from Watson and
Williams becomes
3•2 “ 1 + — (N -2 )
8k
(R, + R2 - R)
(N 1 JO 1 to
f 1 ,N -2 - 1 + — (N -2 ) ( 4 . 3 . 8 )
where for unknown k , k can be replaced by its maximum likelihood estimate,
k - A" 1
given by (3.3.11) or (3.3.10).
For value of k > 10 (R/N > 0.95) the improvement factor 7 is negligible.
Using the likelihood ratio test procedure, discussed in Chapter 3.1, to test H 0 and
H 1 defined by (4.3.6) and (4.3.7) we may obtain a test statistic for small k, assumed
equal, given by
log X - N logI 0( * i )
M * o >
p 2
+ k Q £ £ cos(0 i j - 0)
i-1 j-1
P 2
" k ' I I cos(0 i j - 0j )
i-1 j-1
( 4 . 3 . 9 )
- 63 -
From (1.4.11)
P 2^ ^ c o s ( 0 j j - 0) - R
i - 1 j - 1
P 2
^ ^ c o s (0 } j - 0j ) - R, +
i - 1 j - 1
Using (3.2.11) for small k
k n - —2R
N
2(R, + R2)
N
( 4 . 3 . 1 0 )
where
N - Nt + N2
Using approximation (3.6.4) we obtain the test statistic
“2 log X - jj (R, + R2) 2 - R2 = X] ( 4 . 3 . 1 1 )
In the same manner the likelihood ratio test of the equality of the mean directions
of two samples having unknown concentration parameters, assumed to be equal and
large, may be obtained from (4.3.9).
With k assumed large we may use the approximation (3.3.7) to give
N0 2(N - R)
N2(N - R, - R j)
( 4 . 3 . 1 2 )
- 64 -
Using the approximation (3.6.7) for large k we obtain our test statistic
Nlog ( 4 . 3 . 1 3 )
Mardia (1972) shows this to be a monotonic function of the F-statistic given by
For both the single and two sample cases Upton (1970, 74), using likelihood ratio
techniques, produced significance tests concerning the mean direction and the
concentration parameter for all permutations of k known or unknown, k large or
small, and mean direction known or unknown. Upton also improved these test
statistics using their expectations and associated degrees of freedom, as discussed in
Chapter 3.1
Many single and two sample tests have not been fully reviewed and reproduced here
as this thesis is concerned with the further development of analysis of variance
techniques and only those having a bearing on such development have been
introduced. An excellent review of single and two sample testing can be seen in
Mardia (1972, p 132-).
4.4 Multi-sample Tests Concerning the Mean Direction
Let 6\j (i= l,2 ......p ,j= l,2 ,...... ,q) be q independent random samples of sizes Nj from
M(/x0j,A:j). Let Rj be the length of the resultant of the jth sample, and R be the
length of the resultant of the whole or combined sample.
We wish to test
against the alternative that at least one of the equalities does not hold. We assume
that k y = ... kq = k , where k is unknown.
(4.3.8).
H0 • ^ 0,1 “ , 2 = ^o,q ( 4 . 4 . 1 )
- 65 -
4.4.1 For Small Concentration Parameter, k
For values of k in the range 0 to 1 the likelihood ratio test for q samples extended
from (4.3.11) gives
q- 2 Io g X - ^ ( ( £ R j ) 2 - R2> - X2_ j ( 4 . 4 . 2 )
j - 1
Since this test uses the approximation (3.2.11) for small k, which was shown in
Chapter 3 to be unreliable as k-41, the test may be improved by using the
expectation approximations from (2.5.2) and (2.5.4)
E(Rj ) " n jp + k k > 0
E (R j ) - Nj + N j (N j - l ) p 2
in (4.4.2) to produce the test statistic
q
I « ( ( £ R j ) 2 - R2) ) ( 4 . 4 . 3 )
j - 1
where
k 2 q5 - 1 = 1 ---------+ ( 4 . 4 . 4 )
8 2Nk2
4.4.2 For Large Concentration Parameter, k
Using Stephens improvement (4.3.4) and extending Watson and Williams test statistic
of (4.2.5), the new test statistic, under the null hypothesis (4.4.1), becomes
31 + — -
8 k
(N - q) ( J Rj - R)
j - 1
(q - 1) (N - Y Rj )
j - 1
Fq - 1 ,N-q ( 4 . 4 . 5 )
- 66 -
where k is the m .l.e. of k given in (3.2.2).
Mardia (1972) states that from Monte Carlo trials this approximation is adequate for
k > 1, i.e R/N > 0.45 Stephens (1969, 72), however, shows that the test is adequate
for k > 2, i.e. R/N > 0.7, and requires a further improvement to be satisfactory for
k > 1. Stephens (1972) produced an approximate multi-sample test based on the
exact test given in section 4.2.1, where
R, + R2 + ................ + Rq RZ - ---------------------------------------------- and W — — ( 4 . 4 . 6 )
N N
With test statistic;
z - ( 4 . 4 . 7 )
where
f _ g p (q - * )(N - q)
Let g be the upper percentage point, at level ot, of the F distribution with q-1 and
N -q degrees of freedom, and let D be a parameter taking the following values, for
W between 0.45 and 1.0.
W : 0 .45 0 .50 0.55 0 .60 0 .65 0 .70 0 .80 0 .90 1.00
D : 0 .92 0 .87 0 .84 0.83 0 .82 0 .84 0 .88 0 .93 1.00
If Z > z, reject H 0 at significance level ct
This test is equivalent to (4.4.5) for k > 2, but as (4.4.5) deviates from the
F-distribution below k-2 the factor D is introduced. For values of k > 2 D is the
reciprocal of the factor 1 + (318k).
- 67 -
Upton (1976) produced an approximate test for q samples, known as the G-test,
related to the joint distribution of Z and W, where Z and W are given by (4.4.6).
The G-test is derived from the method of maximum likelihood which gave (4.3.13)
for the two-sample case. Upton states the test statistic
2N2 logi 1 - w1 - z
N(1 + W) + q Xq-1 ( 4 . 4 . 8 )
suitable when all Nj 10 and R/N > 0.6.
In exactly the same procedure as the two-sample case, k 0 and k , are produced
from (3.3.7) to give
kr, =0 2 (1 - W)
k - 11 2(1 - Z)
Using the approximation (3.6.7) for large k and the log likelihood ratio
( 4 . 4 . 9 )
<-r
'w'O
HH
A A
N log + k 0W - k , ZI 0 (£o>.
we obtain the test statistic
-2 logX = N logj ( 1 - W)( 1 i N
( 4 . 4 . 1 0 )
This may be improved by equating the test statistic to its associated chi-squared
expectation. On expanding the logarithm as a power series we may neglect terms
beyond the first two since W and Z will be smaller than 1, to produce
N - | - w 2 + z + § !
- 68 -
using the expectations E(R) and E (R 2) given by equations (2.5.4) and (2.5.2).
E(G) “ (q2k 1 “ A(/c) + 4En
On equating with (q-1) and simplifying we obtain the correction
2NN + R + q
to produce the test statistic (4.4.8).
Throughout Uptons paper, conditions, ranges and tables for the statistic W/N are
given, however, this is a notation error and should be read as R/N or simply W.
The above multi-sample tests will be investigated further in Chapter 9 when suitable
comparisons to an alternative test will be given.
4.5 Multi-Sample Tests for the Equality of Concentration Parameters, ^
In this section three tests for the homogeneity of concentration parameters for
differing values of Rj/Nj are given. The construction of tests (4.5.2), (4.5.4) and
(4.5.5) can be seen in Mardia (1972, p 165-). The composite hypothesis under
consideration is
Ho : - .............- *q - k ( 4 . 5 . 1 )
where j i , , iq and k are not specified.
- 69 -
4.5.1 Rj/Nj < 0.45
Test statistic
q 1 I WJg ’ ( * P
u - “ I - - T 1---------------j - i
q
I wj
j - i
q - i
where
R2R
Ng , ( R j ) = s in - 1 (aRj ) a
1wj 4 (N j - 4)
The test statistic (4.5.2) is based on the approximation
2Rk,
2(1 - a 2k 2)N
which is obtained from the approximation
A(fc) « |
from (3.2.8).
1 - 8
( 4 . 5 . 2 )
( 4 . 5 . 3 )
The functional form of the transformation to normality is
g , ( k ) - s i n " 1 (ak)
under H 0 (4.5.1) the g,(R j) are approximately distributed as independent N(g1(A:),o’j)
where
2 _ 3 _ 1_° j " 4 (N j - 4) “ wj
- 70 -
4.5.2 0.45 < Rj/Nj < 0.70
Test Statistic
U . - I wjS j - 1
R
iNj
f 3v [R j|L WJS2
U - i kJ ■ *= x,
I -Jj -1
q-1
where
( 4 . 5 . 4 )
1 0.7979
wj <Nj - 3>
R 90 1 O
g 2 - = s inh“ 1 -N C 2
c, = 1.0894
c 2 = 0.25789
Test Statistic (4.5.4) is built in the same manner as for (4.5.2).
4.5.3 Rj/Nj > 0.70
This test is the analogue of Bartlett's test for homogeneity of variance and was given
by Stephens (1972) and Mardia (1972).
Let d j = N j - 1
Test Statistic
D = N-q = ^ d j
J-1
Z, - D{ loge £ (Nj - R j ) } - DlogeD - £ d j lo g e (Nj - R j ) + J d j l o g edj
j - 1 j - 1 J- 1( 4 . 5 . 5 )
- 71 -
then U 3 = Z y C where
C = 1 +
n
iJ - i
3 (q - 1)
IL is d is t r i b u t e d as y 2 3 ^n-1
The above tests will be used when techniques in later chapters assume that k is
equal in value for all the subpopulations being tested under the composite hypothesis.
- 72 -
CHAPTER 5
PARAMETER ESTIM ATION LEADING TO M A XIM U M LIKELIHO OD METHODS
FOR LARGER EXPERIMENTAL DESIGNS
In the first four chapters of this thesis, a general review and critique of the
problems, theory, approximations and tests associated with circular statistics have been
discussed. The thesis will now begin to extend these techniques and discuss alternative
methods to construct new tests for experimental designs and their required analysis of
variance.
Gould (1969) gave a regression analysis procedure for use when the dependent
variable is the position of a point on the circumference of a circle or the surface of
a sphere. Gould used an analogue of the normal theory of linear regression for the
circular variable problem, where
0j, i= l,...,N is independently distributed as M ( / i0 + @t'v k) where t j , . . . , tN are known
numbers while f i0, 0 and k are unknown parameters. The maximum likelihood
method, as discussed in Chapter 2, is used to estimate the parameters f iQ and j3.
The logarithm of the likelihood function is
5.1 Introduction
( 5 . 1 . 1 )
N
- constant + k cos( 0 j - / i0 - 0 t i )
i - 1
The maximum likelihood solutions, fiQ and 0 for the parameters are then obtained as
the solutions to the two equations
N
( 5 . 1 . 2 )
i - 1- 73 -
and
N
^ t | s i n ( 0 j - Jt0 - - 0 ( 5 . 1 . 3 )
i - 1
From (5.1.2)
N
£ s i n ( 0 j - ^ t j )
tan fiQ - ^ ----------------------- ( 5 . 1 . 4 )
^ cos( 0 j - 3 t i )
i - 1
P is obtained by a straightforward iterative procedure where P is an initial estimate
of P and fi0 the corresponding value of /z0 from (5 .1.4), for the iteration
N
J t j s i n ( 0 j - ? 0 - ^ t j )
P - P + ^ -------------------------------------- ( 5 . 1 . 5 )
£ t?cos( 0 j “ ?o ~ ^ i )i - 1
From these estimates is developed an appropriate test for p=0. For this model,
maximum likelihood estimation coincides with least squares estimation.
Johnson and Wehrly (1978) showed that the most serious drawback to Goulds
approach is that the likelihood function has infinitely many large peaks.
In this section a similar modelling approach to Gould will be discussed, however, for
experimental design models, as in linear modelling, constraints will be placed on the
estimating parameters. The maximum likelihood estimates of the model parameters
will be produced for the required hypothesis test for the first time. The attainment
of these not only enables hypothesis analysis but greater data appreciation, as will be
shown in the following sections.
- 74 -
5.2 Constraints for Circular Models
For the development of experimental models each angular variate in a one-way
classification may be made up of some overall mean direction f iQ, some effect due to
a particular treatment, /3j, and some random variable representing unassignable
(residual) effect, q j. Hence /3j = (fij - / i0) where /ij is the mean direction of the
jth population.
The form of the model which is used can be expressed as;
As in linear analysis of variance the assumptions on the treatment effects are:
a) the treatment terms add on to the mean direction term rather than, for
example, multiplying
b) the treatment effects are constant
c) the observation on one block or unit is unaffected by the treatment applied to
other units.
In linear analysis it is usually convenient to choose the constraint on the treatment
parameters so that they sum to zero; in circular analysis it is correspondingly
convenient to choose the constraint on the angles specifying the treatment parameters
so that their sines sum to zero. A proof and simple example shows the need for
this constraint; using expression (5.2.1) and assuming zero residual
0i j “ + + 6i j ( 5 . 2 . 1 )
P q p q
i - i j - i i-1 j-1( 5 . 2 . 2 )
- 75 -
ta n 0
q qcos /i0 ^ s in f i j + s in f iQ ^ cos 0j
j-1________________ j ° lq q
cos /i0 ^ cos 0j - s in /i0 I s i n
j- i j - i
under the constraint I s l n h - 0j- i
qs in M0 \ cos 0j
tan 0 - --------------i —5--------------- - tan (iQ ( 5 . 2 . 3 )
COS /l0 I cos /3j
j -1
Example 5.2.1
4
i ) Let fiQ - 100* £ - 0
j -1
and
0, = 48.5* 02 - 19.38* ft - -54 .04*
therefore 0 4 = -13.84* with zero residual
0 M - 148.5* 012 - 119.38* 013 - 45.96* q 4 - 86.16*
Using (1.4.5) to calculate the mean direction gives
0 - 100.582*
4
i i ) However under the co n stra in t ^ s In 0 j — 0
j-1
then 0 4 = -15.745* and 0 14 = 84.255*
to give 0 = 100*
- 76 -
In the first example the sample mean direction differs from the stated population
mean direction when zero residual effect is assumed. When the constraint is that
the sine of the treatments sum to zero, the sample mean equals the stated population
mean direction, i.e. 0 = fiQ. This must hold for all factor effects in larger circular
experimental designs.
Using this notation each observation 0jj is an independent observation from a von
Mises distributed population with a mean direction of / i0 + /3j and whose
concentration parameter is k , i.e.
0 i j * IVM( ^ 0 + 0 j , k ) ( 5 . 2 . 4 )
where q is the number of treatments, and p is the number of observations on each
treatment, with the constraint
q^ s in /3j - 0 ( 5 . 2 . 5 )
J-1
and IV M is read as 'independently von Mises distributed1.
5.3 One-way Classification
Assuming the fly's (i= l,2...p , j= l,2 ...q ) are independently distributed as
M (fi0 + (3j,k), let us construct a test of:
H0 : (3, - P:
H, : at least one |3j / 0( 5 . 3 . 1 )
Let fiQ and ( / i0, j3j) be the maximum likelihood estimates of ( iQ and / i 0, /?j under
H 0 and H 1 respectively. The logarithm of the likelihood function is given by:
P qlog L - constant + ^ cos( f l j j - /*0 - /3j)
i-1 j-1
( 5 . 3 . 2 )
- 77 -
a) Under H 0 the maximum likelihood estimate fiQ of nQ is the solution ot:
P q^ J s i n ( 0 j j - J 0) - 0 ( 5 . 3 . 3 )
i -1 j -1
Therefore
P q
I l sin i j
tan un - - tan 8 . . ( 5 . 3 . 4 )P q
I I 008 9iji -1 j -1
Hence Jt0, under H 0, is the overall sample mean direction, 8.
b) Under H 1 the maximum likelihood estimates fiQ and /3j are the solution of:
P q£ £ s in (0 j j - - j3j) - 0 ( 5 . 3 . 5 )
i-1 j-1
and
P^ s in (0 j j - n 0 - /3j) - 0 'q' equations ( 5 . 3 . 6 )
i=*l
qunder the constraint ^ s in |3j — 0
J-1
Let Aj - £ 0 + 0j
Substituting Aj into (5.3.6)
PJ s in (0 j j - Aj) =* 0 ( 5 . 3 . 7 )
i = l
From (1.4.9) Aj is the mean direction of the jth block i.e. 8 j.
- 78 -
Therefore we may solve for £ 0> 0 ,, (32......... 0q from the q equations:
0- i “ M0 +
0-2 " Mo + h
0 -q “ M0 + 0q
under the c onstra int I s in 3j - 0
j -1
Taking the sine and summing the q equations of (5.3.8) produces
qP q0 . j - s in fL0 2 cos fij
A
+ COS fL0 2 s i n 0j
j -1 j-1J-1
under the given constraint
q^ s in 0.j - s in n0
J-1
^ COS |3j
j-1
Taking the cosine and summing the q equations of (5.3.8) produces
J cos 0.j — cos f l Q
j - 1
^ COS /3j
j -1
Dividing equation (5.3.9) by (5.3.10) gives
qI s in » . j
j -1 s in )t0
q cos /*0COS 0. {2
j - 1- 79 -
( 5 . 3 . 8 )
( 5 . 3 . 9 )
( 5 . 3 . 1 0 )
Therefore
J-1fiQ — tan ” 1 ( 5 . 3 . 1 1 )q
Hence, under H lt jz0 is the circular mean of the q equations, 0 .j
The 3j treatment effects are calculated directly from the q equations of (5.3.8), given
H0 found by (5.3.11) i.e.
Unlike linear analysis the maximum likelihood of the overall mean direction, n Q,
alters depending on the hypothesis test. Example 5.3.1 shows the parameter
estimation and the following hypothesis test at work for a hypothetical data set
concerning animal orientations.
Example 5.3.1
In an orientation experiment four samples of animals were observed: one was a
control group, the other three were experimental groups. Following treatment, their
direction of movement was noted and reproduced in Table 5.3.1.
( 5 . 3 . 1 2 )
- 80 -
Table 5.3.1
ControlGroup
Experim ental Group 1
Experim ental Group 2
Exper imental Group 3
156* 168* 137* 214*
111*
•VOC" 184* 155’
174* 102* 222 * 129’
140* 137* 163* 153*
213* 184* 236* 125’
121 * 112* 133* 228*
2 0 0 * 62* 161* 185’
166* 133* 193* 176*
Statistic:
Resultant
R. j
Sample Mean D i re ct ion
' .J k
Control Group 6.7031 159.9322’ 3.415
Exper imental Group 1 6.231 121.5483* 2 .622
Exper imental Group 2 6.5938 177.9278* 3 .182
Exper imental Group 3 6.5938 169.9278* 3.182
Overall sample mean direction, 0.. = 158.3153*
Overall resultant length, R.. = 24.362
Concentration parameter estimate k - 2.464
Circular mean of the individual _group mean directions 6 - 157.8013*
Each of the sample populations have been tested by Watsons statistic (1961),
using the critical values supplied by Stephens (1964), to show von Mises distributed
data sets. Similarly, the homogeneity of the concentration parameters have been
tested and validated via test statistic (4.5.5).
- 81 -
Parameter Estimates:
Under H 0 of (5.3.1) /t0 = 158.3153* (from (5.3.4))
Under H , of (5.3.1) £ 0 = 157.8013* (from (5.3.11))
0 ,A
^2
e 3
h
. l
. 2
. 3
. A
M0A
M0A
^0y\
Mo
2.1309
-36.253*
20.1265*
12.1265*
I sin 3jj - i
Hypothesis Testing:
The likelihood ratio test for H 0 and H 1 gives the test statistic
N logeI 0C/c0)
p q
+ I \ cos^ i j " M0)i -1 j -1
p q£ £ COS(0f j - f t Q - 0j)
i - i j - i
( 5 . 3 . 1 3 )
which produces the same test statistic given by Watson and Williams and built in
Chapter 4.3 where
P q
£ £ cos ( j j - J*0) “ R. .
i -1 j -1
p q q
I I cos(f l iJ - - 3 j ) - J R. ji - i j - i j - i
Using the approximations (3.3.7) and (3.6.7) for large fc the one-way classification
test statistic is seen as
- 82 -
N l o g eN - R
N - J r . j
j -1
~ Xq-1( 5 . 3 . 1 4 )
which is a monotonic function of the F statistic given by (4.4.5). For Example
5.3.1, applying test statistic (5.3.14) gives:
32 loge7.638
5.87838.38 * x 2
3
x l (5%) - 7 . 8 i x | ( 1%) - n *34
Indicating a significant difference, at the 5% level, between the four treatments
shown in Table 5.3.1 . Having rejected H 0 the above method has shown means by
which the parameter estimates of a one-way classification design may be obtained
and examined further. Figure 5.3.1 illustrates the angular difference between the four
effects or treatments, given in Example 5.3.1, against the model derived mean
direction estimate, n 0. Treatment 1 is seen to possess the greatest divergence from
'0*
- 83 -
E*f> I
* ------------2 0 - 13*
Figure 5.3.1 Angular differences between the treatment mean directions and the
overall sample mean direction found in Example 5.3.1 .
5.4 Randomised Complete Block and Larger Experimental Designs
If the maximum likelihood estimates for other circular experimental design models can
be found in a similar manner to the one-way classification shown above, the testing
of differing hypotheses for 'larger' experimental designs may be undertaken. However,
unlike the one-way classification, the parameter estimates for these designs may not
be found by simple algebraic manipulation.
Let us investigate, for example, parameter estimation for the randomised complete
block design. Here the form of the model may be expressed as
0 i j = /*o + a i + 0j + e i j ( 5 . 4 . 1 )
- 84 -
Assuming the fly's (i= l,2 ,......,p, j= l,2 ,...... ,q) are independently distributed as
M (pLQ +Oj+/?j, k), let us test the 'column' effects for the hypothesis given in (5.3.1)
under the constraints:
P£ s in a j — 0 ( 5 . 4 . 2 )
i -1
qJ s in /3j - 0 ( 5 . 4 . 3 )
j -1
A A
let ( / i0,ay and ( / i0*, cq*, (Sj*) be the maximum likelihood estimates under H 0 and
H 1 respectively. (The starred parameters represent parameter estimates found under
H , ) .
A A
Under H 0 the maximum likelihood estimates / i0 and aj of /x0 and cq are the
solutions of:
P q^ ^s in(f l j j - j i0 - cij) - 0 ( 5 . 4 . 4 )
i -1 j -1
q^ s i n ( f l j j - n Q - a j ) = 0 'p ' equat ions ( 5 . 4 . 5 )
j “ l
under the c onst ra int ^ s in a j - 0
i -1
Equations (5.4.4) and (5.4.5) are in precisely the same form as (5.3.5) and (5.3.6),
and therefore give the same parameter estimates as (5.3.12) and (5.3.11) for aj and
/z0 respectively.
Under H 1 the maximum likelihood estimates / t0*, a * and of f iQ, aj and /3j are
the solutions of
- 85 -
p q£ J s in (0 j j . £ 0* - 5 i * - j3j*> - 0 ( 5 . 4 . 6 )
i -1 j -1
P£ s i n ( 0 j j _ £ 0* - cq* - 3 j*) - 0 'q' equations ( 5 . 4 . 7 )
i -1
Punder the co n stra in t ^ s i n a** = 0
i-1
tr^ s i n cq*
J s i n ( 0 j j _ n0* - cq* - 3 j* ) - 0 'p' equations ( 5 . 4 . 8 )
J-1
qunder the co n stra in t ^ s in 3 j* ” 0
J-1
Unlike the one-way classification of section 5.3, these constrained equations may not
be simplified in the same manner. In 'linear' statistical analysis the equivalent
expressions would simplify to their respective 'column' or 'row' means. This
produces a simple statement where, some overall mean plus a particular 'row' effect
gives the respective row mean; and similarly for column effects. In circular analysis
these expressions will not simplify and require lagrange multipliers to take account of
the given constraints.
For the solution of the system of equations (5.4.6), (5.4.7) and (5.4.8), under the
constraints (5.4.2) and (5.4.3), computer programs have been utilised. Several
algorithms were applied to the constrained optimisation problems in an attempt to
find a global maximum subject to equality constraints. O f main use were sequential
lagrangian methods with the maximisation being solved by quasi-Newton procedures.
The functions, however, are not unimodal in nature and are indeed very heavily
multimodal. Therefore unless the initial estimates are very good approximations,
- 86 -
the algorithms simply find local constrained maximums and not the required global
maximum.
In order to solve merely small randomised block designs a lengthy and inefficient
program has been written using a direct search method. Example 5.4.1 gives a
small hypothetical data set on which the direct search program has been applied.
Example 5.4.1
'Column* E f fe c ts ResultantMean
D i re ct ion
0 i 02 03 Ri . * i .
«i 115* 105* 2 0 * 2.229 83.347
' Row' «2 170* 180’ 65* 1.899 145.343
E f fe c ts « 3 75* 90* 325* 1.761 52.253
«4 120* 150* 45* 2.175 107.632
Resultant
R . j 3.346 3.255 3.202
Mean Di rect ion
* . j
119.517* 130.751* 25.566*
N - 12
R . . - 6.817 ? . . - 97.458*
H 0 • 01 “ 0 2 ™ 0 3 ” ®
Ht : at least one /5j ^ 0
- 87 -
Parameter Estimation;
Under H0 : n 0 - a rc tan
pp
Z s in e i .i -1Pp
Z cos ®i.i -1
A
“ * 1 .A
- V- 0 “ -13, .508
«2 “ * 2 . - ?o " 48,.487
A« 3 “ * 3 . - ?0 “ -44, .603
« 4 “ * 4 .A
- ^0 - 1 0 ,.776
P^ s in a j
i -1
P q PJ £ cos(8 j j _ /t„ - o ii) - 2 Ri
i -1 j -1 i-1
k using approximation (3.2.7) equals 1.838
Under H , : Via the direct search method
?o* " 97.065*
s , * - -11 .78*
S 2* - 46.49*
-48 .49*
« < * - 13.164
? , * “ 23.13*
0 2* " 3 4 .3 5 ’
£ 3 * “ -73 .15*
P
ii -1
^ s in 04 '
^ s in /3j* - 0
J-1
p q£ £ cos( 0 i j _ £ 0* - 04 * - 3 j * )
i -1 j -1
96.856*
0
8.0646
11 .878
- 88 -
Maximum likelihood estimate k , is given by
p q
" " I I c o s ( 0 U “ £o* ~ « i* - 0 j* )M M N i=i j„i
Using approximation (3.3.11) k y = 49.4
Replacing the parameter estimates under H 0 and H t into the likelihood ratio test
statistic (5.3.13) and using the approximation (3.6.7) produces a chi-squared value of
41.88. Comparing this to a x \ indicates that there is a highly significant difference
between the column or block effects.
A A
Under a null hypothesis testing the row or treatment effects the m.l.e. of / i 0 and /3j
are
r qn
I s i n 6-im-a j- i
O il
q
Z 005 ? . jL j - i
- V - i t " 22 777'
- * 2 .A
- " 34 O i l ’
- - i t - -71 . 174"
96.74
I sin hj - i
p q q
£ J cos(0ij - - 3j) = 2 R.j “ 9-i-1 j -1 j-1
8034
k 0 - 3.01
The alternative hypothesis estimates will be unchanged producing a chi-squared value
of 34.74. Comparing this to a x \ indicates a highly significant difference between
the row or treatment effects.
- 89 -
5.5 Summary
In linear statistical analysis the generalised linear model approach may be used to
find the factor parameter estimates for a particular design. These are then utilized
within a maximum likelihood testing procedure to examine for any significant
difference between the factor effects. A similar approach has been used within this
section for circular statistics. In linear analysis the factor effects sum to zero, here
we have seen that for circular analysis, it is convenient to choose the constraint on
the angles specifying the treatment parameters so that their sines sum to zero. From
this, section 5.3 has shown how parameter estimation for the design model of a
one-way classification may be produced, and assist in further understanding the
underlying structure of the data sample under investigation.
For larger designs, however, the solution of the likelihood function under the given
constraints may not be found by the same simple algebraic manipulation. Due to
the complexity of the maximising problems involved many local maxima may be
found and the discovery of the global maxima extremely difficult. Many computer
programs for the optimisation of constrained equations have been tried with little
success and further investigation into improved computer algorithms will be necessary.
If the null hypothesis for a particular test is rejected, the neatness of this approach
helps us to appreciate and to look at the contrasts between the effects within a
factor. Such procedures exist for linear analysis derived by Tukey or Scheffe known
as methods for multiple comparison. If the approach of this section can be extended
to larger design methods using computer algorithms to help find and understand the
parameter estimates, similar procedures of multiple comparison may be derived for
the circular case.
- 90 -
CHAPTER 6
THE EXTENSION OF EXISTING TECHNIQUES TO LARGER EXPERIMENTAL
DESIGNS
6.1 Introduction
From Chapter 4 we have seen how, for tightly clustered populations, Watson and
Williams (1956) developed a one-way classification technique which has been widely
used for 2 and 3 dimensional vectors. This original technique has been refined and
used to produce tests for several samples having common mean direction. Chapter 5
has shown how the parameter estimates may be found for the one-way classification
design. In Section 6.2 this analysis of variance is extended from the one-way layout
to the nested or hierarchical design as an initial step to the analysis of larger more
complex experimental situations. Section 6.3 extends the design further to enable
analysis of randomised complete block designs and the two-way analysis of variance
design, for large k.
It should be noted that within linear analysis of variance the factor components
within an experimental design are correctly referred to as sums of squares, however,
in circular analysis of variance the measures of each factor are not calculated in the
same manner and will be referred to as measures of variation. For examples, the
one-way analysis of variance procedure will be analysed using a total, between and
residual measure of variation.
91 -
6.2 Nested or Hierarchical Design
There are many occasions when a researcher wishes to study the effect of a single
factor such as light intensity or magnetic charge on the displacement of
micro-organisms but because of the nature in which these are obtained a more
complex one-way design is required. The nested or hierarchical model is such a
design.
It is often necessary to measure the response to a treatment on each individual of a
subsample of a unit factor than on the entire unit to which the treatment is applied.
In a drug experiment, for example, the treatments may have been applied to a
particular animal type within different groups, from each group several animals may
be picked at random and their response (perhaps their angle of movement) measured.
To this end a model for the nested design may be built, under the general
assumptions;
i) The samples are drawn from populations with a von Mises distribution
ii) The parameter of concentration has the same value in each population,
that is,
k , = k 2 - ............ - kq = k
iii) The overall and individual parameters of concentration are sufficiently
large, namely
k > 2
Let p be the number of treatments, given by i = 1,2...... ,p, qj the number of groups
or experimental units within treatment i, given by j = 1 , 2 qj, and n the number
of observations within each qj group. Let N be the total number of observations
and U be the number of cells. Extending the expression given by Watson and
Williams (1956), that
- 92 -
p p( 6 . 2 . 1)
i= l i= l
produces:
+ 21c(J R2 j . - R2 . . ) +
j - 2
+ 2k( -t - R p _ ) + 2k(^ 2 n i j . “ Ri j . >
j -1 i -1 j -1
(6 .2 .2 )
with associated independent chi-squared distributions with known degrees of freedom,
for large k :
The first term on the right hand side of the expression (6.2.2) is essentially the
measure of variation among treatments, the next qp terms of similar form are the
measures of variation within treatment i but among the qj groups. The final term
represents the within or residual variation within treatment and within group but
among the sub-groups. As a nested design we may consider differences between
rows, or difference between columns within any one row.
As in the one-way analyses a test statistic for examining the differences between
rows may be given as
P
( 6 . 2 . 3 )p q i
- 93 -
which has an F distribution with (p—1) and (N -U ) degrees of freedom. Similarly a
test statistic for the difference between columns within row i is calculated in the
same manner.
which has an F distribution with (qj—1) and (N -U ) degrees of freedom.
Stephens (1982) produced the same expression (6.2.2) and quotients (6.2.3) and
(6.2.4) in m dimensions for the analysis of data which are proportions of a
continuum such as time or volume. Stephens also gives a good example of the
methodology when studying the proportion of time spent in various activities by 130
students.
6.3 The Randomised Complete Block Design and Two-way Classification with
The randomised block design is a widely used method of dealing with factors that are
known to be important and which the researcher wishes to eliminate rather than to
study. Here the factor is blocked so that each is as homogeneous as possible and
the treatments under study are each used exactly once in each block for the design
to be balanced. The observed differences among the treatments should be largely
unaffected by the factor that has been blocked.
There are many situations where a randomised block plan can be profitably utilised.
For example, a testing scheme may take several days to complete. If we expect
some systematic differences between days, we might plan to observe each item on
( 6 . 2 . 4 )P Qi
Interaction Design
- 94 -
each day; a day would then represent a block. Since each treatment occurs exactly
once in every block, the treatment totals or means are directly comparable without
adjustment.
By again extending the simple expression (6.2.1) and its associated results concerning
the chi-squared decomposition we may now take account of a possible block effect, i,
as well as the treatment effect, j, to produce the randomised complete block model;
P q2/c(N - R ) - 2/c(^ Rj - R ) + 2/c(^ R j - R )
i -1 j -1
p q+ 2k(N - £ Ri - £ R j + R ) ( 6 . 3 . 1 )
i -1 j -1
with associated independent chi-squared distributions, for large k,
XN-1 " Xp-1 + Xq-1 + X ( p - 1 ) ( q - 1 )
Expression (6.3.1) obeys the x 2 decomposition property parallel to the linear form of
the randomised complete block design. The first term on the right hand side of the
expression is essentially a measure of variation due to treatments, the second term of
similar form being a measure of variation due to blocks. The final term represents
the residual variation after variation due to treatments and blocks have been
removed.
From (6.3.1) the test statistic (6.3.2) is produced to test the null hypothesis that
there is no difference between the treatments.
which has an F distribution with (p—1) and (p - l) (q - l) degrees of freedom.
Similarly, the test statistic (6.3.3) will be produced to test the null hypothesis that
there is no difference between the blocks
which has an F distribution with (q-1) and (p - l) (q - l) degrees of freedom.
In the case of significance, we may only state that the mean directions are or are
not equal. The test does not allow for discrimination among single mean directions.
Before the accuracy of the approximations to the distributions of the components of
(6.3.1) and the F distribution approximations Z., and Z 2 are examined, the
associated model for the two-way analysis with interaction will be discussed. A
discussion of interaction in directional data analysis will be given in Chapter 8 .
Using the same approach as for the nested and randomised complete block designs a
two-way classification with interaction may be built;
q
Z i -1 ( 6 . 3 . 3 )2 P q
i - i j - i
p q
R. j . " R . . . >
p q
p q p q( 6 . 3 . 4 )
i - i j - i i - i j - i
- 96 -
The first three terms on the right hand side of (6.3.4) have been shown to be
chi-squared distributed by Watson and Williams (1956) and Stephens (1963)
representing measures of variation due to row i and column j effects and a measure
of residual variation. The final term represents the possible interaction within the
experiment. F test statistics may be produced in the same manner as for the nested
and randomised block designs. (/ represents the number of observations on each
treatment combination).
Testing row effects
P
Z i -1 ( 6 . 3 . 5 )3 p q
F ( p - i ) , p q ( * - i )
Testing column effects
q
j -1 ( 6 . 3 . 6 )P q
= F ( q - l ) , p q ( / - l )
Testing interaction effects
P q P q
Z ( 6 . 3 . 7 )5p q
F(p - i ) ( q - i ) , p q ( * - i )
- 97 -
6.4 Accuracy of the Associated Component x2 Approximations for the
Randomised Block Design and Two-way Classification and their
Corresponding F Statistics
The expressions (6.3.1) and (6.3.4) are based on a sequence of approximations, and
the accuracy of their chi-squared approximations may best be determined by
simulation. An examination may be made by taking Monte Carlo samples from a
von Mises distribution specified with fixed k. Observations from the distribution
specified by the null hypothesis were generated by computer methods outlined in
Appendix B and were grouped into samples of size N. This has been carried out
for 10,000 sets of samples of various size and were drawn from von Mises
distributions with k = 2, 3, 4, 5 and 10. For the testing of the randomised block
model three designs were investigated varying in sizes of N. Tables 6.4.1 to 6.4.5
examine the chi-squared approximations for each of the components within each of
the models. Table 6.4.1 shows the accuracy for the total measure of variation
component, 2k(N-R ), and Stephens improved approximation, 2y(N -R ), where y is
given by (4.3.4). It is clearly seen that cq, the simulated proportion of the
component, approaches a, the x 2 value theoretical proportion or significance level,
when y is applied. This is further illustrated when the first two moments for both
approximations are given in Table 6.4.2. When Stephens improvement is used both
moments approach their expected values and with increased accuracy as k increases.
Accuracy for both approximations increases as k increases.
Similar results are also found when the component of error or residual is also
adjusted by Stephens improvement, y. Table 6.4.3 compares the accuracy of the
X2(p - i) (q - i) approximations, and shows that the accuracy for both approximations
improves as k increases. Table 6.4.4 shows the effect on the two approximations
first two moments when Stephens improvement is applied. As with the total
measure
- 98 -
of variation component, when y is applied, both moments approach their expected
values with increased accuracy as k increases.
Table 6.4.5 gives the accuracy of the two components measuring block and treatment
effects. Comparing the tables of the two effects shows how the number of
observations as well as the size of concentration parameter affects the accuracy of
the chi-squared approximation. As N and k increase the accuracy of the
chi-squared increases.
- 99 -
TABL
E 6.
4.1
Com
pari
ng
the
accu
racy
of
the
'X2
appr
oxim
atio
ns
with
N-
1 de
gree
s of
free
dom
fo
r th
b to
tal
mea
sure
of
va
ria
tio
n
CNJ in CO *0* in CO CO in CO CO CO CM'O’ CO in o o CM CO CO CM v CO □ IX 'T—
CD CD CO CO CD CD CD CD CD CO CD CD CO CD CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . .o □ O o
LD in CO 'O’ CO CD CO CM CM COCNJ CNJ a CO r — CO CD CNJ in CO in CM CO
>( in in *3- 'O’ in in in *0" in in in CO in in/_l CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD« . . . • . . • . . . . . . . . .• a o a □
o:
>_> CO in 'O’ CO T- CO CD IX CM>- in rx CO CNJ CD CO •O’ •O’ *o- CD CM IX CO CDCM □ CD CO CO o O CD CD CD o o l \ CO CD □ □
CD CO CO CO CD CD CO CO CO CD CD CO CO CO CD CD. . . . . . . . . . . . . . • .
o o C3 □
CO rx 'O’ in CO 'O’ CD CO CO in ■crCO a CNJ rx CD CO IX CO CO CD CM r - CO
CD CD rx i \ CO CO in in IN. IN. CO CD CO CD rx COCD CD CD CD CD CD CD CD CD CD CD CO CD CD CD CD
• . . . . . . . . . . . ■ ■ . .O □ a o
CO •O’ CNJ CO IN. CD CO ■sf v~ CO CD CO COCNJ rx CO in CD CO o CO in CD CO CO o CD
. LD in 03 a CNJ CO CNI IX CD CO CM CM IX a CM— _ ) 03 CO CO CD CD CD CO CO CO CD CD IX CO CO CD CD
. . . . . . . . . . . . . . .• O o a a
crz :'—>
rx CO CO •o* CO CNJ in CO IX CO 03 CMCNJ (V □ CO CO CO CD CO IN. CO *0" CO T~
o in o •O’ in CO co IX CNJ *0- IN. CD CM CO CM IXCD rx CO co CO CO CO rx CO CO CO in rx IX CO CO
. . . . • . . . . . . . . . . .o o o o
CNJ CO <3- in a CNJ CO in □ CM CO 'O’ in □T~
cn 0 0-p -P •pc c c0 0 0 0E 0 E 0 E
-P -P -p GID o ID o (D o0 o 0 o □ 0 1—1
m P t—i CO P 1—1 •r* p JO4-> JO 4-> JO -p
LU > in >1 «0* > C3 C3_ j cn JO CO in JO CO CO CM JO in 'r - inQ . LUe : in i CO II ii n CO II II II in II II II< Mcn co PU cr 2 P . cr z : Cu cr 2
- 100 -
The
tabl
e gi
ves
the
prop
orti
on
of 10
,000
No
nte
Carl
o sa
mpl
es
for
whic
h th
e tw
o ap
prox
imat
ions
fell
be
low
the^
X2^
^ va
lue
for
whic
h th
e th
eore
tica
l pr
opor
tion
sh
ould
b
eck
.
CDCo
•H4-> 0(D 0E Z3
•H i--IX 0o >a TJa . 0(0 •P
O
?<0
sCLX
0 0j=-p c.
•H<+- 0o sz
-p0o oc -p0
•H Efn o0 •O> 0
0T3 pC0
4-c O00 0E 0
00 U
JZ bfl-P 0
■QbCC V•H lp0CL SZE •PO •HCJ 2
CNJ
•sfCDUJ_lCD<
0 4--. 4--> 4--1a <—> <—> /—1 4--V r — \ 4—> <—. r— > 4--. 4--1 4--, <—> 'd* CDc CO T- CD CD rN r— CN CO d" CD co *3* CO CO0 CD CD CD IN CO CD d" CO d- IN 'T~ CD CO CD O
•H cn CD V- a CD CO CO in CD • • • •U • ■ • • • • • • • • • CD V- o r -0 LD CD CD CO IN CN in CO IN d - CD a O a O
> CN CN CO CN CN d - d" d - d - CD r~ V r —1 ' '— ' '— ' '— ' '— ' '— ' '— ' '—’ '—' '—' w V-- 4 w '—’ '— ' '—’
D Z
>— ’ CO CD CO CD CO CO CO CD CO r— CD in CO CO> - V CN CN CO in CO in CD CN in CO CO CNCN c d" CN * r - a CD CD d" CD CD CO CN IN o in CD
0 • ■ • • ■ ■ ■ • ■ ■ ■ • m ■ •0 'd' <d" <d" <d* CO CO CO CN CN CN a CD c n CO CO
e= TT— r - CN CN CN CN CN in d* <d* 'd*
0 4--. 4--. 4--> 4--.a I—> r —\ r —\ •—> «—> «—> 4--. 4--> <—» 4--. CN inc CD <—> IN CO CD CO IN. rN r* CD CO v- CO CD0 CN in CD a CO CO in CN CD d* CO IN in IN
•H CD CD IN in CN CO IN. CO IN CD • • • • •C. • • • • » • • • • • in in CD CO0 CD CD CD CN CD cn a IN d - rN CO CO CN r- CD
<—> > CO CO CO CO CN in CD in in d- r~ r~ • 'T- r-• '—’ w w ’ w ' w v—' w w w w w ’ w
c r
<—' CD un CO in d - CO n r - CO CO in CNv CO CD in CD CO CO IN in d- IN CO IN in
CNJ c t - o <d* □ ’d" o CO o CD rN CD CD CD «d-0 • • • • ■ ■ • • ■ ■ ■ ■ ■ • ■0 IN CD in in d - CO CD in d - CO CD in CO CN a
e : r- r— 't— CN CN CN CN CN in in in in in
CN CO *d* in CO CN CO d - in O CN CO "d- in o*” t— *"
CO CD COCN d- CDII II II
■d" CO cn0 CN 0 o <d" 0un O CO O V O
TD ii c TJ II c TJ ii c> 0 0 > 0 0 > 0 0
LU JD •P c ■H JD -p c •H JD -P c •H_J cn a 0 u o 0 U O 0 UCL LU CO 0 0 0 CO 0 0 0 in 0 0 0e : IN I Q . e : > □L e : > CL e : >< 1—1 X X XCO CO LU LU LU
- 101 -
TABL
E 6.4.3
Comp
arin
g the
ac
cura
cy
of th
e*X2
appr
oxim
atio
ns
with
(p-l
Hq
-1)
degr
ees
of fr
eedo
m for
the
resi
dual
me
asur
e of
vari
atio
n
un CD rx CO in CM in co t— CO <3-„_, •3- m CO IX o <r- cn CD CD '3' «3- CD CO COcn cn CO CO CO CD cn CO CO CO co CO IX CO CO CO
\ cn cn CD CD CD cn cn cn CD CD cn cn cn CD cn cnon . . . • . . • . • . . . . . .+ a □ o □
•'“J
cnw CM i \ CO CD CO ix CM CO CM CO rx <3- CO v CO
<3* *3- CM □ cn CO O CO CD IX IX CO CO CO IX1 Vf in CO CM CO <3" <3- CM CM CO CO «3" V O CO <3" 3-
CD CD cn CD CD cn cn cn cn cn cn CD cn CD cn OD• . . . . • . . . . . . . . . . .•H o □ □ CD
cnwCM CO CO CO CO 'T- *3" CO CD in cn *3- co CO
«_i v o CO □ IX CD CM CO in CO rx CO in> a in in CO co □ in M" CO CO cn CO IX CO cnCM CD CO CO CO CO cn CO CO CO CO CO CO CO co CO CO
. . . . . . . . . . . . . . . .□ a o a
v- CO IX CO rx CM CD CM inix CM cn IX CO <3 ‘ CD CD CO CO cn CO CO
J cn CO CO IX ix CO *3* in IX rx CO CO CM in IX COX
CD cn CD CD CD cn CD CD cn cn cn CO cn cn cn CD. • ■ . • . . . ■ • ■ • . . . .
•<—) o o o acnw
CM '3" CO in <3" CM CO CM cn CO COi cn in CO cn o cn in IX CO CD CM CO CM CO
V LD CO ix o r- CO '3' CD O cn a CO cn CM• o cn CO CO cn cn CD CO. CO CO cn CD CO CO CO CO CD•H . . . . • . . . • . . . . ■ . .
cn □ a □ oW12>— > CO rx CO CO CM *3" in CD CO CO CMjn CO CO CO CO CO IX CD CO CO <3* CO <3- <3* OCM a CO CO CM «3- CO CO *3* CO rx in o IX V - CD
CD rx rx CO CO CO CO IX CO CO CO in IX IX CO CO. . . . . . . . . . . . . . . .
o o a CD
CM CO <3* in o CM CO <3" in □ CM CO <3- in CD*“
cn 0 0•p -P •Pc c C0 0 0 0E 0 E 0 E
-P -P -P a(0 o 10 CJ (U o0 o 0 o O 0 i—i
in P rH CO P 1—1 P XD•p n -P X3 -P
UJ > in > <3- > O a_i cn n CO in v JO CO CO CM n in inQ_ UJe: m CO II ii ii CO II II II in ii II ii< Mcn cn Cu a* 2 P-4 c r 2 o - c r 2
- 102 -
The
tabl
e gi
ves
the
prop
orti
on
of 10
,000
Mo
nte
Carl
o sa
mpl
es
for
whic
h th
e tw
o ap
pro
xim
atio
ns
fell
be
low
the^
>(2
va
lue
for
whic
h th
e th
eore
tica
l pr
opor
tion
sh
ould
be
<x.
TABL
E 6.
4.4
Com
pari
ng
the
mean
an
d va
rian
ce
of th
e ap
prox
imat
ions
with
C p
-13
C q-1
3 de
gree
s of
free
dom
to
thei
r ex
pect
ed
valu
es
03O <—* f-- \ t—* 1—\ 1—\ r-» /—s /—\ /—\ /—V ,---v ,—vc CO CO x CNJ <3- x CO un /—. r - CO CM un CO ,—,
• 0 CO CNJ r— un CNJ CO □ CO CO CO o• *H un CO CNJ v- CO 03 CO CM o CO CO CM CO o
cr fn • • . . . . . . . . . • . . •+ (0 un CO CO X un CO CM o o CO CO CO x un CO
'<—> > r - r - r — CNJ CO CO CO CM X co X IX rxcr * w v-- ' >—> »—' *—1’ v_/ '—' '—* *■—' v---' v--> v—> \_* V_t *_/wi
•Hcrw x CO X CO CO CO CO CM IXi 03 *r~ CNJ CO CO CO 03 03 CD a T -
2 c X 03 r - X CO ^— CO X a o CM o^ ca 03 CO • ■'T CNJ o . . . . . . • . • .
> , 03 • • • • . in m ■d" ^f* CO CO rx CO COcnj z : CO CO CO CO co r— r - CO CO CO CO CO
O /—N CO COc CO CO in CO CO .—> CM co ■CJ- CO un CO CO CM
• ID un 03 03 CM a a V“ CM CO CM ,—> CO T“■ •H 03 r - rx CO CO a CO CM IX CO . • CO CO
cr • • • • • . • . • • CO CD • •+ <D r— CO 03 CO CO <r- CD «3* □ a O CM IX
•<-> > CM CM CM "C— CO CO co CO T~ 03 CO COcr ’ w w v_/ w <—1 w V----' >—> v—< \_! V_t V_t \_>
wi
•Hcrw CO CO CO CO <=}- rx CD in un in in CO COi in CO 03 CO CO CM CO IX <cj- IX CM IX CD CO
2 C CO IX CM CO CO CO o CO <=}* co o CO 'Cf CD COV----* 0 . • . • . . . . . . . . ■ ■ .
03 □ 03 03 co co CO IX in in CO CM o CO XCM E l v - V t- '3 ' CO CO
CM CO un a CM co in a CM CO un or— T*
CO CO CMr - CM X
II II IIa CD
CO 0 ^— 0 V* CO 0in G CO G G
TJ it C TJ n C > . TJ n c> 0 0 > 0 0 JO 0 0UJ n 4-> c •H JO +J c •H 4-> c •rH_ ) CO G 0 U G 0 U un G 0 f-.CL UJ CO 0 0 0 CO 0 0 0 0 0 0e : N Q. z : > D . s ; > Q. e ; >< H X X XCO CO UJ UJ UJ
- 1 0 3 -
TABLE
6.4.5
Examining
the accuracy
of the
*X2
approximations for
the
block
and
treatment
measures of
variation
CO CD LD un CN CO CO ■sT CO IX CO COun cn cn CN CD CD IX CN CO CO IX □
CD CO CO CO CO CD CO CO CO CO CD CO CO CO CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . .O a □ O
• LD CN CD CD CO CO CN r~ CO CO IX CN• cn CD CO CN IX CO CO co CD CN *3" a CO
c r LD CO co un CN CO CO ■sT «cr CN un un0 CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
i • . . . • . . . . . . . . . . .O □ a a
•H01W rx rx un CN CD CO IX un CO CN CO'—* □ CO CD CO ix CO CN CO IX CN CO IX CO CO
o ix CO CO CO CD CO co CO CO CD rx CD CO CD onCN CD CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO
. • . . . • . . . . . . ■ . . .o a a a
CO IX IX CD CD IX un CN 'T— CO COCO CO CO CO CO IX CD CD CD CO IX IX co
CD CO CO CO CO CO CO CO CD CO CO co CO CO CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . • . . . . . . . ■ . . .□ o o o
t_% CO CN CN CO CO CD CN CO CO rx CN• CD CN ( \ CO CO un CO □ un o un CN IX CN CN■ > ' in CO •3- <3* un un un *3* un
cr L CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD. • • • . . . . . ■ . . . . . .
i a o □ □
•>“5
cr IX rx un CO un CO CO CNw 'x’ IX LD <3- CN <r~ CO a un CO o a>—< o CO CO CD CD CD CD CD □ CD o □ CD o CD □
CD CO CO CO CO CO CO CO CD CO CD CD CO CD CO CDCN • . . . . . . . . . . . . . . .
o a □ o
CN CO cn a CN CO un o CN CO 'CT un oV
cn cn 0-p -p -pc c cCD 0 0 0E cn E 0 E
-P -P JtC •p 0(0 o (D a (0 o
un 0 o 0 o a 0 rH0 1—1 CO 0 i—i h* 0 JQ
> -P J3 -P JD -pUJ n LD > •sT > o a_j co CO LD H* iJ CO CO CN JO un V unCL UJ COz : m II II II CO II II II un II II II< Mcn az Pu cr 2 CL cr 2 CL cr 2
- 104 -
Tables 6.4.6 and 6.4.7, probably the most important, show the results for the old
and new F-approximations. The tables give the proportion cq of Monte Carlo results
less than the F-value for the appropriate statistics when the theoretical proportion
should be a. The old F-approximations being Z 1 and Z 2 given by (6.3.2) and
(6.3.3) respectively, and the new F-approximations being m Z, and m Z 2 where m is
given by (6.4.1) and is the statistic due to Stephens improvement of the residual
measure of variation.
( 6 .4 .1 )
The conclusions here are the same as for the F-statistic for one-way analysis given
by Stephens (1972). The new tests are clearly very good F-statistics and increase in
accuracy as k and N increase. The approximations are very good, even for N as
low as 10, and improve quickly for larger values of N.
The chi-squared approximations of (6.3.4) is examined in the same manner as above.
Tables 6.4.8, 6.4.9 and 6.4.10 show the chi-squared approximations for all five
components of (6.3.4). The total and residual measures have been adjusted by
Stephens improvements, y, the original approximations of the total and residual
measures were of similar form to those already seen in the randomised block design.
All five approximations, including the interaction measure built in section 6.2, show
excellent chi-squared approximations for k > 2. Accuracy is seen to improve as k
increases.
Tables 6.4.11, 6.4.12 and 6.4.13 give the results for the old and new
F-approximations for testing the two main effects and the interaction component.
The old F-approximations being Z 3, Z 4, and Z 5 given by (6.3.5), (6.3.6) and
(6.3.7), and the new F-approximations given by m Z 3, m Z4, and m Z 5. Clearly all
three F-statistics improve when Stephens improvement is used, similarly and as for
the randomised block design, increased accuracy is seen as k and N increase.
- 105 -
TABLE
6.4.6
Comparison of
the
two
F approximations ZA
and mZ
un IX) CO CD CD CO <3* IX) CD CD COcn CD CM cn o cn CD CM o CD
CD CD CD CD CO CD CD CD CD CD CO CD CD CD CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . .o O □ O
rx CM rx ix CD CO CD CD CD IX) IX)CM IX IX) CM <r* cn <3* IN. O CM rx IX) CM un CM
/ LD CD IX) IX) IX) IX) CD IX) IX) IX) IX) CD un IX) unI CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
M . . • ■ • • . • • • • . . . . .E o o □ o
IX) IX CD CD IX CM CO ■<r- *3- *3*cn CD IX) CD CD IX) cn IX) v- r- rx '3" CD
o r— O □ CD CD CD o O O CM O □ CD aCD CD CD CD CO CO CD CD CD CD CD CD CD CD CO CD
. . . . . . . . . . . . . . . .o O □ □
cn CD IX CM IX cn CM rx un CMCD IX) IX) V cn CD <3* IX) CM CD CD cn cn ■3"
CD CD CD CD CD CD CD CD CD CD CO CD CD CD CD CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . • . . . . . . .CM □ O O □
cn '■
CD rx V“ cn IX CO CD CM IX) COi_> <3* CD IX IX IX) CD CO *3" CD <3" < r IX) IX CO
V IX) rx CD CD LD IX) rx CD co IX) cn CO rx CD un unc C CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CDo . . . . . . . . . . . . . . . .
•H o □ □ □•PCODcr CM CD V“ IX) CO CO CD IX CD CM *3"
LU CD CD CD CM *3- cn IX) cn cn CD un IX ao cn CM V □ CM CM rx CD cn CM
T“ CD CD CD CD CD CD CD CD CD CD CO CD CD CD CD CDIX . . . . . . . . . . . . . . . .
o □ □
CM cn *3" IX) o CM cn *3" un □ CM cn IX) □V-
□UJ un CO r—_ i cnQ_ UJ > >E: rx JO n JO< Mcn cn cn cn un
- 106 -
TABLE
6.4.7
Comparison of
the two F
approximations Z0
and mZ
CN
N
CO CO <3- LD CO co cn *3" IX rx CO*3" CN TT- a — a cn cn a CN CN CN rx
cn cn cn cn cn cn cn CO CO cn cn cn cn CD co cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . . .o o a o
CO CO cn CO IX CO CO CO coIX cn cn CO CO □ IX cn CO CO cn <r~
/ cn CO un <3- IX) CO IX) IX) <3- IX) IX) CO IX) <3* IX)cn O cn cn cn cn cn cn cn cn CD cn cn cn cn cn cn CD
M . . . . . . . . . • . . . . . .E o o o a
IX) CN cn IX) <3* «3- CN CO CN <3* IX COcn CN IX CN CO IX) IX) cn IX cn CN IX) cn CO
□ □ o a TT“ a □ CD cn a >r- a cn □cn cn cn cn. cn cn cn cn cn CO CO cn cn cn CO cn
. . . . . . . . . . . . . . .o □ □ a
cn CO IX co CN CO cn cn CN IX) CO COCO <3- CO CN CN IX) CO CO ’3’ CO CO IX CO CN
cn cn cn cn CD CD cn cn cn cn co cn cn cn cn CD<—> cn cn cn cn cn CD cn CD cn cn cn cn cn cn cn cnCO . . . « . . . . . . . . . . . .
. □ o o oCO
CO<_> IX CO CO <3" CN cn *3" CN rx
a CO □ CO IX CO CO v - CD CN IX a COc > cn CO CO CO 1X1 IX) IX IX CO co CO CO IX rx CO IX)o c cn cn cn cn cn cn CD cn cn cn cn cn cn cn cn cn
•H . • . • . . . . » . . . . . . .-P □ □ o acoZJcr
LLl IX CO CO CO IX) V- CN CO CN COTr- LD cn CO r - □ CO IX IX IX co CO CO cn CO
CN o UO CO TT” v - IX) CO CN TT- IX IX) 3" CO v -N cn cn cn CD cn cn CD cn cn cn CO cn cn cn cn cn
. . . • . . . . . . . . . . . .o □ o a
CN CO "3" IX) a CN CO <3- IX) □ CN CO *3" IX) oV-
oLU un CO_1 COQ_ LU > > >e : m n .□< MCO CO CO CO IX)
- 107 -
TABLE
6.4.8
Examining
the accuracy
of the
X2 approximations for
the
total
and first
main
effect measures of
variation
CO CO in CM CO CM CD CM co CM •3* CO CMCD V CO o a a CD o o r - co a CD a CO
CD CD CO CD CO CD CD CD CD CO CD CD CD co CD CD CD CO CD CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . . . . . ■o □ a a □
—CO in IN CO CD CO CO CO '3 ’ in CO CO CO <3* in CO CM
• CO CD □ in CM CO CD in □ t— CD CO 3* rN CM CO in V in CD• in *3* 'nI ' in '3 ’ in <3" '3" <3" in in in in in 3" in in '3 -
c r I CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD. . . . . . . . . . . . . . . . ■ . . .
o o a a O
•Hc rw'_i in IN *1* CO <3" CM CO CO CO CO CD CM CO in CM CMy in CD co CM IN in in CD CM <3- in «3" in CO CO «3" CO IN V-CM a CD CD CD CD o CD CD CD CD a o o a CD o CD □ □ CD O
CD CO CO CO CO CD CO CO CO CO CD CD CD CD CO CD CO CD CD CO CD■ . . . . • . . . . • . . . . . . . . . .
a a o □ □
CM CM T— IN in CO CO IN CO CO CD INCO CO CD CD CD in IN CD CM CO a CD in o a CM
CD CD CO CD CO CO CO CO co CD CO CD CO CD CD CD CO CO CD CD CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . . . . . ■ .o a a o a
CO CM CM <r- IN CM CM IN t— CD 03 <3- CO COCO a IN CM in <3’ CD CO CO <3* CD CM rN co in IN *3" in CO
in 'c r in in -sf CO <3* in in *3’ <3* *3- '3" in CO CO *3- in in> CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CDC . ■ . . . . . . . . . . . . . . . . • ■
• a a a o a
c ri
in CM rN rN CM CO '3 ’ CD in CO CM CD'—' CM IN CO CD CO IN *3* CM ■3* rN CO in CD o V~ CO> - a CD CO CD □ a CO IN CD a CD CD CD CD □ IN IN CD O QCM CD CO CO CO CD CD CO CO CO CD CD CO CO CO CO CD CO CO 00 CD CD
. ■ • . . . ■ • • . . . . . . . . . . . .a a a o o
CM CO '3" in □ CM CO *3* in a CM CO <3- in □ CM CO '3" in Ov~ V V < r
cJC •rH+J JZ•H 4-33 •H O
3 COc It ----. 4-->
<—> ClO C 21 □ «—« oin •H o •r- in r-
cn •H> 0 4-3 4---. > > >
JO TO ID rH JO JO JOV_/ > rH <_> >_4 V_4
CO I* 0UJ CO 0 o CO O CO in CO a_ ] cn > 0 CD •3- CDCL UJ > n JO sz > > >E ; INI n o G JO II JO n jD ii< H CM incn cn CM in 0 CM 2 CO 2 CO 2
- 108 -
TABLE
6.4.9
Examining
the accuracy
of they
approximations for
the
second
main effect and
interaction
measures of
variation
—CNJ CO CO r - O ' o - CNJ CO in o - IX in O ' CN CO
. CO o CD O CD CD CD CD CO o in in CO CO O ' CO 'T ' v CO
. CD CO CD CO CD CO CO CO CO CO CD O ' CO CO CO CO CO co CD CD COcr CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD+ m . . . . . . . . . . . . . . . . . » . .
. o o o a CD’r~)
crwi CO CNJ in CO CN! IX O ' CNJ O ' IX CO rx CO CO CO CO
. CO CD in CO CO CD CNJ CO o CD ^— CO CO CO CD CD CO CO CD• ?f in CO O ' O ' O ' in O ' O ' O ' in in CNJ O ' CO O ' O ' CO CO O ' o - O"
•H C} cr> CD CD CD CD CD CO CD CD CD CD CD CD CD CD CD CD CD CD CD CDcr . . . . . . . . . . . . . . . . . . . . .wi
o □ a a CD
• o•rH
cr CO CO CO CN! CO CNJ rx CD CN! O ' O ' CNJ CD CO CO O 'w rx CO CD CO CO CD CO rx CO a CO CO in IX CD CO CO rx CO COw o rx CD CO CO CD CD CD CD CD □ rx O ' co CD CD CO co CD CD CD>—> CD CO CO CO CO CO CO CO CO CO CD CO CD CO CO CO CO CO CO CO CO
• . . . . . . . . . . . . . . . . . . • .CM o O o o CD
CO O ' in CN! in in CO O ' CO CO CN O ' CO O 'CD CD CD CO CO V* CD o o CD O CD CO CO CO
CD CD CO CD CO CD CO CO CO CD CO CD CD CD CO CD CO CO CD CO coCD CD CO CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . . . . . • .O □ □ o O
CNJ CO CO CO O ' CNJ CO CO CO CO CO O '• CNJ CO CD CO O CO CD IX rx in CO o in CO CO CO in CD CD r-• lo in O ' O ' O ' in O ' O ' O ' O ' in O ' in O ' O ' O ' o - O ' O ' O ' in• a co CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
cr . . . . . . . . • . . . . . . . . • • • •
i o o o a CD
• o
crw CO in in CNJ CO O ' CNJ CO IX CNJ in O ' O ' CO O ' in>—< o - V- CD CNJ CO CD in rx IX CO CNJ CO in CD CO a CN CO CO
o o CD CO □ o □ a CD CD □ CD CD CD CD CD CD CD a CD CDCN CD CD CO CO CD CD CD CD CO CO CD CO CO CO CO CO CO CO CD CO CO
. . . . . . . . . . . . . . . . . . . . .
o □ a a a
CNJ CO O ' m a CN! CO O ' m o CN! CO O ' in a CN CO O ' in CDr~ V
cito □ ,_, t ^
<—\ b fl > CO o <—» CDin •H u C II V- in
CD CD •H 2> Q) cn JD >> > >
JO TJ JD -P i—1 JD J3 JD<_/ □ •H r—I v__ / V__ / \_ j
CO 2 CDUJ CO in CJ CO CO CO_j cn > cnCL UJ > JD sz c . sz > □ > in > CD£1 M JO -p Q O JD CO JD O ' JD CD< M CN1 •H •H (0 II II Itcn cn CNJ 2 ■P CD CN! 2 CO z : CO Z
- 109 -
TABLE
6.4.10
Examining
the accuracy
of the
X2 approximation
for
the
residual measure
of variation CO cn V IX) IX) COCO cn cn 1X1un ■cr 'TIX) CDCOCO rx <3- COo CO COCDCOCD IX) cn rs. CDv-
cn cnCOCOCOCO co COCOcn CO cn COCOCOCO co COCOCOcnCD CDCDcn CDCD CDCDCDCDCD CDCDCDCDCD CDcn cn CDcn
. . . . . . . . . . . . ■ . . . . . . . .CD CD CD CD a
COCD CDCO CO .. COIX cn cn CO IX) cn CM CMCO cn CDo cn cn CMCMCD cn IX) x IX COCO
cn cn CO«o- 'tf' IX) cn cn IX) cn CMCMcn IX)^ • 2CD cn cn CDcn cn CDCDCDCDcn CDCDcn CDcn CDCDcn cn CD
•>-> . . . . . . . . . . . . . . . . . . . . .•H CD o a a CD
orww~ZL cn CO IX) CM rx COCDCM IX) CDCO CM COcn CM>—> LD COcn ^r V- COCDCD ix X CD '3' x cn IX) X CO>- o rx COCOCDCD IX CDCDCDCD CDx COCDCD 1X1CDr x cn OCM CD COCOCOCOCD COCOCOCDcn COCOCOCOCO COcn COCOcn
. . . . . . . . . . . . . . . . . . . . .o CD a CD a
CMcn <3- IX) o CMcn IX) a CM cn IX) CD CMcn 'T un oV v V"1 O
c co cn <—> ,—i/—\ > n CD 1--V CDLD •rH u c IX)
cn 0 ■H>> 0 0 S Z /-- V > > >XI TJ XI -p 1—1 X3 XI XI«_> Q •H 1—1 >_i u t '
cn 2 0LU cn IX) o cn cn CO_ j cn > 0CL LU > XI SZ C s z > a > IX) > Oe : m XI •p o o XJ CO XI *3* XI CD< H CM•H •H 0 ii II IIcn cn CM s -P 0 CM s cn cn 2
_ 110 _
03JDP
UaH-co
M cE o
•HTO PC (00 O
•H03 P
M •Hcn
cn cnc (0o i—i
•H o-P0 >E cn
•H 3X io Ou 3Cl PCL(0 0
SZU_ p
O c3 •HP
P03 O
SZ 0p P
Pp 0o
cc •Ho 003 E
■Hu "O(0 c□ . QE Oo 0
u 0
3 *
coLU_JCD<
CD CNJ rx IX CN CO CO CO CO 03 CO CN CO IX CN COCO CD r— o CD CD CD r— CD 3 - a co CD CO o O C3
CD CD CD CD CD CD CD CD CO CD CD CD CD CO CD CD CO CO CD cn cn03 CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD cn cn
■ • ■ • ■ ■ ■ • a a a a a a a a a a a a
CD □ a CD o
CNJ in IX CO CD co CN CN CO CNCO a CD CD r - CD T * CO 03 CD CO IX CD 03
> LD U3 U3 U3 LD 3 " 03 03 3 " 03 03 03 03 03 03 3 - 03 3 " 3 * 03 3 "CO t 03 03 CD CD CD CD CD 03 CD CD CD CD 03 03 CD 03 CD CD CD CD 03
M ■ • ■ • • ■ • ■ • a a a a a a a a a a a a
E CD CD CD CD o
CNJ IX in CN V 03 IX CN 3 - CN IX CN CN CO IX rxCNJ 3^ CD 3 " CD IX CO 03 03 CN CO CN CD CN CN CD CD CO 3 " CO
□ CD CD CD CD o O CD CD CD CD O CD O cn O cn □ CD CD03 CD CO CO CD CD CD CD CO CO CD CD CD CO CD CO CD CO CD CO CO
• ■ ■ • • ■ • ■ ■ a a a a a a a a a a a a
CD CD CD CD a
CO U3 3 " CN 3 " s— CO CD CN IX CO CO CO 3"CO 3 * CN CO v~ 03 CO CN CO IX 3 " CO T - CD CO CO CN X
03 CD CD CD CD CD CD CD CD CD CD CD 03 CD CD CJJ cn CD CD CD CD03 CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD cn CD CD CD CD
« • ■ ■ ■ • • • a a a a a a a a a a a a ai—> CD O O CD □LD
CO
CD CNJ cn CO CO CO CN 3 " <3* CO 3 * V~ 3 " CN CO IX<—< r - CO V IX CO 03 CN (X CD CO CO r- CN O CO CN CO o COV 03 IX CO CO LD U3 CO CO 03 03 03 IX rx CD CO 03 IX CO CD 03 03
c 03 CD CD CD CD CD CD CD CD CD CD CD CD CD CD cn CD on 03 CD CDo • • • • ■ • ■ ■ a a a a a a a a a a a a a
•H □ o a o C3P0DJc r
UJ CO CO CO CO IX V CN 3 - CO 3 - V 3 " 3 "rx CO CO CD 03 CO •cr CD CO CD 03 IX CO CO 'T~ CO CO r - CD IX
CO CD CO v - r~ a CO CN CD a a CO CN V v o CO CN CN CD OM 03 CD CD 03 03 CD CD CD CD CD CD CD CD CD cn CD CD cn cn CD CD
■ ■ ■ ■ ■ a • • a a a a a a a a a a a a a
CD CD a a o
OJ CO 3 * CD a CN CO < 3- 03 CD CN CO 3 * 03 o CN CO 3 * 03 □*“ 'T-
,— > CD ,— , Oin < r~ 03 *“
> > > >JD JD JD n
w w
LU CO CD CO □ CO 03 CO □_ l C/D CO CO 3 " CDCL UJ >1 > > >E: m JD II JD II JD II JO II< MC/3 C/D CNJ z: CN z CO 2 CO Z
- Ill _
TABLE
6.4.12
Comparison
of the F
approximations 2n
and mZ
for
the
co
•H-Ptno
•rH4 -•Htncntni—i0>(D31O2 4->0
SZ-P
c•H•PCJ0
4 -<4-0C(DE
-P0P
•HH-
*3- CO CM CM CO 3* X X in X X CO CMCN CM CO cn CM CO o CM CO r— o V“ cn
cn cn CD CO CO cn cn cn CO cn cn cn cn cn cn cn cn cn cn CO cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . . . . . . . .o CD a a o
CO <3- CO X CM in CD 3" CO in CM in COLO in CO CM co CO □ CD in V“ CO CO CO 3" CO X
> LD cn LO in 3- in in in in in in CO in in in in in in in <3* *3"c cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn<3- . . . . . . . . . . . . . . . . . ■ . ■ .
N o CD CD □ □E
CM CO CO CO CM CM CO CO CO CM *3* CO COO CM CO cn CO CO CO X a in CO — CD CM co CO CM
□ v □ □ cn cn a cn cn CD CD t— V o a o a o cn CDcn cn cn cn CO CO cn CO CO cn cn cn cn cn cn cn cn cn cn CO cn
. . . . . . . . . . . . . . . . . . . . .□ o o □ □
CO rx CD CD t— X CO CM CO CM in X CO CM in 3"in <3* v- r— CM X LO CM CO CM LO in 3 CM CO in in *3* CM CO
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . • . . . . . . . . . . . . . . ./--» o □ o □ □CO
onCD in CM cn CO 3 - CO X 3 - CM 3" *3* X CO V CO 3"«— > in CO O *3" CO CM CO CM CO CO cn CO r- X CO CM CO CO co
V CD ix CO CD in in X CO CO CO in X CD CD CD tn X X CO in inC C cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cno • • • . • . . . . . . . . . . . . . . . ••H □ o □ a a-P(0Dcj
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- 112 -
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- 113 -
6.5 Summary
As discussed in Chapter 5 the likelihood ratio function has infinitely many equally
large peaks and may not be broken down under the set hypotheses using the
parameter constraints, as is the case for linear analysis. Because of this the tests
derived initially by Watson and Williams and improved by Stephens have been
extended, for large k , to enable analysis of further experimental designs. The
components within each of the model expressions are shown to be very good
chi-squared approximations and their appropriate quotients to be excellent
F-distributed approximations, both increasing in accuracy as N and k increase.
Simple extensions to the likelihood criterion for the one-way classification with small
concentration parameter showed the test components to be poor chi-squared
approximations with little justification for their construction.
- 114 -
CHAPTER 7
TH E ROBUSTNESS AND POSSIBLE COLLAPSE OF TH E EXTENDED
TECHNIQUES
7.1 Introduction
Chapter 6 has shown how we may extend the original techniques for the one-way
classification with large k, to examine further experimental designs. This chapter
examines the robustness of the test statistics and shows the sensitivity of the
assumptions and possible flaws or breakdowns within the new techniques. Section 7.3
gives further understanding to the construction of Watson and Williams test statistic
and the reasons for its possible failure when larger designs are considered.
7.2 Robustness of Assumptions
Section 6.2 gave the assumptions which must be observed in order that the new
extended techniques may be applied. The third assumption, that the overall
concentration parameter is sufficiently large, namely k 2 , is an extremely restrictive
assumption when analysing larger designs. Equation (1.4.16) gave an angular measure
equivalent to the standard deviation in linear statistics. In order that the assumption
of large overall concentration parameter may be satisfied, the measure of angular
deviation must not exceed 44* i.e. the overall sample data set must be closely
packed. This will not be the case if substantial differences result between or within
factors.
Further and more important problems are noted as the size of the concentration
parameter decreases towards and below 2. During the many thousands of simulation
runs discussed in Section 6.3 many other statistics were collected for each of the
- 115 -
varying hypothesised models. These included the distributions maximum and
minimum values. The simulations for the two-way classification procedure produced
several occurrences within each hypothesised model of a negative interaction
component. The frequency and magnitude of these negative components increased as
the size of the concentration parameter, k, decreased. When simulations were run
to further examine the power of the tests and specified differences were given
between treatments, blocks or cells, (i.e. separate von Mises distributions with equal
concentration parameters but differing mean directions) the frequency of the negative
interaction components increased.
An example of the two-way classification with interaction and its analysis is given in
Tables 7.2.1 and 7.2.2. It demonstrates the situation where the individual factor
concentration parameters (i.e. k 1, k 2, ......) are not significantly different and are
large, but the overall concentration parameter k is small. Other less dramatic data
sets may be shown with a larger overall concentration parameter but producing
similar, although less pronounced, results.
- 116 -
Example 7.2.1
Table 7.2.1 A Two-way Classification
FACTOR
^0 An
61* 177*
48* 139’
Ro 87* R , t . = 4 .7303 155* R 12> - 4 .7604 R i . . = 6 .6 5
102* N11 = 5 1 9 1 ’ N 12< = 5 N i . = 10
73* 164*
FACTOR
318* 240*
353* 281 ’
358 ’ R21 = 4 .8 284 229* R22 - 4 .7623 r 2 . . = 6 .6 5 4 8
328* N 21 . = 5 252 ’ N 22> - 5 N 2 . . = 1 0
344 ’ 2 40 ’
R n = 6.5247 R 2 = 7 .1363 R = 0 .6117
N . 1 . = 10 N 2 . - 10 N = 2 0
P q P q^ R i . . = 13.3058 ^ R j = 13.661 J ^ Ri j < = 19.0814
i= l j = l i = l j = l
p = q = 2, 1 = 5
- 117 -
Table 7.2.2 Two-Way Analysis of Variance Table
Var i at i on Component Value Degrees o f Freedom
Due to Factor A i*i.. - R . 12.6931
i=l
Due to Factor B ^ ^ . j . “ .
j -1
13.0493
( p - 1 ) - 1
(q -1 ) - 1
I n t e r act ion
Res idual
To ta l
P q P
I I Ri j - -i-1 j-1 i-1
q
h - j . + r ...j - i
p q
n - I I Ri ji-1 j-1
N - R
-7 .2727
0.9186
( p - l ) ( q - l ) - l
p q ( Z - l ) - 16
19.3883 N - 1 = 19
It should be noted from Table 7.2.2 that the sum of the two main effects measure
of variation is greater than the total measure of variation and therefore the
interaction measure is negative.
As noted the above example does not satisfy the assumption of large overall
concentration parameter k but has been used to illustrate the associated problems
when k approaches and decreases below 2. This type of data set questions the
independence of the test components in less dramatic cases. [The presence of a
negative interaction measure of variation does not occur if the data is axial in nature
and the general assumptions are upheld.]
- 118 -
7.3 A Further Approach to Watson and Williams Design Using a Regression
Model
Section 7.2 has given an example of where the extended techniques for the
randomised and two-way analysis designs may breakdown. In this section we will
briefly reiterate how the one-way analysis design was constructed and then
investigate, via a regression approach, the structure of the individual components.
From this approach the possible reasons for the breakdown of the present statistics in
larger designs may then be seen.
When testing the hypothesis that there is no difference between a number of
treatments, a basic result given in all texts, for the simple linear one-way analysis, is
P q P P q
where Xjj represents the individual observations, xj the treatment sample mean values
and x the overall sample mean. The left hand side measures the dispersion or
scatter of the whole sample about the true mean. The first term on the right hand
side measures the dispersion of the sample about the estimated mean x, whilst the
last term measures the dispersion of the sample means about the true mean.
It is easily shown that N -R represents the dispersion of the sample of directions
about the estimated mean direction in circular statistics. Equally N -X represents the
dispersion of the sample about the true mean direction, giving the expression
( 7 . 3 . 1 )
(N - X) ■= (N - R ) + (R - X) ( 7 . 3 . 2 )
in our analogue of (7.3.1).
- 119 -
Using the results of (4.2.3)
2k(N - X) = 2k(N - R _ ) + 2k(R - X) ( 7 .3 .3 )
has the associated chi-squared distributions
“ *N -1 + *1
This was developed further by Watson (1956) to examine 2 or more samples with
assumed equal concentration parameter. It may be seen that if the mean direction
of two samples differ greatly the sum of their resultants R , and R 2, would be
much greater than the overall resultant R . Similarly, if the mean directions are
equal R , + R 2 = R . This is then used as a measure of variation between
samples. The variation within the samples is measured by comparing the maximum
length of a sample resultant, N j, to its actual resultant, R j. For a two sample test
the within variation is therefore
(N tl - Re l ) + (N . 2 - R . 2)
This suggests the analysis of variance expression
2k(N - R ) - 2k( N ' t - R .2 + N# 2 - R. 2) + 2k (R. i + R . 2 " R . . >
( 7 . 3 . 4 )
It is easy to see from these analogies how the test for the one way analysis is
derived. Nevertheless, it is not until we investigate the individual components of the
analysis of variance expression that we see an underlying problem that is not fully
appreciated until larger, more complex designs are constructed.
A simple alternative way of building the analysis of variance expression is via a
regression approach using basic vector analysis. Figure 7.3.1 gives the notation that
we will use to construct the components.
- 120 -
j icon oj* sample, neon dii'e.ch’on
Overall sample\ f^ean direohor
.22
JZ
Angular Notation for the Construction of ComponentsFigure 7.3.1
Assuming the 0jj's (i= l,2 ,...,p ; j= l ,2, ,q) are independently distributed as
M(/<i0+/3j,/:), where /*0 is some mean direction and fij is the possible effect due to
treatment j. Let us test
H0 : 0 , - (32 - .........
H1 : at least one /3j * 0
- 0 q " 0
A A A
Let fi0 and (pi0, /3j) be the maximum likelihood estimates of ( iQ and / i0; under
H 0 and H , . Let
P q
S0 - N " I I C0S( 0iJ " " 3j)i -1 j -1
- 121 -
From the statistical results of (4.2.3), 2A:S0 has a x 2 distribution with N degrees of
freedom, for large k. Using approximation (1.4.13), 2fcS0 is equivalent to
P q
k I Ii= l j -1
A 2 ( 7 . 3 . 6 )
Using this approximation we may construct the individual components of the
expression.
7.3.1 Total Measure of Variation
From equation (7.3.6) the component for the total measure of variation is calculated
from the sum of the squared distances between each sample point, 0 jj and the
overall sample mean direction, 6 , as illustrated in Figure 7.3.2.
2.1
22.
12
32
Figure 7.3.2 Distances Between the Sample Points, and the Overall Sample
Mean Direction, 0
- 122 -
From vector algebra and basic directional data properties we have, denoting 10jj | as
the vector length;
P q
k l I ( £ i j - I..)2 ( 7 . 3 . 7 )
i - 1 j -1
p q
- k ^ ^ ( ' i i j * 2 + ^ . J 2 " 2cos(0 j j - 0 m' ) )
i - 1 j -1
= /c(N + N - 2R )
= 2/c(N - R ) ( 7 . 3 . 8 )
Producing the same total measure of variation as in (7.3.4).
7.3.2 Residual or Within Measure of Variation
The residual measure of variation is calculated from the sum of the squared distances
between each sample point, j, and its own sample mean direction, 6 j
Figure 7.3.3 Distances Between Sample Points, and Sample Mean
Directions, 6j
- 123 -
p q
* 1 I < £ ij - £ . j > 2 ( 7 .3 .9 )
i -1 j-1
p q
“ k \ \ j 1 2 + 1 .. j 1 2 - 2cos(0 j j - 0 . j ) )
i -1 j -1
qk{ N + N - 2 ^ R j )
q- 2k(N - £ R j )
J-1
Producing the same residual measure as in (7.3.4).
( 7 . 3 . 1 0 )
7.3.3 Between Measure of Variation
If, as is shown in 'linear' statistical analysis, the total measure of variation is split
into two parts, a residual or within measure and a between measure, then the
between measure may be found from
However, equation (7.3.11) assumes, as in 'linear' analysis, that when circular mean
directions are combined the same overall mean direction is produced. It was first
observed in Example 5.3.1 that this property does not exist for circular statistical
analysis, the following sub-section investigates this prior to deriving the 'true' between
measure of variation.
q q( 7 . 3 . 1 1 )
- 124 -
7.3.3.1 Combining Mean Angular Directions and Resultant Lengths
Expressed algebraically by (7.3.12) a simple proof shows that when angular mean
directions are combined the overall angular mean direction may not be produced, and
is dependent on the resultant lengths found within each of the combined samples.
q P q
I s in e . j 2 I s i n # i j
i —!--------------- ^ 1 y-1--------------- ( 7 . 3 . 1 2 )q p q2 COS e . j I 2 cos 0 j j
j-1 i-1 j-1
An angular mean direction, 0 is given by
cos 0 = ^ s in 0 = ^ ( 7 . 3 . 1 3 )R R
Similarly for 0 j
C j _ S jcos 0 j = -- s in 0 | = ——
s .j 5 -j
Therefore
q q p p q
I s i n ? . j I v T ] l s i n "i j I I s in # i j/ — U — ----------- ( 7 . 3 . 1 4 )
q q p p q
I c o s 9 -j I F T l c o s e i j 2 2 c o s ej-1 j-1 i -1 i -1 j-1
Figure 7.3.1 gave the mean direction of the sample means as T with resultant length
Rfl. This result gives further understanding and an alternative calculation of the
overall mean direction and its associated resultant length. By utilising the resultant
lengths associated with each sample mean direction, the mean directions may be
- 125 -
combined, as in standard statistical analysis, to give the overall mean direction.
Using a rectangular co-ordinate system with X and Y axes and origin 0, let 6 j be
one of the q mean angles with corresponding mean resultant length r y Let x j and
y j be the rectangular components of r j, as seen in Figure 7.3.4. Then by
definition
x.j - r jcos e.j y.j " r.jsin 0.j
Then
( 7 . 3 . 1 5 )
x = — (r,cos 6 + r 2cos 6 2 + . . + r #qCos 0 q) ( 7 . 3 . 1 6 )
y = — (r#1sin 6 ^ + r 2sin 6 2 + .. .. + r qSin 6 q ) ( 7 . 3 . 1 7 )
Therefore
M *1
= [(i I r.jcos ? j)’ + (I I r.jsin ?.j)2]i ( 7 . 3 . 1 8 )
R = Nr
Similarly for p row mean directions
P[(i ^ rj cos )2 + (I ^ri SinOj ) ^
i=l i=l( 7 . 3 . 1 9 )
Giving the overall mean direction as
6 = arc tan
q1q I r - j s in
j= iq
i Y —
q I r . j cos
( 7 . 3 . 2 0 )
- 126 -
Figure 7.3.4 The Rectangular Components of a Non-Unit Vector
This may be further understood by a simple illustration given in Figure 7.3.5. Here
two samples have four observations within each, by taking account of the sample
mean resultant lengths the true overall mean direction is obtained.
- 127 -
Uni^ lenqfil
Figure 7.3.5 Combining Samples Using Angular Mean Directions and their Mean
Resultant Lengths
Note: (a) Fisher and Lewis (1983) considered the problem of forming a pooled
estimate of the common mean direction of several circular samples with possibly
differing concentration parameters. In brief their work discussed the introduction of
some arbitrary non-random weighting factor to each sample defining
'Wi -1 i -1
P2 = + C 2 ( W j > 0 J Wj - 1 )
i - 1
The general pooled estimate for ’ p, given by defined by
- 128 -
cw swcos Aw - — s in Aw = Z“
Rw Rw
Using the central limit theorem Fisher and Lewis produced a confidence interval of
the pooled estimate comparing this to the approximate confidence cone for the single
sample case. (Further approximate confidence intervals for a mean direction of a
von Mises distribution were later given by Upton (1986)). Fisher and Lewis showed
the consequences of specific choices of weights, considering equal weighting of 1/q
and proportional weighting of Nj/N.
(b) An alternative calculation of the overall resultant length from the sample
angular mean directions and resultant lengths is via the dot product rule for vectors
Using the q column statistics
q q qR2 . + 2 ) ) R . R cos( 0* . - 0 J ( 7 . 3 . 2 1 )
.J L L . J . t . j . tj=l j^i t - l
j 7* t
j < t
Similarly for the p row statistics
P P P
R?."IRi. + 2l1=1 i= l t= l
i ? t
i < t
For example, if the overall resultant was to be found from three sets of sample
statistics
R2 = R2 + R2 + R2 + 2R. R , cos(0. - )1 . 2 . 3 . 1 . 2 . ' 1 . 2 .
+ 2R1 R3 cos( 0 t - 0 3>)
+ 2 R 2>R 3 > c o s ( 0 2> - 0 3 > )
R. R cos(0 . - 0. ) ( 7 . 3 . 2 2 )l . t . l . t .
- 129 -
Returning to the calculation of the between measure of variation, it is evident that
this should be calculated from the sum of squared distances between each sample
mean direction 6 j and their combined mean direction T. As an identity this will
leave a further component measuring the difference between the combined mean
direction, 6, and the overall mean direction, 6 , as in (7.3.23)
( 0 i j - * . . > ” < * i j “ * . j > + < * . j - « ) + ( * - 7 . ) ( 7 . 3 . 2 3 )
t o t a l res idua l between d i f fe re n c e betweenthe combined mean d i r e c t i o n and the o v e r a l l mean d i r e c t ion.
Therefore the 'true' between measure of variation is given by;
P q kli - i j - i
p q~ ^ ^ ^ ( l£ . j 1 2 + \8_\2 - 2cos (0 j - 6 ) )
i -1 j -1
- /c(N + N - 2Rd)
- 2k(N - Re) ( 7 . 3 . 2 4 )
7.4 Cross Product Terms and the Analysis of Cross-Classification
In 'linear' statistics the corresponding expression to (7.3.23) will produce cross
product terms equal to zero. On the circle these terms are found to be non-zero;
P q P q p q
I I (£ij - I . ) * - I I<lu + 1 I<Z.j-Z>2i -1 j -1 i -1 j -1 i - 1 j -1
p q p q+ l + J <£i j - I.jXZ.j - I>
i -1 j - 1 i -1 j -1
- 130 -
p q _ p q
+ 2 1 1 + 2 1 1 (- Jj ■i -1 j -1 i - 1 j -1
( 7 .4 .1 )
The left hand side and the first two terms on the right hand side of (7.4.1) have
been simplified in Section 7.3, giving
P q
k ]j ^ ( .^ i j ~ Z . . ) 2 " 2/c<N “ R. . ) from ( 7 . 3 . 8 )i - 1 j - 1
p q q
k £ 2 (- i J " ^ * j )2 " 2/c(N ” 1 R- j ) from ( 7 *3 * 10)i - i j - i j - i
p q
k 1 1 ( - - J " " 2/c(N " R0) from ( 7 - 4 -24)i - i j - i
The following term is the measure of the difference between the combined means
and the overall mean direction.
P qk J J (0 - 0 ) 2 - /cN(|0 | 2 + | 0 _ l 2 _ 2cos(0 - 0 )
i - 1 j - 1
- 2kN (l - cos(0 - 0 ) ) ( 7 . 4 . 2 )
The final three terms are the corresponding cross product terms.
P q _ q R .
2 k l 1 ( - ! J ' Z . j H l . j - £> “ 2fcP I (— - £ . j ) < Z . j - £>i - 1 j - 1 j - 1 p
P q = u _ _
2k I J ( £ . j - £ ) < £ - £ . . ) - 2feN< £ ) < £ - £ . . )i - i j - i N
- 131 -
P q R R q _2k J J ( 0 ; : - 0 0 ( 0 - 0 ) - 2/cN(— * ------- ) (0 - 0 ) ( 7 . 4 . 3 )
. - . - " N Ni - l j - 1
Summing the cross product terms of (7.4.3) equals
q- 2k (^ R j - 2N + Rfl - R + Ncos(0 - # . . ) ) ( 7 . 4 . 4 )
j - 1
Producing the circular model
q2k(N - R . . ) - 2/c(N - J Rmj ) + 2k(N - R0) + 2/cN(l - cos(0 - 0 . . ) )
j “ l
(T o t a l ) (Res idual) (Between) (D i f fe re n c e betweencombined and o v e r a l l mean.)
q+ 2k(N - J R j - 2N + Re - R + Ncos(0 - 0 ) )
j - 1
(Cross product terms)
Reducing
2k(N - R ) - 2k(N - y R j ) + 2ic(N - R0)
j - 1
(T o t a l ) (Res idual) (Between)
q+ 2 * ( J R . j + Re - R . . - N) ( 7 . 4 . 5 )
j - 1
(C orrect ion)
Unless the residual term does not exist the correction will always be negative since
the between measure in (7.4.5) will always be greater than the between measure in
(7.3.4) from Watson and Williams.
- 132 -
When larger designs are considered in 'linear' statistics the between sum of squares
can be broken down into three components measuring differences between rows,
differences between columns and the interaction within the design. In directional
analysis, from extending Watson and Williams, this splitting produces
P q P q
( I 2 Rij . - R. . .> - ( J r i . . - R . . ) + c 2 r j . i -1 j-1 i-1 j-1
(Between rows) (Between columns)
P q P q
+ ( I I Ri j . - - ] r . j . + R . . . )i -1 j-1 i -1 j-1
( I n t e r a c t i o n ) ( 7 . 4 . 6 )
If the first and second terms on the right hand side of (7.4.6) are calculated their
sum may be greater than the total measure of variation, as was illustrated in
Example 5.3.1. In this situation the value of the interaction will be negative,
whether interaction exists or not. It is only when the between measure is broken
down in this manner for larger designs is the problem within its derivation fully
realised.
- 133 -
7.5 Summary
Chapter 7 has shown via a simple alternative approach to Watson and Williams
how,as k decreases, the model components begin to breakdown. The most obvious
consequence is seen when analysing two-way classification designs when a negative
component may be produced. These are shown to be a consequence of the way in
which circular mean directions combine. Unlike linear statistics, if the overall mean
direction of a sample is found, 6 , and then the same sample is split into equally
weighted samples and the overall mean direction re-calculated from the resulting
combined means, a different overall direction, 0, may be produced.
For the one-way analysis this does not cause any major problems. However, when
the technique is extended to larger analyses the sum of the main effect components
may give a result greater than the total measure of variation.
It is important to re-emphasise that the above only occurs as the overall
concentration parameter k decreases towards and below 2. The new extended
procedures, developed in Chapter 6, are satisfactory for highly clustered data sets
since the larger the value of the overall concentration parameter the smaller the
correction value of Section 7.4, and similarly the closer the two mean directions, 0
and 0, become.
- 134 -
CHAPTER 8
DEVELOPMENT OF A NEW ANALYSIS OF VARIANCE PROCEDURE
8.1 Introduction
Chapters 6 and 7 have shown (1) how the modified components for the one-way
classification have excellent chi-squared approximations for large k (2) how the
extended components for further design models also follow good chi-squared
approximations, for large k (3) how the underlying structure of the components may
breakdown as k decreases (4) the dependence of the components as k varies and (5)
since the overall concentration parameter must be large as well as the sample
concentration parameters, the tests may be good approximations only when applied to
highly clustered data sets.
This chapter is concerned with developing new test statistics which may be
generalised across all values of k and are based on likelihood ratio statistics for
testing both the mean direction and the concentration parameter, k.
The main reason for the breakdown of the extended models as k diminishes is
concerned with the combining of the angular mean directions, discussed in Section
7.3. A new procedure is shown to overcome this problem and therefore enable
components such as interaction to be investigated as k decreases.
Section 7.33 showed how the overall mean direction will remain unaltered if the
sample resultant lengths are retained when combining sample mean directions. Using
this fact, Section 8.2 constructs new components via a regression approximation
approach previously utilised to show the construction of Watson and Williams original
test statistic.
- 135 -
Following the discussion of the cross product terms in Chapter 7, Section 8.4
discusses the interpretation and calculation of interaction for the procedures.
8.2 Minimising Chord Distances to Build New Test Statistic Components
Using exactly the same approach as Section 7.3 a design procedure may be built via
a regression approach using basic vector analysis but also taking account of the mean
resultant lengths. This method will minimise the chord lengths between points within
the circle.
Assume the 0jj's (i = 1, 2 ,...., p; j = 1, 2 ,...., q) are independently distributed as
M (/t0 + /3j, k) where /*0 is some overall mean direction and 0j is the possible effect
due to treatment j. The null and alternative hypotheses are
H0 'Pi @ 2 ** ......... “ Pq — k 2 — .............— kq
against
H1 * 0 2 * ......... * /3q k , * k 2 * .............* kq
or the testing of different populations. To overcome this, the equality of the
concentration parameters must be examined prior to any examination of the main
effects, in a similar manner to that undertaken in standard 'linear' analysis of
variance.
8.2.1 Total Measure of Variation (TM V)
It was shown in Section 7.3 that 2kS0 is equivalent to
P q
k I 1 (6U " Vo ~ Pj>2 (8.2.1)i= l j - 1
- 136 -
Hence under H 0:
P q P q
kl I <#u -f» -ep2 = kl Ii - 1 j - 1 i -1 j - 1
Using the above expression the total measure of variation will be calculated from the
sum of the squared distances between each sample point, 0jj, and the overall sample
mean direction, 0 , as illustrated in Figure 8.2.1.
22
3Z
Figure 8.2.1 Vector Lengths for the Total Measure of Variation
Here the mean resultant length of 0 _ is utilised rather than effectively extending it
to the circumference of the circle. From vector algebra and basic directional data
properties we have, denoting 10jj i as the vector length:
- 137 -
p q
kli -1 J- l
(8 .2 .2 )
p q
[ *^.ij 1 2 + !£. . ,2 “ 2 *£.ij 1 *£... 1 c o s i j “ 0. . ) ] i -1 j - l
p q
‘ 2 2i -1 j - l
R2 2R1 + —— cos(0js - 0 )
N2 N
R2 2R2N + -
N N
TMV - kR2
N -N
( 8 . 2 . 3 )
8.2.2 Residual Measure of Variation (RMV)
The residual measure of variation will be calculated from the sum of the squared
distances between each sample point, 0jj, and its own sample mean direction, 0 j, as
illustrated in Figure 8.2.2.
P q
k l h i a - t . p 2i - i j - i
( 8 . 2 . 4 )
p q
k J J [ *£.i j 1 2 + ' i . j ' 2 ■ 2 ,£ i j l lL. j 1 c°s i j - 0 j ) ]
i - 1 j - l
p q
‘ 2 2i - 1 j - l
RJ.i 1 + - JR .(N1
Nf j N . j .
c o s ( 0 j j - 0 j )
r q R2 . • J
q R2 . • Jk N + ) - 2 yL
L J - l N - j .L a
J - l
- 138 -
r q R2 . • JRMV - k n - j
LL J-i N - i
22,
32.
Figure 8.2.2 Vector Lengths for the Residual Measure of Variation
8.2.3 Between Measure of Variation (BMV)
By incorporating the resultant length with its respective angular mean, as discussed in
Section 7.3.3, we may produce the same overall resultant length and angular mean
when they are combined.
Let us now construct the between measure of variation in the same manner as for
the residual and total measures. The between component will be calculated from the
sum of the squared distances between the sample angular means and the overall
angular mean. This measure is illustrated in Figure 8.2.3.
- 139 -
Figure 8.2.3 Vector Length for the Between Measure of Variation
P Q
k l hi-i - i j - i
j( 8 .2 .6 )
p q
i - 1 j - l
p q
* 2 2i - 1 j = l
R2 . R2• J
N2 . N 2 • J
R . •J
N Jj
Rcos(0 #j - 6 ' )
3 R2 . • J
R2 R2y + — - 2 — ( 8 . 2 . 7 )L
U - 1 N - j . N. . N ' * 4
f 3 R2 . •J
R2BMV - k I _ _J_L
LLj-i N - j . N. .
( 8 . 2 . 8)
- 140 -
Lemma. To show ^ Rjcos(0j - 8 ) - Nr -j “ l
fo r equat ion ( 8 . 2 . 7 )
Proof
J R^ jcos(0 j - 0 ' ' )
j “ l
q* J N j r j (cos 0 cos 8 j + s in 0 s in 0 . j )
j - l
■ ij - i
N . j r . j
X X • J
r r •Jj
y . . y . j
lr,- r-jJ
X y . .
r * * -
Nx +r
Ny
N[x2 + y 2 J - N r
8.3 Cross Product Terms
As discussed in Section 7.4, in standard 'linear' analysis the cross product terms sum
to zero, however, with the extended models for circular statistics a non-zero value is
obtained. By taking account of the mean resultant lengths this property is
re-established. Expression (8.3.1) shows the components within the design
- 141 -
p q p q
k l h e n - I - » . j > +i - i j - i
( T o t a l )
i - 1 j - l
(Res idual )
p q p q
k 1 1 ( $ -J ‘ 6- - )2 ‘ 2k 1 1 ( 9 iJ ‘ ( 8 . 3 . 1 )i - 1 j - l i - 1 j - l
(Between) (Cross-Product)
From (8.3.1) the cross product term may be broken down as follows;
P q
2kl I <*IJ - ‘i - i j - l
p q
“ i ii= i j - i
! (£ j j I l£ . j |cos(0 j j - 0 j )
( 8 . 3 . 2 )
- i£ i j I 10 |cos(0 j j - 0 ) - l £ . j M 0 . j l + 10. j I l£ . . lcos(0 j - 0 _ )
p q
“ i ii - 1 j - l
R• J
N JJ
c o s ( 0 j j - 0 . j ) “R
Nc o s ( 0 j j - 0 )
R2 . • J +
R . •J
R
N2 . •J-l ln j J N . ^
cos(0 < j - 0. . )
2kr ^
yR .
• JR2 q
- IR2 .
• JR2
LU - i N J . N. .
Lj - l N - j . N. .
i.e. The cross product terms sum to zero.
- 142 -
Hence the one-way classification design model can be decomposed as:
R2 r qr» R2 . •J
R2 r qr* R2 . •Jk N ------ — - k i - — — + k n - y
N LL j - i N . i N . # L
L j - i N J .
(TMV) (BMV) (RMV)
8.4 Analysis of Cross-Classification and Interaction
8.4.1 The Interpretation of Interaction on the Circle
The extent of the problem within the original approach was not fully appreciated
until extensions were made to larger designs. Before constructing the statistical test
components for larger designs using the approach of Section 8.2 the understanding
and interpretation of interaction on the circle will be discussed.
To illustrate the meaning of interaction on the circle a simple example giving
population angular means for a particular design may be show. (Table 8.4.1).
Table 8.4.1
Treatment Level
A B C
1 356* 6* 357*F a c to r
2 0* 14* 1*
Unlike standard 'linear' statistics a simple subtraction of level mean directions may
not be used to indicate the presence of interaction since here periodic values are
given and not quantities. The true angle between each of the directions must be
calculated in order to observe any differences. (Table 8.4.2).
- 143 -
Table 8.4.2
Treatment Level
A D i f f B D i f f C
F a c to r
1
D i f f
2
356*
( 4 * )
0*
( 1 0 * )
( 1 4 * )
6*
( 8 * )
14*
( 9 * )
( 1 3 * )
357*
( 4 * )
1*
We note that the difference in direction between the two Factor types is larger for
the middle level of treatment than for the low and high levels. Similar differences
are seen between any two treatment levels and the two Factor types. With unequal
differences in sample angular means we may state that interaction exists between
factor and treatment levels.
As in linear statistics graphing the cell mean directions can be an aid in interpreting
the interaction. For example, consider an experimental design that involves three
levels of treatment and two levels of a factor. Lines are used within a unit circle to
represent the cell mean directions for the experiment. Table 8.4.3 gives the sample
means for the design, with equal concentration parameter within each cell.
Table 8.4.3
Treatment Level
A B C
1 270* 320 ’ 40*F ac to r
2 170* 220 ’ 300*
- 144 -
Figure 8.4.1(a) indicates no difference in the length of the line segments between
level A and level B of the treatment. Figure 8.4.1(b) shows no difference in the
length of line segments between level B and C of the treatment. Similarly Figure
8.4.1(c) shows no difference in the length of the line segments between level 1 and
level 2 of the factor. Here Figure 8.4.1 illustrates an experimental design where no
interaction is present.
(a ) D i f ferenc e between (b) D i f fere n c e betweenlev e ls A and B o f treatment leve ls B and C o f treatment
—
2Z
(c) D i f ference between leve ls 1 and 2 o f f a c t o r
Figure 8.4.1 Mean Responses without Interaction
- 145 -
Alternatively Table 8.4.4 and Figure 8.4.2 illustrates the same design layout but with
interaction present.
Tab1e 8 . 4 . 4
Treatment Levels
A B C
1 270* 240’ 340*Factor
2 280* 360’ 60*
23
(a) D i f ference between (b) D i f feren ce betweenlev e ls A and B o f treatment lev e ls B and C o f treatment
22.
23
(c) D i f ference between leve ls 1 and 2 o f f a c to r
Figure 8.4.2 Mean Responses with Interaction
- 146 -
As in 'linear' statistics when the data indicates that large interactions exist, it is
important to consider whether large interactions actually are present in the sample
means or whether there may be some other explanation for the occurrence of the
interactions in the data.
Unexpected interactions may be caused by a problem in the data, there may be an
outlier or an erroneous response. Possibly another effect may be taking place which
has not been accounted for. In experiments involving animals, for example, where
room or time of day may give an apparent interaction when none exists. In such a
case, the errors can no longer be said to be random or independent. Thus an
unexpected interaction may be a clue to a failure in meeting the assumptions of the
model being used.
8.4.2 The Calculation of Interaction on the Circle
The between cells sum of squares in 'linear' statistics can be split up to produce a
between rows, between columns and interaction sum of squares. For a two way
analysis the individual cell interaction values are made up of three components, the
distance from the overall mean to the cell mean (x - xjj ), from the row mean to
to the cell mean (xj - xjj ), and from the column mean to the cell mean
( \ j . - % ) •
(x - x j j ) - (Xj - X i j ) - ( x j i - x i j . )
Giving
X i j - x j - x j + x ( 8 . 4 . 1 )
For the overall interaction this distance is squared and summed over all observations.
For interaction on the circle Figure 8.4.3 illustrates the sample mean direction for
the three vector lengths involved.
- 147 -
Figure 8.4.3 The Components of Interaction
Interaction on the circle may then be given by;
p q m
* 1 I I < £ i j . - £ i . . - i . j . - L . Ji = l j = l Z=1
( 8 . 4 . 2 )
p q m
- * I I Ii - 1 j - l Z- l
R? . R? + - 1
N? . N?. i j . l
R2 . R2+ - 1I 1 + —
N2 . N2
- 2R. .
i jNi j
R.1 cos(0 j j^ - 0 i m' ) - 2
‘i j
L« i j
R
NcosC^jj^ - 8 j )
- 2R.
1
Nicos( 0 j - 8' ' ) - 2 VJ_
LN. j - j Ncos (6 j - 8 )
- 148 -
+ 2R
NL L
i J . J
cosN
‘i . .
N i . .
•J
Ncos(0j - 0 s )
j .
( 8 . 4 . 3 )
R.i j
Njj
R .J
It can be shown that
p q in
1 1 1i - 1 j —1 Z—1
p q m
2 2 2i -1 j - l Z -l
p q m
2 2 2i -1 j - l Z -l
p q m
2 2 2i -1 j - l Z -l
p q m
2 2 2i -1 j - l Z -l
p q m
2 2 2i -1 j - l Z -l
R, , R* Pn R2i j . i . . c o s ( 6 j j ^ - « i . . > - )
l . .
Ni j * . Ni . . .Li
i - 1 Ni . .
R
lNU
R.l . . R .• J.
_N i . . NJ ,
c o s (0 i j . - 0 . j . ) " \
j - l
R2
C O S ( 0 j # # - 0 5 )
N
R2
N
R2
N
R2
N
R2
N
Replacing these into (8.4.3) gives
r p q
y y R2 .IJ .
p_ y R?l . .z zLi-1 j - l Ni j . .
zi - 1 Ni . . .
Ij “ l
RJN
R2 + —-
N( 8 . 4 . 4 )
- 149 -
A comparable breakdown of the between measure of variation for the circle can now
be seen as
r p q
y y N j .] R2 r p y R? . . ' R2+ kr qy fRJ . l R2L Il i - i j - i Ni j . . N. . .
zLi - l Ni . . . N. . .
zLj-i NJ . . N. . .
(Between Factor 1 (Between F acto r 2 or Rows) or Columns)
+ k
r p q
I I fR?J.lp
- IR?
i .. q
- IR2 .
•JR2
+ ■ - - -La La
L i - l j - l [ " t J . JL
i - 1 [ N i . JL
J - l k j j N( 8 . 4 . 5 )
( In t e r a c t ion)
In the same manner as Section 7.4 the cross product terms may be found for the
two-way classification with interaction design. Following lengthy vector algebra we
may show that the cross product terms listed below all equal zero.
p q m
2k 1 1 1 " Z - - - ) ( I-J* " I . . . * “ 0i-1 j - l Z -l
p q m
2k I I I (I i . . " Z. . . } (Z i j . - Z i . . “ Z. j . + Z. . . > * 0i -1 j - l Z - l
p q m
2k I I I (Ii.. “ " l i j^ "°i - 1 j - l Z- l
p q m
2k 1 1 1 (^ -J - " Z . . . > < i i j . ’ I * - . " I . J . + " 0i -1 j - l Z- l
p q m
2k 1 1 2 (Z . j . - Z . . . ) (i . j z - Z i j . ) - oi - 1 j - l Z- l
p q m
2kI I I (*ij* - Zij.xZij. -Zi.. -Z.j. +Z...> - 0i - 1 j - l Z- l
- 150 -
Finally the two-way analysis of variance model with interaction may be given as
R2XT * * * — U
r p y R24- k
r qy fRJ . l R2
N. # # z. i -1 Ni . . N. . .
zL j - i N J . . N . . .
+ kr p q
I I Ri j -p
- 1
1 i
q
- 1R2 .
•J-R2
+Li Li
L i - l j - i LNi j J Lii - 1 [Ni . . J
UJ - l k jJ N . . .
p q R2 .+ k n ■ I 1 i j -
Lt uL i - i j - i Ni j * .
8.5 Other Design Models
8.5.1 The Randomised Complete Block Design
Section 6.3 discussed the requirement and structure of the randomised complete block
design where the blocks are formed so that each is as homogeneous as possible.
Within this design no interaction exists. The vector difference (0jj - 0 ) i.e. the
total measure of variation, may be expressed as the sum of three terms
<®ij “ " (Z i . " I . .> + (Z . j - + (i i j " Z i . ~ Z . j + Z. .>
( 8 . 5 . 1 )
Both the first two terms on the right hand side of (8.5.1) have been seen in Section
7.3.3 and give
P q
k l I <Iti - 1 j - l
p q
k l I < * .i - i j - i
r p R2 R2- £ _ ) 2 - k y l . - —l-l
Li. i -1 Ni . N . .
r qn R2 : R2 'j - Z . . ) 2 “ k I
• j
L j - i N - j . N . .
( 8 . 5 . 2 )
( 8 . 5 . 3 )
- 151 -
Note that (8.5.2) provides an estimate of the measure of variation between p
treatments and that (8.5.3) provides an estimate of the measure of variation between
q blocks. The third term on the right hand side of (8.5.1) is an estimate of the
residual measure within the design. Using the same procedure as Sections 8.2 and
8.4 it can be shown that
i -1 j - l
r pv qV R2+ 0 ) 2 - k N - ) - / + — —
Li -1 Ni .
L»
j - l N-j. N . .
( 8 . 5 . 4 )
In the same manner as for the two way classification with interaction the cross
product terms all equal zero. Hence the randomised complete block design may be
given as
R2 r pV’ Ri . ‘ R2 r[R’ j]
R2k N -----— - k y - --— + k y - ——
N LLi-i N‘ -. N. .
LLj-i ."•J. N. .
PN - I
P2Ki.
q- I
R2R-j
R2
Li-1 Ni*.
Lj - l N-j. N_ _
8.5.2
+ k
Latin Squares Design
( 8 . 5 . 5 )
In the randomised complete block design, the effect of a single factor was removed.
It is occasionally possible to eliminate two sources of non-homogeneity simultaneously
in the same experiment by using the Latin square design. Such designs were
originally applied in agricultural experimentation when the two directional sources of
non-homogeneity were simply the two directions on the field, and the "square" was
literally a square plot of land. Its usage has been extended to many other
applications where there are two sources of non-homogeneity that may affect
experimental results, for example, machines, positions, operators, runs, days. A third
variable, the experimental treatment, is then associated with the two source variables
- 152 -
in a prescribed fashion. The use of Latin squares is restricted by two conditions:
(i) the number of rows, columns and treatments must all be the same
(ii) there must be no interaction between row and column factors
The analysis of Latin squares is based on essentially the same assumptions as the
analysis of randomised blocks. The essential difference is that in the case of
randomised blocks we allow for one source of non-homogeneity (represented by
blocks) while in the case of Latin squares we are simultaneously allowing for two
kinds of non-homogeneity (represented by rows and columns). As with the
randomised complete block design the relatively simple but lengthy proof of
construction via vector algebra has not been reiterated, however, all cross product
terms can be shown to equal zero, producing the model expression:-
R2N -
N
if
Ii -1
Ri . .
Nl . .
R2
N+ k i
■J-l
R2 . •J
N J-J
R2
N
+ /cr p
yR 2R. . z R2
zlZ-1 N. . .
+ kP
N - Ji - 1
R? l ..Py R2 .
•J*Py r : . z' R2
+ ? * * *
Ni . . .Z
j- i NJ . .z
Z- l n . . * . N. #( 8 . 5 . 6 )
- 153 -
8.5.3 Nested or Hierarchical Design
Section 6.2 discussed the structure of the nested design; as with the randomised
complete block and Latin square designs the nested design may also be constructed in
the same manner to produce the model expression;
8.5.4 Three-Way Classification
Clearly larger and larger designs can be constructed in the above manner, here a
three-way classification design has been built to illustrate the generalised nature of
the approach;
R2N -
N
+ k
r p y M*
K) R2Ll i - i Ni . . N . . .
' my R! . z R2
zlz - 1 N. . .
+ k iLj-1
R2 . •J
N
R2
N
+ kr p q
I I
R? . i j -
P
- 1 R?l ..q
- 1 R2 .• J.
R2 + — ■La La
Li-l J- i iNijJ Li -1 [Ni . . j
LJ- l k j . J N. . .
+ k
p m
i i ^ - i=1 (_1 i . I
+ k
+ k
i - 1 Z- l
q m
i i .
P m q
i i ii - 1 Z- l j - l
i - 1
q
ij - i
R?1
Nj
R2
N
m
- I
J-J
Z- l
m
R? . i
N . . Z
R2
- 1Z- l
r : . z
N . . Z
N
R2
N
R? . . i j *
p q
- y yR? .
i j -q m
- 1 1.Ni j I.
La La
i - 1 j - l k j . JL L
j - l Z - l k j / J
p m
I Ii - 1 Z- l Ni A i - 1
R?l * IJ - l
R2 . •J
N ■J.J
m
iZ -l
R?.z' R2
N. . * . N. . .
+ k
p q. m
- i l li —1 j —1 Z—1
R2 . .i j *
lNi j / J
- 155 -
( 8 . 5 . 8 )
Chapter 11 shows further use of this generalised method with the implimentation of a
split plot design to real data arising in an angular form.
8.6 Summary
It was seen in Chapter 7 how the combining of angular mean directions may give a
false overall mean direction. Taking account of the mean resultant lengths together
with their corresponding mean directions has been shown to eliminate this problem
and has helped to indicate a new approach to the circular analysis of variance. It
has been emphasised that the approach is, in the first instance, an analysis of
differing populations, via the maximum likelihood ratio test, rather than differing
mean directions. The requirement for the testing of the equality of concentration
parameters is essential in order that a true test of mean directions may be
undertaken.
Using the knowledge of angular mean combinations, the use of vector algebra and
directional data properties, new components for the total, residual and between
measures of variation have been built. The significance of this method has shown
how the cross product terms, found to be non-zero for the original approach in
Chapter 7 and requiring a correction factor, are now zero.
This method has then been extended to discuss, explain and illustrate the construction
of interaction on the circle and hence build the new procedure for the two-way
classification design, showing zero cross product terms.
Finally this generalised method has been further extended to construct other larger
designs such as the Latin square and three-way classification design.
- 156 -
CHAPTER 9
TH E DISTRIBUTION OF TH E NEW COMPONENTS AND TEST STATISTICS
Having developed the generalised procedures in Chapter 8 it is necessary to examine
the theory of the associated statistical tests for all experimental situations which may
occur. Some attempt was made to evaluate theoretically the exact distributions of
the test statistics. However, as was found by Upton (1972) and Stephens (1969, 72),
even for simple single sample tests the numerical integration involved was extremely
tedious. R does not have a simple density function and a direct evaluation of the
significance points is not straightforward. Stephens (1969) gave the upper and lower
1% and 5% points for several values of k and N for R/N and X/N. Stephens
(1972) has also evaluated the exact two-sample test given by equation (4.2.2) for
differing values of k and N. In 1969 Stephens discussed the problem of obtaining
the exact theoretical distribution of
for the sphere and circle developed by Watson (1956) and Watson and Williams
(1956). Stephens also stated that the analysis is not so straightforward as for the
Normal distribution; with the distribution of the test statistic being intractable. This,
unfortunately, is also true for the distribution of the test statistic
from (8.3.3). However, the asymptotic results, as for the test statistic by Watson
and Williams, may be investigated.
9.1 Introduction
9
u
- 157 -
Sections 9.2 and 9.3 will examine the component statistics for small and large k
respectively. In addition the first two moments for each of the components are
found and used to improve the associated approximation. The accuracy of the
chi-squared distributions and their corresponding F statistics are examined via the
same simulation techniques used in Chapter 6.
As with standard 'linear' analysis of variance, it is important to reiterate that because
of the way in which the new procedures are constructed, the equality of the
concentration parameters are examined prior to any analysis of variance.
9.2 Small Concentration Parameter, k
As given in Chapter 8 the new procedure is based on the likelihood ratio test for
the null hypothesis /31 = ...... = /3q, k 1 = .... = kq against its general alternative
hypothesis. Let X be the likelihood ratio criterion for this problem, as in (3.1.5),
giving the test criterion
-2 logX = 2 J f c j R . j - i j l + l N . j l o g
j - l J- l
10( ^ 0)
M * J ) .
A A
where k 0 and kj are given by
R RJA (k 0) = - A ( k j ) =
• J
NN-j.
j = 1 , 2 q
( 9 . 2 . 1 )
(9 .2 .2 )
The power series for the ratio of Bessel functions I Q(k) and I,(A:) for small k are
given in (3.2.8). The first term approximation, A (k) = kl2, has been shown to be
tolerable for k z 1. Using this in (9.2.2) gives
^0
2R
N
2R
N Jj
j = 1 ,2 , . . . , q ( 9 . 2 . 3 )
- 158 -
also given in (3.2.11) as approximation k 7. Hence for small k (9.2.1) reduces to
( 9 . 2 . 4 )
This same procedure is given by Mardia (1972) for his S statistic. Identity (9.2.4) is
seen as the between measure of variation (8.2.8), for small k.
From Rayleigh (1919) and Lord (1954) it has been shown that 2R 2/N has a
chi-squared distribution with 2 degrees of freedom. This result was given by
Rayleigh after finding the form of the p.d.f. of R, equation (2.3.14), for large N.
This result has been used to test whether the population from which a sample is
drawn differs significantly from uniformity. We cannot, however, show 2 (N -R 2/N ) to
be chi-squared in a similar manner. Following preliminary investigations and
simulation routines both 2 (N -R 2/N) and
are found to be what may be termed as negative or reflected chi-squared
distributions (or random variables), negatively rather than positively skewed. With
the inclusion of N the chi-squared is effectively transformed to a reflected
chi-squared with its variance decreasing as k increases.
For small k we may only investigate the between measure as produced from
maximum likelihood. This chi-squared may be improved following the equating of
expectations, using
E(R2j ) - N . j + N j ( N j - l ) p 2 from ( 2 . 5 . 2 )
M*>where p = A(k) = and
I 0W
E< * I ( q - l ) > "- 159 -
Producing the improvement factor
1( 9 . 2 . 5 )
1 - p 2
Therefore we may approximate the distribution
2 r 9 y [R‘jl R2
1 - p2 zLj-i N
( 9 . 2 . 6 )
by a x 2 variable with 2 (q -l) degrees of freedom. When k is unknown, the
maximum likelihood estimator k , given by (3.2.2), will be used.
9.3 Large Concentration Parameter, k
Watson and Williams showed that 2k(N-R) is approximately distributed as a
chi-squared with (N - l) degrees of freedom. For the procedure discussed in Chapter
8 we are required to show that fc(N-R2/N) is distributed as chi-squared with (N - l)
degrees of freedom, for large k. As a simple initial proof we may show that
n - N k(N - R)
For large k, R/N—>1, therefore
2k(N - R)
1 + I
R2N - — N
Alternatively we may adapt the approach of Mardia (1972, p 114) where the
distributions of 2fc(N-C), 2fc(R-C) and 2fc(N-R) are found. Here the distributions of
& (N -C 2/N )), fc((R2/N )- (C 2/N )) and fc(N-(R2/N )) are required.
- 160 -
9.3.1 Distribution of k {N - C 2/N)
Let 0 be distributed as M(0,fc), then for large k we wish to show that
~(N - C)k(N + C) - * 2 ( 9 . 3 . 1 )
Let
N
€ — J /c(l - COS 0 j ) ( l + COS 0 . j )
j - l
N
i -j=l
N
i -j “ l
1 -
e 2 . e A .i +
2! 4!1 +
®?j 6U 1 1 + —
2! 4!
2 e A . e A. 0 G.2 l i _ • J _ -j• j 4! 212! 4!2!
For large k, 0 j is small
N
e * J
j“ l
From (2.2.2) 0 j may be approximated by N(0 ,k~%) and therefore kd j 2 will be
distributed as the square of the standard normal variate which is approximately a
chi-squared distribution with 1 degree of freedom. Hence, by the additive property
of chi-squared
N
ij “ l
kd2 . « k • J
C2N " N~ * *N ( 9 . 3 . 2 )
- 161 -
9.3.2 Distribution of fc((R2/N) - (C2/N))
It can be seen that
£ (R + c )(R - C) - k (1 - cos0) (1 + cos0) ( 9 .3 .3 )
It has been shown by Mardia (1972, p 98) that the conditional distribution of 0
(sample mean direction) given R is M (0 ,kR). On using (9.3.1) the conditional
distribution of (9.3.3) for given R is x? which does not depend on R.
9.3.3 Distribution of k{N - (R2/N ))
Following the identity
c 2 R2 C2 R2N =■ k N " N + k N - — N
and using (9.3.2) and (9.3.3), by the additive property of chi-squared it can be
shown that
where k ((R 2/N )- (C 2/N )) and k (N -(R 2/N )) are independently distributed.
In the same manner as Watson and Williams original expression, (9.3.4) behaves like
the similar form found in 'linear' analysis of variance.
Using a similar approach, it follows that for large k
( 9 . 3 . 5 )
k N ( 9 . 3 . 6 )
- 162 -
Therefore
H - I
j “ l
R2.J
N • j jXN-q k
r v R?j' R2) - — fSL
U-i N.j. N» V 2
Lq - i( 9 .3 .7 )
As for small k these approximations may be improved following the use of
expectation (2.5.2). For large k , equating expectations gives an improvement factor
1( 9 . 3 . 8 )
k ( l - P2)
Therefore an improvement, when k is unknown, is made by replacing k by (9.2.5)
1N -
R2
N N -l ( 9 . 3 . 9 )1 - p 2
The remaining components also require the same improvement factor, giving;
1 R2N -------
N
1 r *y R?j' R2
- P2 1 - p 2 LL j- i N . j . N
1
1 - p » - 1j -1
R2 . •J
N.JJ
( 9 . 3 . 1 0 )
with associated chi-squared distributions
*N -1 " Xq-1 + XN-q
9.4 The Variance of the Component Chi-Squared Approximations
9 . 4 . 1 Variance o f
Var R2 R22
R2N - —— N - E N - ——
N E N - — N
= E(N2) - 2E(R2) + ER4
N2- [ ( N - l ) 2 - 2(N - l ) 2p 2 + (N - l ) 2p4 ]
- 163 -
Using expectations E (R 2) and E (R 4) given by (2.5.2) and (2.5.3)
- 2(1 - N ) p ’ - 1 - (N - 1 ) V + 575 ■(N -" 4 ) |P/I
+ (T ^ 3 ) T p2(2 + + ( N ^ 2 ) I (2 + 4p2 + p V + N ( 9 .4 .1 )
Let (9.4.1) be represented by T . For large k, equating T to the variance of its
associated chi-squared
k 2T - 2(N - 1)
Let S be the improvement for variance, therefore
S2 2(N - 1) k 2T
Re-equating expectations gives
( 9 . 4 . 2 )
( 9 . 4 . 3 )
N - N + (N - 1 ) [1 - Sk( l - p 2) ] « ( 9 . 4 . 4 )
This lengthy expression has been tested on simulations for large k (> 2) and
excellent chi-squared approximations produced. However, with the adjustment for the
expectation given in (9.4.4) it is possible for negative values to be obtained.
Similarly, as part of an analysis of variance procedure (9.4.4) would be impracticable.
- 164 -
In order to study the variance (9.4.1) further, the asymptotic expansion of p has
been used. For reasonably accurate approximations the first four terms of the series
A(k) are required; for large k
1 -
1 1 1
2k 8k 2 8k 3
1
from ( 3 . 3 . 2 )
1 9 1 11 + + + -------
k 8k 3 64k4 32k s 64k6
2 1 1 1
k k 2 4 k 3 4k4
( 9 . 4 . 5 )
When these equations and those of p i and p 4 are substituted into (9.4.1), the
variance of the chi-squared approximation,
Var R2 1 1 -53 89 5NM - __N ~ *£ t(2N 2 ) + N " N + k 4 32N + 32 “ 32
Therefore
1 -9 ^ 7 5N 1 -1 7 9N 1k 5 m + 8 ~ T T + k 6 T3n " 32 32 + 0 k 7
( 9 . 4 . 6 )
VarD 2
N - — N 2(N - 1) + £ ( 9 . 4 . 7 )
This interesting result shows that the variance of the component is equal to the
variance of the chi-squared approximation plus further smaller terms. These terms
are found to be negative and heavily dependent on the size of k and N. Higher
terms may only be neglected when k and N are relatively large. Further proof of
this is discussed in Section 9.5.
- 165 -
r q[R f j l9 . 4 . 2 V ar iance o f n - yLa
j - 1 N -j .
Following the same procedure as Section 9.4.1, let q be the number of samples and
c the number of observations within each sample and of equal size
Var ■ - 1 j - i
R2
lN-jJ-q - p 2[ 2 (c - l ) q ] - p4[ q ( c - l ) 2 ]
c! 2c!P4 + --------------- P2( 2 + p 2)
(c - 4 ) ! (c - 3 ) !
c!
(c - 2 ) !
Using the series equations of (9.4.5), for large k
(2 + 4p2 + p 2) + c
Varr qN - I [Rj ]
1------- ( 2qc
1- 2q) + —
q-qc + —
Uj -1 N -j . k 2 /c3 c
( 9 . 4 . 8 )
1 -5qc 89q 53q 1 -5qc 7q 9q '+ — ------------------ + -------------- - -------------- + — ------- + — - -----
32 32 32c /c5 16 8 16c
1 9qc 7q lq 1+ — ----- + — _ ----- + 0
k 6 32 32 16c k 7( 9 . 4 . 9 )
Therefore
r q v 1 q
Var< k N - ) - 2q(c - 1 ) + - -qc + —La
J-1 N . j . k c- - ■
( 9 . 4 . 1 0 )
As with the total component of variation the variance of the residual component is
equal to the variance of the chi-squared approximation plus smaller negative terms
dependent on the size of k and N.
- 166 -
r q R2 . • J
R29 . 4 . 3 V ariance o f ) - —
LiL j- i N J . N
Let q be the number of samples and c be the number of observations within each
sample of equal size
Varf q
y R?j' R2 1 (qc) !
zU - i N-j. N (qc) ( (cq) - 4 ) !
2 ( q c ) ! (qc) !+ --------------------- p 2( 2 + p 2) H--------------------------(2 + 4p2 + p 2) + c
( ( qc) - 3 ) ! ( (qc) - 2 ) !
c ! 2c!P4 + --------------- p 2( 2 + p 2) +
(c - 4 ) ! (c - 3 ) !
c !
(c - 2 ) !(2 + 4p2 + p 2) + c
+ (q - 1) - 2(q - l ) p 2 + p4 (qc2 - (qc ) 2 + q - 1) ( 9 . 4 . 1 1 )
Using the series equations of (9.4.5), for large k
r qv *■ } R2 1 1 -1 1Var y ------- ------- ( 2 (q - D ) + — q - -L>
L j- i N -J. N k 2 R3 c q
k A
-53 53q 89q 89
32qc 32c 32 32
1+ ---
/C5
-9 9q 7q 7
16qc 16c 8 8
+ 0k G
( 9 .4 .1 2 )
- 167 -
Therefore
r * R2 1 -1 l 'k y ------- . - 2 (q - 1) + - — q - - +
L ,
L j- i N - j . N k c q -
Again it is important to note that the variance of the between component is equal to
the variance of the chi-squared approximation plus smaller further terms. However,
Section 9.5 shows how the size of k and N within the components have substantial
effect on the size and accuracy of the variance, particularly so on the between
measure of variation.
9.5 The Adequacy of the New Procedure, for Large k
As we have discussed in Chapter 4 and showed in Chapter 6 , the approach by
Watson and Williams, and adapted by Stephens, is the principal approach for testing
differences between mean directions of different samples with assumed equal
concentration parameter. The procedure developed in Chapter 8 is ultimately
designed to analyse larger experimental situations where extensions to Watson and
Williams have been shown to breakdown. However, it is necessary to compare the
accuracy and power of the new test statistics with the alternative tests for the
one-way classification before investigating its suitability for larger designs.
As in Chapter 5, for testing the adequacy of the extended techniques from Watson
and Williams, simulation techniques have been used to examine the accuracy of the
approximations to the distributions of the components of (8.3.3). The observations
from the von Mises distribution specified by the null hypothesis were generated by
the computer method outlined in Appendix B. 10,000 sets of samples of various size
were drawn from the von Mises distribution with k = 2, 3, 4, 5 and 10. For the
sake of conformity and in order to further check that the simulation techniques
- 168 -
being used were satisfactory, the same sample sizes are analysed as in Upton (1974).
Five multi-sample experimental designs are examined varying in size, with N ranging
from 10 to 60.
To verify the findings of Section 9.4, the first two moments of the components for
total and residual measures of variation are given in Table 9.5.1. The mean of the
chi-squared approximation value for each component is seen to be an excellent fit
increasing in accuracy as k increases. If k is unknown and the improvement factor
(9 .2 .5) is used the accuracy of the chi-squared approximation mean value is
maintained. As shown in Section 9.4 the variance of each component is seen to be
below its chi-squared approximation value and dependent on the size of N and k.
As N and predominantly k increase, the accuracy of the variance increases.
This deficiency within both components is reflected in Table 9.5.2 and 9.5.3, showing
the accuracy of the upper percentage points to the x 2 distribution with associated
degrees of freedom. It is clearly seen that cq, the simulated proportion of the
components, approaches a, the x 2 theoretical or significance level, as N and k
increase. Similarly, due to the size of the chi-squared approximation variance values
the probability of accepting the null hypothesis when in fact it is false, a type II
error, is increased; and unacceptedly so for k=2 and small N. Tables 9.5.2 and
9.5.3 also give the comparable accuracy of the Watson and Williams component
improved by Stephens. For increased understanding the same 10,000 sets of
observations for each set of samples were used. It is seen for both components that
while the new procedure components overestimate the theoretical proportion, following
Stephens improvement Watson and Williams components slightly underestimate the
theoretical proportion or significance level of the associated x 2-
- 169 -
TABLE
9.5.1
Comparing
the mean and
variance
of th
e^2 approximations
for
the total
and residual measures of
variation
against
their
expected values
0o t--V /—\ t—\ *—\ /—v t—» r i /--N t—\ /—* /—\ ,—* ,—% r-vc CO V CD V CM CM CO CD r - CO CD CO <3- CO CM
«—> (D CO CO O o V O CO CO IX) CM CO CO IX) CM CM•H CO *d- IV V CO CD IX) <3* CO CD IX) *3"
• P . • . • . • * . . . . . . . .s (0 o CO CO cn CM v~ CM CO ^d- -cf rv r — CO
*> ; > V V- CM CO CO CO CO CO CO CO CD CD*_/ '—' >—> V_! >—> >—* V---! <—> >—> <—i '—> v_/ w \---! A_>
CM •OHV_/w1 CM CO CO CO CO *d- IX) CO CO CO CM CO
z c CO ID r - »r~ CO CO IX) CO rv V CD <3- rv CO<—1 (D V V- CD a <3- CM o CD CO IV abz CD . . • . . • • . . . . . . . . -
e : CO CO CO V CO CO CO CO CO CO CD CO CO CO COv~ *” r_ 'T *d* «3- <3* 'd '
CO CO COV* CO CD
11 II IICO CO
CO 0 r - 0 *3" 0G G G
TD II c TD II c TD II CCD 0 0 0 0 0
LU -P c •H -P c •H -P C •H_i cn O 0 P o G 0 P IX) G 0 PQ_ UJ 0 0 0 0 0 0 CO 0 TB 0e : m IX) CL e : > * CL e ; > % CL e : >< H * X CD X IX) Xtn cn IX) LU M“ LU V IU
0G /—\ t—\ /—\ e—\ r - t /—> /--N ,—« t—\ r - \ t—^C CD r~\ CM CO CD CM V- CO CD IV CM CM CM IV ,—,0 IX) CO IX) CO V CO «3- CM IX) CO CO «3* CM
•H rv <3- IV CO CD CM 1X1 IX) V CD CO CD CO IX) IVP . . . . . • '• . . • . . . ■ .(D IX) tn IX) V d* CO *3" IX) CO CO CO CD CO IX)
<—> > r - V- V r - CM CO co CO CO CO CO cO CD CDz: V-- ! v---! w v—/ V_/ l-- ! >—/ w w V_/ w \_1 * ' \_j V_> V_>\CM •OHI CO CO CO CO CM «3* CO rv 'd* CD CO
Z c CD CO cn V- CO V CO r - V IX) V— CO CM d-V_> CD 'T~ <r* CD a 'd ' <3* CM CM a CM r - CO a
CD • • • • • • . • . • . . . .E : CD CD CD CO CD CD CD CD CD CD CD O CD CD CD
r- v - t - IX) IX) '3* 'd* <3*
CM CO «3" IX) O CM co «3- IX) CD CM CO *3* IX) Ov
CO CO 00V - CO CD
II II IICD CD
CD 0 v~ 0 <3" 0G G G
"O II C TD II c TD II C0 0 0 0 0 0to -P c •H •P C •H -P C •H
_ j cn G 0 p O G 0 P IX) G 0 Pc l to 0 0 0 0 0 0 CO 0 0 0e: m IX) Q. s : > % CL e : > * D . e : >< M * X CD X cn Xcn ai IX) to V“ LU V IU
- 170 -
TABLE
9.5.1
continued:
Q1O /—i <—> <—i 4—> 4—> <—. i—> CD CO <—, ,—> 4--> CD r *C CD in CN <d- V rx CD CO <3* CD CO CD CD CNto CD CO in co co in CD CD IX CD CN X CO CN CO4—> •H CD CD CO CD x CD CN . • . CD . . .
•*“ 1 Ch • • • • . • • CO CD a • . a CO a• to CO in CD o V- r x a a in in o CD r -
s > 'd* co CD x x rx CD r— v <r“ IX CD r — v\ \_4 '_t *—> V_/ »—/ >_i >__i >_* <_< <_i >_< <__ i__ V_4 i_i
•r~in •Od<_> v CO V CN in rx CO CD X X COW LD CO CO co CO V- CN CO 'Cf CD X CO CO in1 c O o in CN a V- CN CN CD V
2 CO . . . . ■ . . . . . . . . . .<— > 0 CD CD CO co CO CO CO CO rx rx X X CD CD CObZ E : CO CO CO CO CO in in in in in in in in in in
CNCD r-x r - V
n ii IIco IX in CDCO 0 in 0 r— in 0
o o •> 0TD 11 c CO TO n c in n ii C0 0 CO 0 0 r- 0 0
LU •p c •H •> -p c •H •> -p c •H_I C/1 O o 0 U a o 0 U in 0 0 UCL LU C\J 0 0 0 CN 0 0 0 V“ 0 0 0e : m * CL s= > * D . e : > * CL e : >< M o X a X in XC/1 C/1 CM UJ t - UJ UJ
^0o <--V <—> 4—. <—> i—> CO CD CO CO t—> CO COc x in X CO CO 'd- CO 'd ' CD CN CO CN CN0 CN co in CD CN CN CN CD CO CO CD CD CD in
•H in CD CO r - . . . . *d" . . . .Ch . . . • . • CO ^d" in • V- CO CO *d*0 o X CN CO CO in CD CD V t— CD a a
<—- > in CO X X IX rx V- V r - r — X v V“ ■>r-~ZL V__/ V__4 \__* V__/ w \__* \—> V__4 \__> V----4 V__4 V__t V__4 *__4 V__4X
(M*QC co CN in in CD IX CD T- CO <r CO *$■ CO COI CD CO o CO CO CD T“ ^ r CN <r <sr CO CN in
2 c O CD in CO O CO o CD r— in CD CD r—v—t 0 . • . . . . . . . . . . . . .
bE 0 o CD CD CD CD a o CD CD CD CD CD a CD CDz : *d* CO CO CO CO CD CD CD in in CD CD CD in in
JZ CN CO in □ CN CO in a CN CO *3* in o
CO COCO V-IX <r“ T-
II II IICD CD in CDCO 0 in 0 in 0
a a * 0TO II c a TD it c in n ii c0 0 CO 0 0 0 0
LU •P c •H % •P c •H * -p c •H__1 C/1 O O 0 U a a 0 U in a 0 uCL LU CN 0 0 0 CN 0 0 0 V- 0 0 0e : m * CL e : > % a e : > % CL e : >< M □ X CD X in XC/1 C/l CN UJ UJ V UJ
_ 171 _
TABLE
9.5.2
Comparing
the accuracy
of the
two
X2 approximations with
N-1 degrees
of freedom
representing the
total
measure
of
co•H-P<U’Hto>
rv <3- cn 3* CN V CO CO CD CD UOCO rv V 0 rv CO CD co CO CO rv cn
CO cn CO cn cn CO cn CO CO CO cn cn CO cn cn COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . • . • . . . . . . . ■ • . ■o O 0 0
CN CD UO CO rv CD CO UO CN CDCD '3' cn CN cn CO CO CO UO CO <3_ CO T- UO
CD 3" 'T ■cf UO <3* '3' CO <3" «3* UO *3" 3* uo UOI i cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
■ . ■ . . • • • . . . . ■ ■ ■ .cn o o 0 0
'— ’ ^r cn CO CD v CD IV CO cn CN IV T- CN CO> - IV CD rv «3* CO CO CO <3* cn rv co «3_ CDCN o CO cn cn a cn cn CO 0 cn 0 rv CO cn O O
CO CO CO CO cn CO CO CO cn CO cn CO CO CO cn cn. • • ■ ■ . . . . . . . . . . .
o o 0 0
<3" cn CO CO CO CO CO UO cn cn T- CO COcn rv *3" <3" cn cn 3* CN CO CO CN CO CN 0
cn cn cn cn cn co cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. • ■ . ■ . . . . . . . . . . ■o o 0 0
CN CD rv UO cn IV UO cn rv UO <3" CO<—> UD CO CO r - <3* <3" rv <r- 0 CO 3* CN <3-un CO CD UO UO UO CD UO UO UO UO IV UO UO UO uocn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• ■ . . • ■ . • . . . . . . ■ . .M* o o 0 0cn
CO <3- V UO r - V cn UO cn CN UO UO IV-—i o rv *3" UO 0 cn rv CD CO CO CD CN CN
o CO a CD 0 0 CN cn a cn 0 CN cn cn CD acn cn cn cn cn cn cn CO cn CO cn cn CO CO cn cn
. . • . . . . . . . . ■ • ■ . ■o a a 0
CN CO UO 0 CN CO '3" UO 0 CN CO •3" UO 0'T- T~
LU_ i cn O uoCL LU COz : m UO *< M ► O UOcn cn UO t -
- 172 -
The
table
gives
the proportion
of 10,000 Monte
Carlo
samples
for which
the two
approximations fell below
the/*
value
fpr which
the theoretical
proportion
should be
TABLE
9.5.2
continued:
x lo in CN X CO in in cn cn in CDo CN cn □ CN ^— X a CO o X X CO cn
cn cn cn CO cn cn cn CO cn CO cn CO CO cn CO COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . . .o o a o
X cn X cn CD <3- CN «3- CO CNy a X CO X cn X cn X V cn cn cn <3"
‘ ‘ )f in *3" *3* in in «3" cn <3- «3" in *3* cn *3" m. c cn cn cn cn cn cn cn cn OD cn cn cn cn cn cn cn. . . . . . . . . . . . . . . . .c n o a a □2v—/ IX CO CO X CD CD X in CD X *3" in CO>- c n in CD CN CO in o X <3* in cn a CNCM CD cn CO cn □ a cn CO cn a a CO CO □ a r -
cn CO CO CO cn cn CO CO CO cn cn CO CO cn cn cn• . . . . . . . . . . . . . . .
a o CD CD
cn CO cn cn cn cn cn CN X COCO CN v~ CN CN X v cn CD 'T- CO CN V cn
cn c n cn cn cn cn cn cn cn cn OD cn cn cn cn COcn c n cn cn cn cn cn cn cn cn cn cn cn on cn cn
. . . . . . . . . . . . . . . .□ a □ a
CO cn X in cn CO r— r - cn CD cn in in X<—> <3- cn CN CD cn CO CD cn X CN CO r - cn CNS V/ in X in in in in X n - in *3" in X *3" in *3" in\ < cn cn cn cn cn cn cn cn on cn cn cn cn cn cn cn
• . . . . . . . . . . . . . . . .<N ■ □ a CD CDc n
~ZL x • in V“ <3* cn c n cn in CN in>—> CO n - CO cn <3* * — cn cn CD CN CN OD CD in
o CN cn cn cn o CN CO CO cn a CN CO CD cn CDcn cn CO CO CO cn cn CO CO CO cn cn CO cn CO cn
. . . . . . . • . . . . . . . .a o o o
CN cn in □ CN cn <3" in a CN cn <3" in ar— V "
O incn
QJ •i %_ i cn □ □ in□_ UJ CN CN <r-e : m * % *< M a O incn cn CN V“ 'T-
in
- 173 -
a300PbO0TJCT
cSO o-P •H•H2 0
■Hcn pc 0o >•H-P q-0 oE•H 0X Po CJp 0CL 0CL 00 E
•T* - 1—1X 0ZDQ TJ2 •H+-> 0
00 PSO-p 0JCq- 4->o bfl> Co •H0 4->p OZD 0o 0G 00 P
CL0 0SO P4->
Ebfl □C TJ•H 0P 00 PCL q-EO q-CJ o
CO
CD
cnLU_JCD<I—
X CD CO co in CD CD CO CD co CD X COCN «3" CD CO CN CT CO CO a X cn a CD
cn cn CO cn cn CO cn CO cn CO cn cn CO CO cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• • • ■ ■ • • ■ • • • m • • ■ •
o o o a
*3" CO CO CT in CN in t - CO X ct CO CO COin CN CT V X CO CD X CD cn CO X X cn CD
^ \ / in «3" CT in *3* CT CO CT <3* CT CT CD CT CT in•>—) cJen cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• • ■ ■ *s" • • ■ • ■ • ■ ■ • mcr ’ o □ o aw
cn CO X X CO CD CD CT CD <3* CN X cn cn CO<— < <3* CT CO CD co in in CO CN X X C T CN o CD>- o CO CO cn O cn co X cn cn cn X CO cn □ OCN cn co CO CO cn CO co CO CO CO CO CO CO CO cn cn
• • ■ • • ■ ■ • • ■ • ■ ■ • • •o □ a o
CO CN X CO cn CN CT CD cn C T COCO X CD CT cn CO in co CO X CN CO CN
cn cn cn cn cn CO cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• ■ • • • • • • ■ ■ • • • • • •y - \ o o a a
X in CN CO CN X CO in CN in CT CT COCN o O o CO CD CN m >3* <3* v - T;— in
• i—>\ cn CO CD CD CD in CO in in in in X in in in in(N • I cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn • • ■ ■ • • ■ ■ • ■ • ■ ■ ■ ■<—< o o a aw
in <r- CN CO CO CN CD CO CO CD CN CT CD CO>—< CN r— CD *3" a CT X X CO CN CO in CT cn CN□ CO r - O V- o CO cn O cn a CN cn cn cn acn cn cn cn cn cn cn CO cn CO cn cn CO CO CO cn■ • • • • • • • ■ • m ■ • • ■ •o o a a
jS CN CO "3* in o CN CO CT in a CN co CT in CDv- >r-
UJ_ i cn □ inCL LU COe : m in * %< M * a incn cn in V
O2 C-P o•H0 4->SO P4-J OCLSO Ou P•H CLJZ2 i—1
0P OO•Hq- -P00 P0 OrH 0Q. SOE 4-100 0SOo -PrHP SO0 oCJ •HSO0 24->C PQ oe: q-CD 0a ZJCDi—i
0a >—
CTq- 1o nZc Xso 0•H JZ4-1 -pPO 2CLoO 1—1P 0□. S20 1—1JZ i—!4-1 0
q-00 0> c•H o obfl -P0 0 0i—1 E JZJZ •H0 X TJ4-1 O i—I
P ZJ0 D. OJZ CL SO(— 0 0
- 174 -
TABLE
9.5.3
continued:
cn X x in <3* CO CN CD CO V“ CO CO□ in x CN o X a cn CO CO cn co cn
cn c n CO CO cn cn cn CO cn co cn CO CO CO CO COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn• * . . • • ■ . * . . . • . »□ o o a
CO cn in x CO CO CO X CN COun x CO T- x CD CO in CO CO X X □ cn inCO in in *3* CO <3" in CO CO in ■sf in
J ° j cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn• » • . . • . . . . . ■ . .t r ' □ o a ow
2 CN V CN x CO cn *3" CN cn COs--> CO in cn o cn r - cn cn *3* cn CN CO X X> - o CO CO cn a o CO rx cn cn a X X cn cn oCN cn CO CO CO cn cn CO CO CO CO cn co CO CO CO cn• ■ • . ■ * • . . . » . . . . .
o a □ □
N <3* CO r- cn CO in CO in "vfCO CN CO CO CO CO CN CN T- CO CO CN
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn• • . . • • . . . . . . . . . .□ □ a □
t—\ V" CO cn in CO CO CO CN CO CO!--V x CO CN in CO CO o cn CO V in <3"• 1—V in x in in in in x in in in X in in in in
• o cn CD cn cn cn cn cn cn cn cn cn cn cn cn cn cn53 * • • • • • • • . . . • • • ■ .X a a a a
•r-)(M •c n x X X <3- CO CN V” CO CO CN COw CO in CN t- CO □ CO V CO O CO in X in CO>■1 • a CO cn o a O CO CO cn cn O CN CO cn cn □1 CD cn CO cn cn cn cn CO CO CO cn cn CO CO CO cn• • . . • . . . • . . . ■ . . .'—' □ a □ ov :
CM CO <=r in a CN CO <3* in □ CN co <3* in CDr— V
in
%□ inCO V
UJ %_1 C/3 a o inCL UJ CN CN21 M % * *< M a o incn cn CN T- v—
- 175 -
Table 9.5.4 gives the first two moments for the between measure of variation
component. Once again the mean value of the component chi-squared approximation
is a very good fit increasing in accuracy as k increases. The deficiency in the
accuracy of the chi-squared approximation variance is again shown. This is slightly
worse than for the total and residual components.
Following examination of the fitted components percentiles to the upper 50% of the
theoretical x 2 distribution, several improvement factors were tested to increase the
accuracy of the chi-squared approximation variance without greatly impairing the
accuracy of the mean. The improvement factor 0, (9.5.1), was chosen as a balance
between the two moments, acceptable for all large k
1 1 1“ " 1 - — ------ ~ ( 9 . 5 . 1 )(3 5k 10k2
Table 9.5.4 gives the comparable moments for each set of sample sizes multiplying k
by (3 and shows how the mean of the chi-squared approximation has increased
slightly from its desired value whilst the associated variance has improved appreciably.
Table 9.5.5 gives the accuracy of the improved between measure of variation to the
upper percentage points of the associated x 2 distribution with comparable statistics for
Stephens measure. By introducing (3 the accuracy of cq the simulated proportion, is
greatly improved, increasing in accuracy as k increases.
It is important to note that to acquire this accuracy with the component distribution
variances being below the desired theoretical values implies that the accuracy of the
lower percentage points have deteriorated. However, the accuracy of the whole
component distribution is still very good, although Stephens measure has the superior
fit across all percentage points.
- 176 -
TABLE
9.5.4
Comparing
the mean and
variance
of th
e>f2
approximations
for the
between
measure
of variation
against
its expected
values
2 r—\V . 0
• oIN* c 4-- \ t—. 4---. 4-- \ 4-- \ 4-- \ 4---. 4-- \ 4-- \ 4---\ 4-- .c r 0 r \ CO in rv in cn CO in CN •3" CN CN cni •H CD o cn cn CN CO rv *r- cn cn a CD CN4--> U in rv. rv. rv cn CO CO CO IV un CO cn rv cn
(0 • • • • • • • • • • • • • . •• > t - ^— ^— ^— r - r - V“ 7 - V“ r— r—
&N
•I- )a •c r CO CN CN CN IV cn CO v - IV IV v ->—> c r \ CJ) cn CO ’T" cn CO a V o CO cn CNw 0 o a o r - r— o a a o cn O■—» 0 • • . • . • • • . . • . . . .CD e : v t— V t - v~ V V V o
r—\2 i—>\ 0
• o(N* c t—\ 4-- \ 4---. 4-- . 4-- \ 4-- . 4-- \ 4-- . 4-- . 4--\ 4-- \ 4-- .c r 0 CO CD V 4-- \ cn in CO v CO 4--> 4~ \ COi •H CN CO o CD a a rv cn IV cn cn CO CN rv
/—. P CO rv CO V CO CO a cn ^1- IV CO•f- ) 0 • • . • • . ■ • • • • • . • •
• > ^— V- ^— *— ^— V V" ^— T“ V ^— ^—
\•<-1
ct •cr r — CN in CN rv CD co cn I V V rv in>—< c cn cn cn cn CO O o IV V O CO cnw 0 r — □ cn cn o a a CD cn a v o cn cnw 0 . . . . ■ . . . . . . . . . .v : e : r - V" o a v V " r — V o <r- o CD
CN cn in a CN cn <3* m o CN cn in ar —
CN CN CN
II II II
0 0 0G a o
TD ii C TD ii c TD n c0 0 0 0 0 0
LU •P c •H -P c •H -P c •H_j cn o 0 P a o 0 p in a 0 uCL LU 0 0 0 r— 0 0 0 cn 0 0 0£= M un CL S I > % CL e : > % OL e : >< M * X a X in Xcn cn in UJ UJ UJ
- 177 -
TABLE
9.5.4
continued:
\ 03• o
CN • c /—> r—\ /-- \ t—\ /—\ /—\ /—s t—\ ,---V r-» ,—* ,—^cr (U O cn CO CO 03 IV CO V CO ,—, un CO CO COi •H CM CM o CO V r — 03 <3- CO i—> 03 CO CN un IV
«—v Ch in CO cn CO 03 CNI CO in CO CO o CN CO CO•<—> (0 ■ • • . • . . . . . • • • • •
• > r - v~ CD CO CO CO CO '3‘ un un un unw w \_/ %_/ »■-- ' V_> \_t \_f \_/ \_! i_> * . *_, * .
\
CN “ l—!cr • CO CO CM K~ CO K~ CO IV '► T“ CO <r~ IV un'—i c CO a IV a CO <c CO CN LD CO rv CO un «sfw 0 O o CNJ CN <c— V a CN r~ □<—> 03 • • ■ . . • . . . . • • • • •CD e : T— r— CNJ CN CN CN CN CO CO CO CO COx :
,—,0
• oCN • c r - \ t—\ i—i *—\ t—\ ,—\ i—\ t—\ r—\ r—\ (—1 r—.cr 0 CO CM ' <—* n 03 CN V ID 03 IV V“ COi •H V“ IX CO CN 03 rv CO V «cr *3" 03 un rv 03 CD
<—i L. r— CO rv CO CN CO 03 LD CO 'Cf 03 'tf ’ CN0 » • ■ • ■ • • • . . . . . . .
> V“ r - r — CNJ CN CN CO CO CO CO un unw
« *r_jCNcr CO *3- i \ CNJ CNJ rv 03 rv 03 CN r— 00 <3*'—> C O rx CM 03 a 03 CO o r — CO V 03 ^ r Vw (0 o O 03 CJ NT* CD 03 □ o CN CN a o aw 0 • • • • . • ■ . . • . . . . .v : E= o CNJ CN T- CN CN CO CO CO CO CO
CM CO <sj- un a CNJ CO LD □ CN CO un av
CNJ CO
II II IICO
T- 0 CN 0 CO 0a G * Gu II c O •a II C LO ■a ii C
CD 0 CO 0 0 0 0LU -P c •H * +j C. •H * -p c ■H_ i cn a o 0 u o G 0 Li un G 0 UCL UJ CM CD 0 0 CN 0 0 0 T~ 0 0 0E l N * a e : > % Q. e : > * CL e : >< w o X o X un Xcn cn CM LU UJ r - UJ
- 178 -
TABLE
9.5.5
Comparing
the accuracy
of the
two
*X2
approximations with q-1
degrees
of freedom
representing
the between
measure
of variation
CN CD 03 CO CO un rx un CO CD un CDCD rx CO r - 03 rx r \ CD o V CD CN CD CN CD
cn 03 CO CO CD CO CO CO CO CD CD CO CD CO CD COCD CD CD CD CD 03 CD CD CD CD CD CD CD CD 03 CD
• • • . • . • . . . . ■ ■ • ■ •o o CD a
N un CD CO I X m CD *3* CN T“ rx* cn CD CD CD •3- CO CO V- un CO 3 * 03 [ X• in *3- 3" '3" *3" 3" 'O’ un 3 - in *3* un un
01 )/CD CD CD 03 CD CD CD CD CD CD CD CD 03 03 CD CDt c • • • • • » • . . . . • ■ . . ■• I—1 □ O a a
CT *w V CO v CN CN *3* CN ^3" rx CO v— CO rx CO'—' CO CD CO CN 3* CO □ CD CO rx CO rx CO r— CD
□ CO CD CD a CD CD C3 CO CO CD CD CD CD CDCN CD CO CO CO 03 CO CO CD CO CO CO CO CO CO CD CO
• . • • . • • • . . . . . • • ■O o a a
CD CO in CD CN CD CN CO un rx 03 V rxCO CN CN CO v 'O’ un V- CN CN 3 " 3 - a CO
<—> CD CD CD CD CD 03 CD CD CD CD 03 CD CD CD CD CDCD CD CD CD CD CD CD CD CD 03 CD 03 CD CD CD CD
\ . • . . • . . . . . . . . ■ ■ «• O □ a O
(N*c r
i CD U3 CD CD CD rx CO rx. CN V V r xCD CN *3" <3* un CD *3* CO 03 un un rx CO O
•r—l in un un un LD <3" un un *3- ■O’ un un un un CO un_ • > CD CD CD CD 03 CD CD CD CD CD CD CD CD CD CD CD
. • . ■ . . . . . . . . . . • .□ a □ a
•>-)N •QC 03 CN CN CN <3* CO CO CO un CO CO>—i a CD CD <3* ( \ rx t X O’ CD rx CD CD oW □ CD CD CD □ CD CD 03 CD CD CD cn CD CD □*—' CD CO CO CO 03 CO CO CO CO CO CO CO CO CO CD CDCO ■ . • . • • . . . . . . ■ ■ . .
a □ o a
CN CO un □ CN CO 3" un □ CN CO «3* un at- 'T-
UJ_ j cn a unCL UJ 'T- COz : m un %< M * o uncn cn un T— v ~
- 179 -
The
table
gives
the proportion
of 10,000 Monte
Carlo
samples
for which
the
two
approximations fell below
thep
<2_v
alue
for
which
the theoretical
proportion
should be
<x.
^
TABLE
9.5.5
continued:
r \ CNJ cn CD CNJ r- in co CO in in COCO a i \ cn cn cn cn cn cn cn cn CNJ cn cn
cn CO cn CO CO CO CO CO CO CO CO CO cn CO cn COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• . . . . . . . . . . 9 . . . .o o CD a
. in cn CO CO CNJ cn cn CO UD CO cn rx• CO o CO CO CO CO CNJ co CD CD cn
c r ■ U1 in s}- <3- in in in UD in in in ){cn cn cn cn cn cn cn cn cn cn cn cn cn 03 cn cn* . • . . . * • . . . . . ■ 9 .
• o □ CD ac rw>—> CO in CO CNJ CO CNJ <r- V- CD V- rx cn
V- TT- CNJ □ CO a CO r- CD CO co rx in aCNJ CD a o cn cn CD cn CD o O CD cn o cn 03 CD
cn cn cn co co cn CO cn cn cn cn CO cn CO CO cn» ■ . . . • . . . m . . . 9 .o o CD a
V co CO in CO CO CO CNJ CO CNJ in<—> in CNJ r— a <3- CNJ CNJ o CNJ CNJ CN] CD
CD cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• • . • . • . . . . . . . . . . .rl* CD CD CD ao:A CD CO V- CO rx CO cn cn CO r - cn rx
• i-i CO CO T- cn CO CO CO V- r - CO CNJ CO111 LO in in in in <3- in in m in in in un incn cn cn cn cn cn cn cn cn cn cn cn cn cn CD cn
\ . ■ . . . . . . . . . ■ . 9 .a CD a CD
N •cr'—< CO v cn in CO m V T“ i \ CNJ in COw cn rx cn cn in CO V CO CO CO r ->—i o cn CD cn cn o cn cn CD a a cn cn cn cn CDCO cn co cn CO co cn CO CO cn cn cn CO co CO CO cn
■ . ■ . • • . . • . . . . . . .CD CD CD o
CNJ CO in o CNJ CO in CD CNJ CO in a
in'T~
CD inCO r -
UJ * *_ i cn CD o inCL UJ CNJ CNJ t-e : m * * *< M CD a incn cn CN] V V-
- 180 -
Following the introduction of (3 the test statistic components can now be expressed
as;
R2 r qv RJ r qv R?j'N ------- - k(3 y - --- + k n - y
N Li
L j - i ."•J. N L.
j - i N-j. •
( 9 . 5 . 2 )
The associated test statistic, Q ,, for the null hypothesis of no difference between q
treatments or samples is given as;
Qi
r qV 'R?J RJ
*0 (N - q) ) - —L i
Lj-1 N - j . N
k (q - 1 ) N - I
j = l
R2
N JJ
( 9 . 5 . 3 )
which has an approximate F-distribution with (q-1) and (N-q) degrees of freedom.
This may now be examined and compared to the associated test statistics from
Stephens (4.4.7), and Uptons G-test (4.4.8). (See Table 9.5.6.) Test statistic
(9.5.3) and (4.4.7) are compared to the theoretical significance levels of an
F-distribution, while (4.4.8) is compared to those of the x 2 distribution.
Test statistic (9.5.3) is seen to be accurate for large concentration parameter and
increasing slightly in accuracy as N increases. This dependence on the size of N is
more pronounced for k equal to 2. As stated by Upton (1974) the G-test is greatly
affected by the size of N and is inappropriate for small sample sizes. Stephens test
statistic is again seen as the 'best fitting' statistic when examined across all large k
and varying sizes of N.
- 181 -
TABLE
9.5.6
Comparison of
the three
test
statistics;
q and
Stephens compared
to the
F
distribution
and Uptons G-test
compared to
the x2
distribution
'If-rv CO CM in rv CO CO CM CO
<_1 CD CM CO o rv CO in CO CD r— rv r - —CD r \ CO CO CO CO CO CO CO CO CO CO CD co co CD
c CD CD CD CD CO CD 05 CD CO CO CO CO CD CD CD CDo . . . . . . . . . . . . . . . .
•H o CD CD a-P(0Dcr
IL1 CM CO CO rv CO rv V CM CM CO rv CM CMun rv un CO CM in CO CD rv CD in t* rv
1-- V/ un CM CM CM CM T~ <cr CO CO CO 'd- 'd- «d- in *d-cn c) CD CD CO CO CD CO CD CD CD CO CD co CO CD CO CDUJ . . . . . . • • . • . . . . . .h- a CD CD CD
LD
CO2 V UD CO un CM CO '3- CO CM CM CO unO rv CO 'Cf* M" o CD CM co CD un CO IV CO CO COf - o CO CO CO CO CO CO CO CO CO CO CD CD CD a CDQ_ CD 03 CO CO CO CO CO CO CO CO CD CO CO CO CD COZ3 . . . . . . . . • • . . . . . .
o o a CD
CO CO UD co r - CO '3* in CD in CO CMCD CO CD CM o CD rv a CM CD 'r- 03 CO r—
rv. CD CO CO CD CO CO CO CO CO CO CO CD CD CO CO CD. CD CD CO CO CD CO CD CO CO CD CD CO CD CD CO CO
t* . . . . . . . . . . . . . . . .O CD CD O
«cr
co rv CM CD CM r— CO CD in IV rv CO CO
•H V" CM CO CD (V CD CO un IV CO o a IV in COt un un UD 'cr un '3" un un 'sf '3* 'S' un un <d- un ^r
(U * CD CD CO CO CO CD CD CO co CD CD CD CO CD CO coID . . . . . . . . . . • . . . . .c r □ CD CD CD
UJ
CO2UJ CO «cf- CO CO IV rv in CD CO COX CD CO un un CO CO un CO CO CM CO IVQ. o CD CO CO CO CO CD a CD CO CD CD CD CO ’T* COLU CO CD CO CO CO CO CO CD CO CO CO CD CO CO CD CO1- . . . . . . . . . . • . . . . .co □ CD CD CD
CO <3- CM CO CM •st- CO r— CO <r~ IVCM '3- CM '3" CM O CM 'T- 'd- *d- CD CO CM
CD CD CD CD CD CD CO CO CO CD CD CD CD CO CD COCD CD CO CO CD CD CO CO CO CO CD CO CO CO CD CO
. . . . . . . . • . . . . . . .,—> □ CD CD aCO
un
CD CO CO un IV CM T - CM CO T“ in un>__- CM CO CO CO CD <3* CO CO co V CD CD *d- CD 'ct-
un CO un un un «=t* CO in in UD un CO CO un un inc * CD CO CO CD CO CD CD CD CO CO CO CO CO CO CD CD□ . . . . . . . . . . . . . . . .
•H o CD CD □-P(UDCT
UJ CM rv CM rv CM CO CO rv *7~CO CO CO in CO CD CD 'sf- IV T- CO CO un 'd- CM
f * o CD a □ CO ■ T— V a CD CD a o CD r- aO ' CD CD CD CO CO CO CD CD CO CO CD CO CD CO CD CD
. . . . . . . . . . . . . . . .CD o a CD
■Si CM CO ’st un CD CM CO ^r un a CM CO 'd* in CD<r"
UJ un_J CO a COQ_ UJ VE l N un % *< M * a unCO CO un V
- 182 -
TABLE
9.5.6
continued:
'—> CJ) CO CO CO CO CO cn CO CO CO CO CO cn cn co COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
c . . . . . . . . . . . . . . . .o o o o CD•H+J(UD)c r CO V- CM CM CO co rx rv 3" IV rv
LU cn cn rx *3* in 3" r - □ CO rx CO in cn cn COLD 3" 3* 3" ^r 3" in in in *3" 3" 3" in 3* 3* 3"
> CO cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn . . . . . . . . . . . . . . . .LU □ □ o af -
LD
CO rx CO CM CM in T- in cn in rx in CO cn in2 V in rx. r— in 3" rv CM cn CO CO in inO o o a cn CO cn o a a cn cn cn a cn cn cnh- cn cn cn co CO CO cn cn cn CO CO CO cn CO CO COX . . . . . . . . . • . . . . . .ZD o □ a □
LD 3 - rx CO rx V cn in CO CO CO<—> □ cn CO CO cn CM CO CO □ v~ □ r~ orx. cn cn CO CO CO CO cn CO cn CO cn cn cn cn cn cn
■ cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn3^ . . . . . . . . . . . . . . . .
. □ a CO a3*
co CO CM *3* in CO CO CO CO CO CO CM CM CO in
•H co CM CO rx CO CO CM cn r— □ rv CM CM COLD in in in 3 - 3" in in in 3* in in in in in 3*
(0 c cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnD . . . . . . . • . . . . . . . .cr □ o a CD
LU
cnxLU in co 3* "3* r - r— in CM CM 3" CDX CO CO 3" cn V CD CO CO cn rx in CO CM OX □ o a cn CO O o o o cn cn CD CD O CD CDLU cn cn cn CO CO cn cn cn cn CO CO cn cn cn cn cn1— . . ■ . . . . . . . . . . . . .UD o a a a
CO CO 3" CO rx CO cn cn in3 - CO v- V in CM CO CM CD co 3" CO CO a
cn cn cn cn cn an cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . . .□ □ a CD
CO
in CO rx CO rv 3" t- r— rv co cn• V CO in co CO CO cn in 3" a CO co
cn { LD co in in in in CD in in in in CD CO in in in>—' c cn cn cn cn cn cn cn cn cn cn cn cn cn c n cn cn
. . . . . . . . . . . . . . . ■c a o a oo
•H•P<0CJ V“ CO •3" CO CO CO CO CM cocr rx 3 - co 3" □ r— CO CO CO cn CO CO 3"
UJ a o rr* o cn O V a □ a CD CD CD acn cn cn cn CO cn cn cn cn cn cn cn cn cn cn cn
’T- . . . . . . . . . . . . . . . .o - □ o o CD
CM CO 3" in a CM CO *3* in o CM CO 3* in a
in
□ inCO r -
UJ *_J CO a o inX LU CM CMx m % * *< M a CD inCO CO CM V—
- 1 8 3 -
9.6 Comparing the Power of the Tests, for Large k
In previous sections the components and test statistics introduced have been compared
with each other principally on the basis of their apparent goodness of fit to their
respective distributions. All these tests involve the use of many approximations in
their construction;
a) The approximation of the ratio of Bessel functions in the likelihood ratio
test as a short power series (Chapter 3.6).
b) The approximation to obtain an explicit value for the maximum likelihood
estimate of the concentration parameter (Chapter 3).
c) The use of the asymptotic result about the distribution of -21og\
(Chapter 3.1).
d) Occasionally an approximation is introduced to simplify the test statistic.
e) The modification of the statistic by correcting its expected value
(Chapter 2.5).
It was similarly noted by Upton (1970) that because of all these approximations it is
often surprising that so many of the tests derived using this procedure are good fits
to their respective x 2 anc* F distributions. It is for these reasons that it is important
to examine the statistics relative goodness of fit and establish their range of validity.
What is also important is to examine the relative power of the statistics against any
available alternatives. In order to compare the test statistic (9.5.3) against alternative
multi-sample tests given by Stephens (4.4.7) and Upton (4.4.8) it is unnecessary to
conduct a complete investigation of all the different sample situations we have already
used. Table 9.6.1 gives the relative power of the three tests by studying two sets of
samples, with one sample mean direction set approximately 30* from the true value
- 184 -
of the other samples. It is clear from the table that Uptons and Stephens tests are
virtually identical in power, with the Q-test slightly less powerful improving in
comparison as k increases. The power of all three tests naturally increases as the
underlying distribution becomes more peaked.
As has previously been noted likelihood ratio theory produces tests that are
asymptotically uniformly most powerful. Therefore the tests derived in this manner
examining the required hypothesis will be at least as powerful as any other test.
The test statistic (9.5.3) initially constructed by likelihood methods but requiring a
further test to eliminate an assumption of unequal concentration parameter would not,
expectedly, be quite as powerful as that derived without this restriction. Nevertheless
this loss of power is not very large and is compensated by its increased range of
application as a generalised approach for larger experimental designs.
- 185 -
TABLE
9.6.1
Comparison
of the
power
of three
alternative
tests, for
large
concentration
parameter
CDo J ZCO •p
3* 3* rv. 3* r - CNJ CO 3"II o CO CO CO CO CO CD CO CO TD
CO 3" CO CD CO CNJ O CD cn CO CDco <D v— T- X ~ 3" 3* 3~ rv CO CO TD
x o\° • . • . . . . . . CDii r~ □ O CD 0
OX□ CNJ
CNJ © 3" IV un un un CO CO CO CO 0CO CD cn CNJ CO o CO a o
II II CNJ CNJ v cn CO rv CO CO CNJ JZCO CO CNJ CO CO CO CO CO CO CJ
CNJo o\° . • . . . . • • . •H2 O in □ o CD J Z
CO 3□ 1 cn •
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X V“ o o CD cn jo•H
O P1—1 4-3v P cn
ra *hCJ TDCNJ cn CO cn un 3" CO un un
CO CD o r— CNJ CO CNJ 3* 0 4->3" 3* cn CO CO o IV rv un -P cCNJ CNJ r— CO CO CO cO CO CO C <0
o\° • • • . . . . . . o >O CD CD X 0
»-HCNJ a 0
O 0 'c— 3" CD cn CNJ CO 3* 3 - o PCNI o LD rv cn 3" CO CO IV aII IV v 3" 3 - 3 - CNJ CO CO un ■> 0II 3" 3* ’3* CO CO CO cn cn cn O J Z
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X CO <4- q -II 1
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c cnv r" CO 3 - cn CO CNI CD CNJ CNJ CO □ -p©N
o O rv T“ CD CO CO rv •H co\° CO CO un cn cn cn cn cn cn -P -HO ■ • ■ • • . . . • P o
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CDCO
CDCO
CDco
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CNJ 2 J Z CO 3* X CO CO z X CO •p p!— o CL UJ □ CL UJ o a . UJ 0CO II h - UJ J - II J - UJ 1 - II 1— UJ J - 0 CLUJ CL 1— 1 CL 1— 1 CL i— 1 J Z CL1— ZD CO C D ZD CO O ’ ZD CO C D h - ZJ
- 186 -
9.7 The Adequacy of the New Procedure, for Small k
As previously stated, the new procedure for small k is based on the likelihood ratio
test for the null hypothesis /31 = ... = /3q, = ... = kq against its general
alternative hypothesis. For one-way analysis problems Mardia's likelihood ratio test
(4.4.3) for the null hypothesis /31 = ... = /3q with assumed equal k will be
asymptotically uniformly most powerful as a direct test of the mean directions. As
with test statistic (4.4.3) the comparable test statistic (9.2.6) only involves the main
factor effect and cannot take account of the total or more importantly the residual
or error effect. However, unlike (4.4.3) the loss of power of test statistic (9.2.6)
may be compensated by its possible application in larger experimental situations to be
discussed in Chapter 10. Table 9.7.1 compares the accuracy of the two x 2
approximations (4.4.3) and (9.2.6) following 10,000 simulations of varying
concentration parameter and sample size. The sample sizes range from 10 to 60
using the same sample sets as for large k. Before the tests are applied the equality
of concentration parameter is checked using the appropriate tests from Section 4.5.
Ranges of small k are given as its accuracy cannot be assured for such small values.
From Table 9.7.1, test statistic (9.2.6) is seen to be less susceptible to smaller
sample sizes than (4.4.3). As the sample sizes increase the accuracy of the two tests
become more comparable with both showing good fits to the upper percentage points
of the x 2 distribution with associated degrees of freedom. For both tests cq, the
simulated proportion of the component, approaches a, the x 2 theoretical proportion
or significance, as N and k increases.
- 187 -
TABLE
9.7.1
Comp
arin
g the
ac
cura
cy
of test
stat
isti
cs
(9.2.6
) and
(4
.4.3
) as
X2 ap
prox
imat
ions
with
2Cq-
1)
degr
ees
of fr
eedo
m
CN CO m CN CD i \ CO <r- in CNCD CO CO CO *3" IV . *3" l \ in CD CD rv.
CD CD CD CD CO CD CO CD CO CD CO CD CO<—> CD CD CD CD CD CD CD CD CD CD CD CD CDCO . . . . . . . . . . . . .
. o CD a CD CD o CD<3-
'3 'v--- /
in <3* CD CO r \ ' in CO CO 'r~ <3"CJ CD O CN V r \ CO r — CD CN COM s/ in CD CO CO in 3* in «3" CD CO ^ r1 - C CD CD CD CD CD CD CD CD CD CD CD CD CDCO . . . . ' . . . . . . . . .M o o CD a a CD ab-<f—CO
f \ in CN <3* CD CN CO CN rv.1— OJ V- «3" CD co CD CO CD CO CD a COCO o l \ <3* CO CD CO o CO r \ CN rvLU CD CD CO CD CO CO CO CD CO CD CO CD CO1— . • . . . . . . . . . . .
CD o O a a O CD
r \ CO CN CD r \ CO CD r \<3* <3" a CN CD a V 'T- CO CN CN
CD CD CD CD CD CO CD CD CD CD CD CD CD•—> CD CD CD CD CD CD CD CD CD CD CD CD CDCD . . . . . . . . . . . . .
. □ □ CD a O CD aCN
cn
CD v - CN in LD CD CO VCJ CN «3* CD *3" CO CO CD in CN *3- COM in CD in in in <3" in <3* in in in >3*' CD CD CD CD CD CD CD CD CD CD CD CD CDCO L . . . . . . . . . . . . .M □ CD a CD CD o a1—<h -co
CN CO CD '3 ' CO CO CO *3*1— O CO CO «3* CO in CO CD iv .CO CD r — CD CD CD CD O CD o CD CD OUJ CD CD CO CD CD CO CO CD CO CD CD CD CDH - . . . . . . . . . . . . .
□ O CD o O a a
in o in a in a in CD m CD in CD. . . . . . . . . . . .
CD T“ o o o V a oO in CD in a in o in CD in CD in
. . . . . .o □ o □ o o
inV
*CD inCO v -
LU •« %_1 CO CD in o CD inCL LU V- CO CN CN <r—e ; m m * % * * %< M % O in CD a inCO CO in V CN
- 188 -
Table 9.7.2 takes the tests statistic (9.2.6) out of its proven range and compares it
to equivalent tests in the concentration parameter range 1 to 2. The comparable
tests are those stated by Stephens and Upton given in Section 4.4, and used
previously when examining tests for large k. Stephens test is shown to be suitable
for k as small as 1, while Upton's test is suitable for R/N as small as 0.6 (fc~1.5).
Table 9.7.2 shows test statistic (9.2.6) to be comparable to the equivalent tests, whilst
all three tests show good fits to the upper percentage points. Once again it is
important to note that the equality of concentration parameters has been tested prior
to examination of mean directions.
- 189 -
TABLE
9.7.2
Comparison
of the
three
test
statistics;
C9.2.6)
and Uptons G-test
compared
to the X2
distribution and
Stephens compared
to the
F
distribution
CO
*3"
co
•H-P
cn03
o
90
96
*0
0.9
83
7
0.9
87
8
0.9
90
3
0.9
87
1 COCOco03
O
(0ZJcr in IO CO IOLU CN rx CO CO 3"
\ IT) CN CO <3- <3" *3- <3*+> c 03 cn 03 03 cn 03 03cn . . . . . . .CD o o a o o CD CD
-pi
CD
co CO CO CN CN CO ! \2 CO in a CO CN r -o o CO CO cn cn cn 03t - cn CO CO CO CO CO COCL . . • . . . .=D □ a o o □ o O
CN '3" <3* CO CN CN r - CO r \<—i CN CN t * i \ CO cn cn CO CO a
cn 03 03 cn CO CO CO CO cn CO co cn cn• cn cn cn cn 03 03 cn cn cn 03 cn cn cn
*3" . . . . . . . . . . . . .. o o a o o CD CD
<3"'— '
Co *3" CO CO CO CO CO i \ CO CO 03 CO•H CD co Tr- CO IO CO o CO CO CO r --P . in in IO <3- «3* «3* IO in '3" <3“ in in03 £ 03 cn cn cn 03 03 cn cn cn 03 03 cn 03D . . . . • . . . . . . . .cr □ o C3 o CD CD a
LU
cn
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i \ CN rx ^r IO <3* TT" IO COV- CO CN CO CO co 03 l \ CO CO
cn 03 CO cn CO CO CO CO CO CO CO 03 COcn cn cn cn cn cn cn cn cn cn 03 cn cn
*—* . . . . . . . . . . . . .CO o o o a a CD o
CN.cn<_i "3- CO r— to CN CO in CN
IO in CN CO ** in CD r \ Tr *3* a° > to *3" IO <3" «3* IO io *3* IO *3*•H < cn cn cn cn cn cn cn cn 03 cn cn cn cn-P . . . . . . . . . . . . .cn o a a a o a CD
•H•P(D
-Pcn cn «3" cn CN TT- TT- cn Tr* <3*
in CO CN 03 CO l \ 03 <3" cn cn CN-p o CO cn cn cn CO cn cn cn O CO cn cncn cn CO CO CO CO CO CO CO CO cn CO CO CO0 . . . . . . . . . . . . .
i— CD o o a CD CD o
in in IO to IO IO in IO to in toCN CN i \ CN CN r \ CN CN l \
• • • • • ■ • • . . . .V V- Tr- TT" V” t- TT- V
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- 190 _
9.8 Comparing the Power of the Tests, for Small k
As for large k , it is important to examine the relative power of the test statistic
against any available alternatives. To compare test statistic (9.2.6) against Mardia's
test statistic (4.4.3) for k < 1, and against Stephens (4.4.7) and Uptons (4.4.8) for
k > 1 it is unnecessary to investigate all the different sample situations of Section
9.7. Tables 9.8.1 and 9.8.2 show the relative power of the four tests by studying
two sets of samples, one with equal sample sizes of 20, the other with differing
sample sizes of 10, 20 and 30. Table 9.8.1 examines the power of the test with
one sample mean direction set approximately 30* from the true value of the other
samples, as in the power examination for large k. Table 9.8.2, however, gives a
further test when one sample mean direction is set 90* from the other samples. A
larger displacement has been used for the examination of small k , since to acquire a
significant difference between samples when simulating almost randomness is difficult
and larger fluctuations in test statistic would be expected. In both tables it is clearly
seen how the detection of a significant difference is increased as k increases. When
a displacement of 90* is used with k = 1.75 all three test statistics indicate the
presence of a displacement on almost 100 percent of occasions.
Examining the power between the tests for k < 1 shows Mardia's statistic (4.4.3) to
be slightly more powerful than test statistic (9.2.6) in both situations and across all
percentage points. For k between 1 and 2 Stephens test statistic dominates close to
1 while all three tests appear to be identical in power for k increasing towards 2.
Test statistic (9.2.6), constructed using maximum likelihood techniques but requiring
examination of the sample concentration parameter, is seen to be almost as powerful
as test statistics (4.4.3) and (4.4.7) but with the possible added application to larger
experimental situations.
- 191 -
TABLE
9.8.1
Comp
aris
on
of the
power
of three
alte
rnat
ive
test
s,
for
smal
l co
ncen
trat
ion
para
mete
r
□ o\°cn c - 1!rr\
0.00
57
.010
5
0.02
75
.018
7
0.05
38
.052
5
0.11
99
.108
2 .
.120
1
l T JX CO
© CXI LD v~ cn CO CO CO CO COcxj a cn in <3* CO CM
II CO LO o CO in CO CO rx□ CXI □ □ r - o CM CM CMcxi <D o\° • . • • • • • . .II LD □ a a □
CXI IIx
0oCO CXJ in <3* cn CO I \ in
O 1 l \ in in V“ in cn in oCO o CO in r \ xj* □ cn CO
0 % □ V CM CM CO COo . . . . . . . . .
X r - o □ □ □
CO CM in CO in in inl \ □ CO [X. CO t- CM CO i \o T- '3 ' CO cn cn cn V r -o a □ o o a r — CM CM
o\° . . . • . . • • •t - o □ CD o
OCXI CXJ CXJ CO CO in i \ v~ CO
© r— o ( \ CO cn CO in a v -II CD CO CM CO o CO
II o o T - V- CM CM •sj* «3*CXJ 0\° . . . • . . • . •
N O 5 o a a oo
II CO1
GN [X, in CO rx.CO o CM <sj" CM CO 'c f o CM
CXJ CO o CD CO CO in CMo\° V CM CM CO CO in in inO • . • • . . . • ."T- □ a o o
,_, ,_, ,_, ,_, ,_, (_( (_>CO CD CO CO .--V CO i \ CO
. . • . IX . . <—> •CM "st" CM . CM CO CM
. • . • . . . .CD cn . cn cn
'—' >—' w >—' >_i >_> . \—j<_'
u a CJ CJ CJ h - >_< cjM M o M i—i I— H co M1— i— . J - i— CO h - UJ i— I—
cn co CO co CO UJ co I— CO CO. M M M M h - M UJ M
□ i— I - M- h - I— h - - H - I—< < < < in CO < in CO <V i— I— I— CM X h - X - I—CO co CO CO • UJ CO . UJ CO CO
V X v X Xf - 1— I— I— X I— X □ J—CO CO co in CO CO n UJ CO n UJ J - COUJ LU LU • UJ UJ J— UJ I— X UJ1— o J— a h— I— CO t - CO X I -
- 192 -
TABL
E 9.
8.2
Com
pari
son
of th
e po
wer
of th
ree
alte
rnat
ive
test
s,fo
r sm
all
conc
entr
atio
n pa
ram
eter
to 00 LO r—1 cn 00 r-HvO LO * 00 vO LO cn cn toc rH vO cn vO CN 00 LOo o CN r—H vO vO cn cn cn
o\° • • • • • • • • •to »—1 o O o o
©oto IIII 1—t vO 00 r—H 00 to
to CN "3- to to 00 I4- CN I—t to LO
N 0 LO 00 vO 00 vO LO 00 00 00c o -3- to 00 00 cn cn cn
ii o\° • • • • • • • • •o LO o O o oCN OII O
cn cnA
1 to LO to VO rH cn rH CN vOo to oo LO o rH CO CN to tof—1 r—I CN cn r r CN o cn cn cnII <D o\° rH 1—1 LO LO cn cn cn cn cn
O • • • • • • • • •2 r—I o o o o
rH CN vO »—i CN t4- r-HvO to LO VO t4- CN 00 00i—i CN o VO LO 00 00 00o O *3- to 00 00 cr, cn cn
o\° • . • • • • • • •(—1 o o o o
O CNCN (T) vO *3- rH LO r - CN cn
rH cn LO CN rH o LO r - vOII II cr. cn to cn VO LO cn cn cn
o o o vO LO cn cn cn cn cnCN O o\° . . • • • • • • •
2 cn LO o o o o
-ti irH rH
Z <D r-1 CN LO CN t4- vO LO cn00 00 00 «-t o LO 00 cn 00f - vO •*3- cn 00 e' cn cn cn
o\° rH »—i vO cn en cn cn cnO • • • • • • • • •f-H o o o o
/—\ /_^ /—\ /—\ /—\ /—\ /—s f —•>to vO to vO vO r -. 00 vO
■'3" CN •*3- CN 3" CN ■*3" *3- CN
r t cn *3" cn 43" cnv—' v—' \_/ v_> '__/ v_> '_/ V__> v_>
c j u u u c j CJt—1 HH o t—1 HH E-1 l-H E— i—iE-1 E-1 E-1 E-> CO E-1 CO E-4
LO CO CO rH CO CO UJ CO UJ E— CO• 1—1 t—t 1—1 HH H 1—t E- CO HH
o E- E-* V E-1 E-4 E-r UJ E-;< < < < LO CO < LO CO E-4
V E-1 E-> E-* E-4 CN 2 t " 2 E-CO CO CO CO • UJ CO • UJ CO CO
V *—t X I—I X 2E- E-1 E- 4 E - E-* a. E- 1 Oh o E- 4co V CO CO LO CO CO II UJ CO II UJ E-4 COu j u j CIJ • UJ UJ E-4 UJ E-4 Cu UJE - O E- 4 " O E- 1 E - CO E - ^ CO X E-4
- 193 -
9.9 Summary
As the exact distribution of the new test statistic components have been seen to be
intractable by an exact theoretical approach for both large and small k , other
methods to examine their distribution have been used. A comprehensive step
approach was initiated for both situations of large or small concentration parameter
(1) to test the components validity by use of approximations in a theoretical
manner
(2) to obtain the expectation and variance of each chi-squared approximation
component via Bessel function approximations
(3) simulation testing of the components to examine their accuracy to the
upper percentage points of their associated chi-squared approximation and
obtain and examine their first two moments
(4) to compare the new F or chi-squared test statistic against available
alternatives, and finally
(5) to examine the power and robustness of the test statistics against these
alternatives
Following the production of the first two moments, via the Bessel function
approximation and simulation, an improvement factor /3 was derived to increase the
accuracy of the distribution approximations for large k and therefore improve the
final test statistic. The comparison of the new test statistic (9.5.3) to associated
one-way classification tests was favourable although as it requires the prior testing of
sample concentration parameters it is not a uniformly most powerful test.
Nevertheless the new approach lends itself to an increased range of applications
unlike the alternative techniques.
- 194 -
For small k only the associated component chi-squared approximation can be used as
a test of difference since the corresponding total and residual components for small
concentration parameter do not form chi-squared distributions. The test statistic
chi-squared (9.2.6) also compared favourably to the available alternatives for small k.
As for large k, the tests of (4.4.3) and (4.4.7) are uniformly most powerful although
generalisations of these tests to alternative experimental designs is not feasible.
- 195 -
CHAPTER 10
THE RANDOMISED COMPLETE BLOCK AND TW O -W A Y DESIGNS V IA TH E
NEW APPROACH
10.1 Introduction
A comprehensive analysis of the distribution functions for the generalised approach
has been carried out in Chapter 9. Here the new test statistics for the randomised
complete block and two-way classification designs, together with their associated
improvement factors, are produced. The relevant component statistics and test
statistics are compared with their respective chi-squared and F distributions to
ascertain their reliability and robustness for larger designs. In order not to repeat
this analysis for every possibility the randomised complete block and two-way
classification designs are used to represent and examine the adequacy of other larger
more complex design situations. Although the components and test statistics are not
investigated here, Chapter 11 extends the approach and analyses experimental
situations using such methods as Latin-square and Split-plot designs.
10.2 Randomised Complete Block and Two-way Classification with Interaction
Designs, for Large k
In Section 8.5 the randomised complete block design for the new approach was
constructed via vector analysis and was shown to produce zero cross product terms.
We may now test for any possible block effect, i, as well as any treatment effect, j,
to produce the randomised complete block expression;
- 196 -
R2 r pr» R? i .
R2 r q [R' j ] R2k N -------— - k y - ---- — + k y - ---- —
N LL i - 1 N i - .
N LL j - i N - j .
N
r pv R? qv fR j ]
R2+ k N - J
l .- ) + —
La
i - 1 N i .L
J - l N - iN
with associated independent chi-squared distributions
X (N -1 ) “ X (p -1 )+ X( q - 1 ) + X( p - l ) ( q - l )
As noted in Section 8.5 the first term on the right hand side of the expression is
the measure of variation due to p treatments, the second term of similar form being
a measure of variation due to q blocks. The final term represents the residual
variation, where we assume that the experimental errors are independent and von
Mises distributed.
Equation (10.2.1) produces the test statistic Z g to examine the null hypothesis that
there is no difference between the p treatments
r p R2l . R2( p - 1 ) ( q - 1 ) y ----- L -L
L i. i= l [Ni . J N
( P - Df P N - J R?l .
q
- J R2 . • J
R2 + —
Li = l Ni* .
L i N
( 10 . 2 . 2)
which has an F distribution with (p-1) and (p - l) (q - l) degrees of freedom.
Similarly, the test statistic Z 7 is produced to test the null hypothesis that there is no
difference between the q blocks
r q
fR' J]R2
( p - 1 ) ( q - 1 ) I ----- L -LL i
L j - i k J N
r pr* R? ' q R2 . • J
R2( q - D N - J 1 .
- I + ' *La
i -1 k J laj - l k j J
N
( 1 0 . 2 . 3 )
- 197 -
which has an F distribution with (q-1) and (p - l) (q - l ) degrees of freedom.
Test statistics Z 6 and Z ? indicate whether or not the mean directions are equal, they
do not allow for discrimination amongst single mean directions which we were able to
derive for the one-way analysis in Chapter 5.
Prior to examining the accuracy of the component chi-squared approximations of
(10.2.1) and the associated F approximations Z 6 and Z ?, the test statistics for the
two-way analysis with interaction may be constructed.
Using the same approach as above, the two-way classification with interaction may be
seen as;
R2M - * * *
r p y RL' R2 r v
y R2 .•J-
R2
Nft L
. i - l Ni . . N LLJ-l N - j , N
+ k - 11 i-1 J- l
R?. i j
lNU.J
+ k i ii-1 j - l
R? . U
lNU.J- I
i=l
R?1l
J- l
R2 .•J
N J-J
R2
N
with associated independent chi-squared distributions
( 1 0 . 2 . 4 )
CN-1 X (P -1 ) + X( q - 1 ) + Xpq(m-1) + X( p - 1 ) ( q - 1 )
where m represents the number of observations within each cell. F test statistics are
built as for the randomised complete block design;
- 198 -
Testing for differences between rows effects ( i= l,2 , p)
p q ( m - l )
r p R2 R2y i . .z. i - 1 Ni . . N
8
(P -1 )
r p q
- - I I
> 90t-i.
JO
N i j . .L i - 1 J - l -
~ F ( p - 1 ) , p q ( m - l )
Testing for differences between column effects (j= l,2 q)
p q ( m - l )
f Vy R2 .
• J-R2
zU-i N - j . . N
(P -1 )
p q
■ - 1 Ii - 1 j - l
R?.i j
NiJ-J
” F ( q - ! ) , p q ( m - l )
Testing for difference between interaction effects
p q ( m - l )
r p q
y y R?. i j -
Py
f—(S*«-iz zL i - i j - i Ni j - .
zi - 1 Ni . .
lj - l
R2 . • J
N J-J
R2+
N
( 1 0 . 2 . 5 )
(1 0 .2 .6 )
“ 1 0 r p qV V R2 .
i j .
"
( p - i ) ( q - i ) N ■ I ILi LiL i - 1 j - l iN i j J
( 1 0 . 2 . 7 )
F ( p - 1 ) ( q - 1 ) , p q ( m - l )
- 199 -
10.3 Accuracy of the Associated x 2 Approximations for the Randomised
Complete Block and the Two-way Classification Designs with their
Corresponding F Statistics, for Large k
The accuracy of the expressions (10.2.1) and (10.2.4) are determined by simulation
methods. Monte Carlo samples from a von Mises distribution with fixed k are made
for the distribution specified by the null hypothesis. The computer method used for
the generated observations is outlined in Appendix B. As for the testing of the
one-way classification and the extended techniques of Watson and Williams in
Chapter 6, 10,000 sets of samples of various size were drawn from the von Mises
distribution with k = 2,3,4,5 and 10. The same experimental designs investigated in
Chapter 6 are used here to enable comparison between tests. For the randomised
block three designs are examined varying in size of N.
10.3.1 The Randomised Complete Block Design
Tables 10.3.1 to 10.3.6 examine the chi-squared approximations for each component
within the randomised complete block design. Table 10.3.1 gives the first two
moments of the components for total and residual measures of variation. The
simulated mean value of the chi-squared approximation for each component is seen
as a good fit, although slightly over-estimated, and increasing in accuracy as k
increases. This follows the findings of Section 9.5. When k is unknown and
equation (9.2.5) is used as its replacement the accuracy of the simulated mean value
is increased. Again, as in Section 9.4, the simulated variance of the chi-squared
approximation of each component is seen to be below its expected value and
dependent of the size of N and k. As N and predominantly k increase the accuracy
of the variance increases.
- 200 -
Table 10.3.2 shows the accuracy of the total and residual measures of variation
components to their associated chi-squared. The goodness of fit in the upper 10
percent significance levels, for concentration parameter greater than 2, is seen to be
very good, and increasing in accuracy with increasing size of k. For concentration
parameter equal to 2 the approximations under estimate their associated significance
levels. Although not shown within the tables, the fit of the simulated distributions in
the lower percentile levels is not as good as in the upper tails, but this would be
expected given the simulated means and variances of Table 10.3.1.
Tables 10.3.3 and 10.3.4 give the first two moments for the treatment and block
measures of variation components, respectively. As for the one-way analysis the
simulated mean value of the component is approximately equal to the mean of the
chi-squared approximation and increasing in accuracy as k increases. Similarly the
variance is seen to be rather poor, most noticeably for small k. As in Section 9.5
the improvement factor (3 is used and multiplies k in order to increase the accuracy
of the chi-squared approximation variance and therefore the distribution fit. The
comparable moments are given in adjacent columns in Tables 10.3.3 and 10.3.4 and
show a slight increase in the simulated mean value but an important improvement in
the accuracy of the variance. This is reflected in the accuracy of the improved
treatment and block components to the upper percentage points of their associated
chi-squared distributions given in Tables 10.3.5 and 10.3.6.
- 201 -
TABLE
10.3.1
Comp
arin
g the
mean
and
vari
ance
of
the X2
appr
oxim
atio
ns
for
the
tota
l and
re
sidu
al
meas
ures
of
vari
atio
n ag
ains
t their
expe
cted
va
lues
2\rj • «—«cr m+ o ,—, ,—% ,—, ,—, r-» t—\ *--\ t—* * ' 1 ' r '
c CD *3" CO CD r- ,— , cn l \ cn cn l \ cn <r cn IX)•r- ) <U cn CO 'd' O r- cn CO CD o CM cn CO CO
• *H CD o ur O o cn cn cn o O r— cna a a a a a a * • • * • *
\ CD CD CM cn 'd' l \ CM cn CD l \ CM CM tn V
r - CM CM CM CM CD CD CD l \IN • ' »__/ >_» »--- ' «.—j <--- ’ •— > '— ' '— ’ '—' *—' '—' %__/ '— 1 '— ’c r*—•w
i
CO cn CO CO CM LD l \ CD CO r - CO l \
•H C o CD V CO CO r — cn cn ID CO IX) ( \
S CD CM r~ o O cn <cf V- CD o cn IX) O\ 0 . a a a a a a a • 9 • • ■ • •
21 CO CO CO CO CO <" *d" <d_ CD CD CD CDV t - V- v ~ cn cn cn cn cn
c rww
s\__/X L
CD 'd' CO CD CMCO <r- T- CM cn r \
II II II II n ii
0 0 0G G G
TD C TD C TD cCD 0 0 0 □ 0 0
UJ CD -P c •rH co -P c •rH -P c •H_1 O co p O 0 P G 0 PQ_ UJ > CD 0 0 > 0 0 0 > 0 0 0e: m JD Q. s : > JD CL e: > JD Cl z : >< M X X Xcn cn CO UJ cn UJ LD UJ
I do ,—* /—s /—1 t—\ /—\ <---1 * v * '
c CD rv. V CO CO r~ CD CO CM CM <a- CDCO CM cn CM cn CD V- CM CO CM r \ CO «d- IX) a•H CD CD in in V- r - LD «c~ cn cn CO ’d" CO COP . a a a a a a a a a a a * a ■(D r x cn CD LD CD CO cn CM cn CM CO ( \ o CM i \
2 > T- CM CM CM CM CM cn <d" CD CO cn cn cn\ w %_' w w v—/ w '—' 1— * '—' v---* v__/ '—’
CM •cr
i t- CM «d" cn <d" cn cn ID CDZ L cn r " l \ o cn LD T - CM cn cn cn O CO
' C CM CM V o LD CM CM o 'd - cn co cn o-*£ (0 a a a a a • • • • ■ • • ■ • •
CD 'd' "d " «d" cn cn cn cn cn o CD cn cn cnEl r - V V~ CM CM CM CM CM LD IX) 'd-
X L CM cn «d" LD O CM cn "d- I D o CM cn «cr IX) or - ’t- T-
<d" CO cn CD cn COCM CM cn
II II n II n n
0 0 0O G a
TD c ID C TD cCD 0 0 0 □ 0 0
UJ LD -P C •H CO -P c •H r~ •P c ■H_ ) O 0 P O 0 U G 0 UCL UJ > 0 0 0 >> 0 0 0 > 0 0 0e: m JO CL e: > JD Q. e: > JD Cl e: >< w X X Xcn cn CO UJ cn UJ LD UJ
- 202 -
TABLE
10.3.2
Exam
inin
g the
ac
cura
cy
of the
X2
appr
oxim
atio
ns
for
the
total
and
resi
dual
me
asur
es
of va
riat
ion
2\
fN •c n+ cn CO CO CO CO LD in CM CO CO CO cn in
«--V CO CO cn CO <3- CD CO CD cn CM aCO cn CO cn cn co CO cn cn CO CO cn cn cn cn CO
• cn cn CO CO CO cn CO cn CO cn cn CO cn cn CO CO53 . . . . . . . • . . . . . . . .
CD □ a CD
IN •c nv-- !w CM •3* CM CM CO CM CO COI CO a CO *3" cn CO a CO re l\ . CD in CO V
1—1 cn co CO cn CD in in CO in in CO in in in in•a cn cn cn cn cn CO cn cn CO cn CO cn cn CO CO CO
•H 0 . . . . . ■ . . . . . . . . •53 CD o a CD\
cn *Hcn'—> CM CO CO CO ■3* CM CO CM CM IN CO CM CMw r - CM a 3" CO cn CO in in IN. CO CO cn CDi CD cn V CD a CM CD a CD CD CM CO CD o O
CD CO cn cn cn cn cn CO CO cn cn CO CO CO CO cn*—* . . . . . . . . . . . . . . . .XL CD CD o CD
rN CM 'd- CO un CM CO CO CO COCO in . V cn CO 'Ct’ cn CM CO *— cn T—
CD CO CO cn cn cn cn cn cn cn CO CO CO cn CO cnCO cn cn CO cn CO cn CO cn cn CO cn CO cn cn cn
. . . . . . . . . . . . . . . .CD CD a o
CO CM CO CM <3- CO CM CM CO COCO CO cn <3- in CM in cn v - l \ cn cn CM
ID l"N CO in CD in r \ LD in in in rN in incn cn cn cn cn CO CO CO cn cn cn CO cn cn CO CO
I ■ . . . . . . . . . . . • . . ■2 CD o a □
CN •c ni CO CO CO CO CO CO CM *3*
CO □ CO IN. a CO CM in cn r— r \ I \ CM CM>—> □ CM □ CO o o CM □ CO o CD CM CO 00 CD CD
cn CO CO CO cn cn cn cn cn cn CO cn CO CO cn CO. ■ . . . . ■ . . . . . . . . .
CD a o CD
CM cn CD o CM cn in a CM cn in CDV t—
cn cn cn-p -p -pc c cCD CD CD cnE cn E cn E xL
-P AC -P a C -P uCO a 10 G in oCD o 0 O CD g 1—1
LU CD P r—1 CO P i—1 T- p JO_i cn -P JO -P JO -pQ . LU > in > > o Oz : m JO cn cn v - JO cn CO CM JO in in< i— n ii n n II II ii II iicn cn cn M •“ ) 2 cn H 2 in M •n 2
- 203 -
TABLE
10.3.3
Comp
arin
g the
mean
and
vari
ance
of
the
y2
appr
oxim
atio
n for
the
trea
tmen
t me
asur
e of
vari
atio
n ag
ains
t its
ex
pect
ed
valu
es
m<—• oz c /—\ t—\ r-* /—\ \ /—s !-- \ /—t *—» f--N /---V /—» r-> /—\\ (0 i x T- CD IX cn CO CD V IX i n CM r -
. •H LD r - LD CO <—i CM CO CM CO CM O IX)N • Li LO IX IX CO r x r x cn CM LD LDc r (0 . • . . . • . . . . . . . . .i > CO CO CO CO CO CO CO CO CO CO CD CO r x IX IX
*—» W *.— ’ *—; S-- ! w w v-- > v-- ' w V---' *— > v---' v— f v-- / V__ >
s
•<“ 5PI • IX T— LD 'tf- v— CD CM <d- *}* IX CO CMc r c CD cn IX) CD IX) CD CO □ 'T ‘ •ct" IX) CO r — cn CD<— i (D CM CM v CD a CM CM V o <3* CM CM Ow 0 . . . . . . . . . . . . . . .>_t e : OJ CM CM CM CM CM CM CM CM CM 'TM&
<—> 0Z o\ c t—s t—\ /—\ !-- \ t—\ t—* t—\ t—\ t--N r i !---\ t---\ !---\ !---\
. 0 CM <—■> v CD co IX CD CO CO CO CO IX IX -PI • •rH CO CM V“ a IX o CD O ■sT cn CO IX) oc r u O CO IX CO CD cn CO IX) CO CO ^ r CO CD
i 0 • . . . . . . . . . . . . . .<—> > CM CO CO CO CO CM CM CM CO CO LO CD CO IX
•i- ) v—' *.—t v---' v—' w w w w w w w w v—' w
s
•i- jpi •c r CO i n CO CO LD CO CM CM CM CO CD<—' c LD CD CO CM LO CD CO IX) CD •cr IX CDw 0 □ O a o O a a CD a o a CD a o cns---> 0 . . . . . . . . . . . . . . .j r e : CM CM CM CM CM CM CM CM CM CO
CM CO CD □ CM CO LO a CM CO i n CDr - T- V-
CM CM co
II ii II II n II
0 0 0G G G
TD C TD C TD c0 0 0 0 O 0 0
LU lt > •P C •H CO -P C •H •P c •rH_1 O 0 Li a 0 Li G 0 LiCL LU > 0 0 0 > 0 0 0 0 0 0e : m JO P . z : > JO □L e : > JO □L E l >< M X X Xcn cn CO LU CO LU LD UJ
- 204 “
TABLE
10.3.4
Comp
arin
g the
mean
and
vari
ance
of
the
•x2- ap
prox
imat
ion
for
the
bloc
k me
asur
e of
vari
atio
n ag
ains
t its
ex
pect
ed
valu
es
CDr-> o 4---\ 4---\ 4--\ 4-- \ 4---\ 4---* 4---\2 c <—> <—. /—I #—i <—> CD CO i—> in CO 'J - CN ,—, CD CD\ to cn cn in cn < - O CD x o cn CD in CN CD
. •H CN cn cn CD CO cn cn a in in CD CN cna • L, t- r— in in x • . . . . . • • . .c r (0 • . • • • □ CN cn cn cn 'd- in CD X Xi > CD x x x x r— V V t— V T-
’ w w w v—’ *_4 w w ’ *---4 w w X_4 w
•H
(N.pH in V- CD in CO CO co v CO X ■cf CD Xc r C CO O >r- CD CD o 'd* CD CO in <3* V- COV_/ (0 's f cn CN o a CD cn CN CN CO in CN <r _w CD • . . . . . . . . . . • • . •>—> e : «cf* CO x rv. X IX O CD CD CD CD
pq&
,—i 0gc r—\ <—> <—> r—\ /—i 4---1 i—i 4—» 4--> ,---> 4---> 4--<
• to i—. X CD r — cn CO .—, CN CO CN in cn cn V -* •H 'c n CO CD in 'Cf rx cn CD CO CO
c r P CD CD x cn o cn in CD cn CO cn X COi to . • . • • . . • . . . . . . .
>—> > 's f CD CD CO x CO o t— CN CN r - CN in in. ’—' 1—' >—> '—' >—’ '—' T- T“ r — r — r—
•H V-- 4 V_4 V_4 V_4 v_4 \_4 V_4 V_4 V_4
<N*Hc r cn CO *3r cn *r~ CD CO cn CO V <r V- CO CN>—> c in CD CD CD CN O co CN cn X CO CD cnw to o O O o CD V- CN CD a CN O av—' 0 . . . . . . . . . . . . • . .X L e : ^ r cn X X x x X cn CD CD CD CD
X L CN CO '3 ‘ in CD CN cn in a CN cn «Cj- in av " r —
* “
•3* CD CO'd* CO x t - V-
ii II n II II II
0 0 0a G G
TO c • a C • a C0 0 0 0 o 0 0LU in -P c •H CO -p c •H rr- -p c •H_J O to p o 0 P a 0 uCL LU > 0 0 0 > 0 0 0 > 0 0 0E: m JO Q. e : > JO CL e : > JO CL e : >< M X X Xcn cn cn LU cn LU in LU
- 205 -
TABLE
10.3.5
Exam
inin
g the
ac
cura
cy
of th
e^2
appr
oxim
atio
ns
for
the
trea
tmen
t me
asur
e of
vari
atio
n
"vf- CO '3' CN CD CD <3" 3* CN CO CO CD CD CO□ a o cn CO CN CO CN a a O
cn cn cn cn cn co cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
/—\ . . . • . . • » . . • • • . . .2 □ a a a\
a •cri CO un CO CO CO CO CN CO *3* «3- CO CD CO *3"
<—> r— cn CO CO un LD cn CO co i-n cn CO IX)•«“> un un <3* un un un un IX) un 1X1 •3" <3* *3- IX)
• s,cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnS £ • ■ • • • . . • . . . ■ . . . .\ o a o a
•i- )M •cr\_/w CO CO CO «3" <3" «3" CN CO CO CD CO COi_< in . CO CO CO cn CO CN CO v~ CO t— <r~ IN. t— COpq o CO CO cn cn cn CD cn CD O CD cn cn cn cn CD
cn CO CO CO CO CO cn CO cn cn cn CO CO CO CO cn» . . . • • . . . . . . . . . .
□ o a □
<3- c\j CN CO *3* CO CO CO CN CN CO *3" CN COCO CN V r \ LD CO CN <3- CO CN CN
cn CD cn cn cn cn cn cn cn cn cn cn cn cn cn cnCJ) cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . m * . . . . .a □ CD CD
s\
. CO CD CO CO CO «3‘ CN CO CN CDCN • •3* CN r— in . <3“ CD <3" CO IX) IX) CO cn LO cncr LD CO CO un un l"N CO CO un IX) IN. CO LX) IX) IX)i >/O ) cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
^ c • . . • . . . . . . • . . . . .•i—i o a a CD
55
•«“)CN • CO CN CN CO *3" CN CD CO CO CNcr ^r <3- CO CO CD CO LD cn CD CO T~ CD CD CO CO<_i a CN r— a O O CO V- v CD CO "T* V a CDw cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn\_/ » . . . . . . . . . . . . . . .iT o o CD a
CN CO <3- un o CN CO *3- un a CN CO «3" IX) □T- r—
cn cn 0-p -p •Pc c a .CD 0 0 0e cn E 0 E
-p 4-> •P ara a (0 a in ocn o 0 □ CD 0 i—i
llJ un p i—i CO P rH V p JD•p JD •P JD -p
Q_ UJ > LD > > a CDe : m JD CO un TT- JD CO CO CN JD IX) 'T- IX)< M ii n II II II II ii II IIcn cn CO H r ) Z CO M •“) 2 IX ) M •“J 2
- 206 -
TABLE
10.3.6
Exam
inin
g the
ac
cura
cy
of the
X2
appr
oxim
atio
ns
for
the
bloc
k me
asur
e of
vari
atio
n
<3* 1* CO CM CO <3* CO X CO CM CD CM XCD 'T- a CD V“ CM CM a CD V“ CD a CD a
CD CD CD CD CD CD CD CD CD CO CD CO CD CD CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. • . . . . . . . . • . . . . .z o □ a oX .
<N •cr
CD x CM CD CO V —n CO CD x CD CD CD CO CO CD CD o CD CD V“i—> LD LD LO in in CO ^r in CO in *1* '3" in
• ]rcn c n CD CD CD CD CD CD CD CD CD CD CD CD CD CD•P f . • . . . . . . . . . . . . . .
o a CD a
cr>—« V CO x CM CM 'T-w a CO X CD CM IX CO CD CD CD CO CD*—> a CD CD CO CD CD CO CO CO CD CD CD CO CO CD CDCO CD CO CO CO CO CO CO CO co CO CO CO CO CO CO CD
y j . . . . . . . . . . . . . ■ . .r-Z* a a CD CD
CM CM 3" UD CO CM CD CMco ID CO CM CM CO CD IX CM X un in V*
CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CDcn CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
w . • . . . . . . . . . . . . .o CD O CD
Z\
(N • CO CM CM CD X CD CM CO CM CM v-cr CD CO x CO CD CD CD CD CD v CD CD CO <3*i LD X CD ud UD in x CD CD LD in X CD CD in in
' ' V c n CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD. • . • • . . . % . . . . . . . .
•H o CD o CD2
m .pcr CO <3- CM <cr CO CM CD CD CO iny—j CD CD CO x <ct* in X □ in in CD Xw a CO <r- CD O CD CO V a V- a CO CM □ o'— ' CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . . . . . . . . . . . . .CD a o CD
CM CO ID CD CM CO in o CM CO <3- in CD,r" r—
CD 0 0p p Pc c c0 0 0 0E 0 E 0 E
4-> p P o(0 a CD 0 0 o0 o 0 o a 0 1—1
UJ LD U i—i CO u r—1 V U JD_J -P JO p JD PCL UJ > in > > CD CDE : m JO CO in JD CO CO CM JD in V in< M II n ii II li II ii II itcn cn CO M • - ) z CO H z in H r~3 z
_ 207 _
The accuracy of the F distribution approximations are provided in Tables 10.3.7 and
10.3.8. Table 10.3.7 examines the F distribution statistic for the testing of the p
treatments Z e, against its improved F distribution where the term (3 has been
included. Table 10.3.8 examines the corresponding test for q blocks, Z ?. The
improvement tests show better fits for both factors across all significance levels,
although it is important to note that it still slightly under-estimates the proportion a
leading to an increased probability of a type I I error.
In Chapters 6 and 7 the extension of Watson and Williams and Stephens approaches
to larger experimental designs was shown to breakdown due to the combination of
sample means. Examination of the chi-squared approximations for randomised
complete block and two-way classification designs were, however, carried out since
the combination problem was negligible for very large k and did not affect the
associated test statistics. On the assumption that the extended techniques were valid,
direct comparison may be made to the corresponding tables in this section. The
associated tables for the randomised complete block design are Tables 6.4.1 to 6.4.7.
For all the components within (10.2.1), incorporating the correction factor /3, show a
comparable fit for large k and an improved fit for small k for both the chi-squared
approximation and moment calculations.
- 208 -
TABLE
10.3.7
Comp
arin
g the
ac
cura
cy
of the
test
stat
isti
c Z, a
gain
st
its
impr
oved
st
atis
tic
for
exam
inin
g tr
eatm
ent
effe
cts
in m CO in in CD CN in X COin CO CO o CN 'T CN ■cf V CN CN o CN
CD CD CD CD CD CD CD CD CD CD cn CD cn cn CD cnCD CD CD CD CD CD CD CD CD cn cn CD CD CD CD CD
. . . . . . . • . . . . . . . .o a CD O
in CO CN TT- CD CN r - cn X X COCN T— CO CO CN V“ CO in CN t— o X CD
in CD CD in in in CD in in in in CD in m *3" in>/CD CD CD CD CD CD CD CD CD cn CD CD CD CD CD cn
CD I • . • . . . . . . . . • . . . .N o o O □CQ
in V— CO CN CN CN CN X COCO CD in CD CO CD '3" CN in CD CN
a a a o <r- a a O O □ o CD oCD CD CD CD CD CD CD cn CD cn cn cn cn CD CO CD
. . • . • . . . . . . . . . . .a O O □
in CD in CD CD CO CD cn CO
■
T— XCD in CO <r“ CN in in CO V O in CO CN CN
CD CD CD CD CD CD CD cn CD CD CD CD cn CD cn CDCD CD CD CD CD cn cn CD CD CD cn cn CD cn CD CD
j—t • . . . . . . . . . . ■ . . . .CN □ o □ o
CN
ar - in CO <r- IX CO CO T- CO X X CD
CO CO O x in CO cn CD CO X in CD COin CO CD CD in in CD CO in in in CD CD in in in
c > CD CD CD CD CD CD CD CD cn CD cn CD CD cn cn CDO c . . . . . • . . . . . . . . . .
•H o a a a•P(DDcr
LU CD CN V ix CD CO CD CO T— CD '3'a a CO V- CD CO CD □ CO
CD □ CO CN r- □ CO CN t- □ □ CN CN CN v □X CD CD CD CD CD CD CD CD CD CD CD cn CD CD CD CD
. . . . . . . . . . . . ■ • ■ .□ o o □
CN CO <3* in CD CN CO in a CN CO <3- in a*" V
OLU in CO V-_ICL LU > > >e ; m JC X3 JD< Mcn co CO CO in
- 209 -
TABLE
10.3.8
Comp
arin
g the
ac
cura
cy
of the
test
stat
isti
c Z-,
agai
nst
its
impr
oved
stat
isti
c for
exam
inin
g block
effe
cts
CM ’3" CM in CD CD CO <3*CD CO CO V“ CO CM — CM t * CO CM CD v-
CD CD CD CD CD CD CD CD CD CD CD CD CD CD CO CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
• » . • • • . . . . . . . . .□ O CD CD
CO s}- CM t - CM CO Xr * o CO CD in co CM CO a — CM
V in IX CD un in in CD in in in in in CD in in in0 CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. . . . • . . . . . . . . . ■ .x o □ o a
X IPQ
CO CO CM CO CD CM CO COCO o CD CD in o in CD ^r — o a CD X
o CM CM □ CD CD □ V o O V □ CD □CD CD CD CD CD CD CD CD CD CD CD CD CD CD CO CD
. • . . • . . . • . . . . . . .o o o CD
CD X X X CD CD CM COIX CO CO CD CO CM CO V“ CD CM CM
CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CDCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
/—\ • . . . . . • . . . . . . . . .CO O □ CD a
C\J
oCO in CD CO X CM CM CD X CM
' — • <3* X V- CO X CD CD CD in CO CD CO CDun IX CD co LD in CO CD CO in in CD CD CD in in
c ICD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
□ *- . . . . . . . • . . . . . . . .•H o a CD a+-><0DD"
LLI CO CO t— CO CO CO <r* v- CO X XCO T— o CM CD X CO o in CD CD □ CD
IX o CO CM CO CM CM r— CD CO CM CM □X CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
. m . . . • • . . . . . . . . .□ o □ o
CM CO <3- in a CM CO <3* in □ CM CO in o
aUJ in CO V_JCL UJ > > >2 Z NI -Q n JO< 1-cn cn CO CO in
- 210 -
10.3.2 Two-way Classification with Interaction
As for the randomised complete block comparable statistics for the two-way
classification with interaction design, examining the component chi-squared
approximations, are produced in the six tables from 10.3.9 to 10.3.14. Here the
chi-squared approximation moments are not reproduced as similar terms from other
models have already indicated their adequacy. Similarly the correction factor 0 has
been included for the main effect and interaction terms.
There are four two-way designs simulated, each varying in size from N=30 to N=90
and varying in concentration parameter from k=2 to k=10. Table 10.3.9 indicates
the accuracy of the total measure of variation and the first main effect to their
respective chi-squareds for 10,000 simulations. Table 10.3.10 for the second main
effect and interaction terms and finally Table 10.3.11 for the residual term. The
main effect and interaction terms show excellent fits across all values of k, while the
total and residual terms are relatively poor fits for concentration parameter as low as
2.
The accuracy of the three test statistics Z 8, Z g and Z 10 are shown in Tables
10.3.12, 10.3.13 and 10.3.14 respectively, with the corresponding improvement factor
/3 applied. The F distribution approximations for the main effects show similar good
fits as in the one-way classification and randomised block designs, with the accuracy
decreasing at 2. The final Table, 10.3.14, indicates the accuracy of the
interaction test statistic to the F distribution and shows an excellent fit across all
concentration parameters.
Comparing the Tables 10.3.9 to 10.3.14 with the corresponding Tables 6.4.8 to
6.4.13 of Chapter 6, and on the assumption that the extended techniques are valid
for large k, shows once again an improvement in the chi-squared accuracy for the
new approach.
- 211 -
TABLE
10.3.9
Exam
inin
g the
ac
cura
cy
of the
total
and
first
main
effe
ct
comp
onen
ts
to their
resp
ecti
ve X2
dist
ribu
tion
s
CO CO CO CN *3* CO CN CN t- IX CO CO CO un CO CN cn•3- CN CO V CO CO r - CO •3- V CN CN CO CO a
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . • . . . • . . . . . . . . . . . . .2 o a a a o
CM •cr co CO CO CO CD CO r— CO CO co CO CO CO uni Nj cn LD <3- cn •st •3* CN cn in •3* r x IX •3* CN CO un a o
\ LD LO in in <3- un un un '3' un un CO un un un un un un un un uniJ cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn. . . . . • . . . . . . . . . . . . . . .
•H o o o a oZX .
CM *H *3- <3- ^r <3* <3* CO 3- IX co IX CN CO cn CO CO CO cn •3-cr CD rx un CO <3* in o CO un un a un o CN *3" cn CO>_i o cn O o cn a cn cn cn a a o o a o a cn O O cn ow cn CO cn cn CO cn co CO CO cn cn cn cn cn cn cn CO cn cn CO cnV_/ . . . . . ■ . . . . . . • . . . . . . . .CO a o a o a
*3" <3* •3- <3- CO co CO r~ CO •3" CO r x <3" IX COCO LD *3- CN cn IX □ V“ cn CO CN co CO a CO co CN
cn CD cn cn cn CO cn cn cn cn CO cn cn cn cn cn cn cn cn cn cncn CD cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn CD cn cn. . . . . . . . . . . . . . . . . . . . .o a a o □
CN CO in CN •3" CN un CN CN CN •3* CO COV CO IX CO cn IX cn CO co cn CO CN O <3- CO CO CO cn CN
) lh tx CO LD LO un CO 3- *3* un un CO un in un un CO <3- •3* *3* unt cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
t—\ • . . . . • . . . . . . . . . . . . . . .2 o o □ o aX .
CM •cr CO <3- CN V- CO r— CO CN •3- CO CN CN CN t -i cn ^t- •3* un cn CO •3- cn IX in CO IX in cn a cn V O co
□ CN cn a O a CO cn cn a CN cn cn cn cn CO cn □ □>—> cn cn CO cn cn cn cn CO CO CO cn cn CO CO CO CO cn CO CO cn cn
. . . . . . . . . . . . . . . . . . . . ■o o a o o
CN CO un □ CN CO •3- un o CN co <3" un a CN CO •3* un o•?-
ci(0 □
<—> b0 > CO <—i i—\ oin •H u c II a un V-
cn 0 •H> CO cn JO i—i > >
JO TO JO •P i—1 > JO JO<_> o •H i—1 JO i_> >_1CO 2 0 >_iLU CO un a CO CO CO_J > cnCL LU > JO _c c j i > O > un > O
s : m JO -p o o JO CD JO •3" JO cn< H CN •H •rH 0 II II ncn cn CN 2 - P 0 CN z : CO Z CO z
- 212 -
TABLE
10.3.10
Exam
inin
g the
ac
cura
cy
of the
second
main
effect
and
inte
ract
ion
comp
onen
ts
to their
resp
ecti
ve X2
dist
ribu
tion
s
Cd
on
<N "H C d
WI
•H
(M ‘HC d
WW
O
CD CD rx r— ix V- cn cn CO «3* CD CD *3* cn cn*3" cn cn in CM CM o r- CM □ o o cn cn cn CMcn cn cn cn cn cn cn cn cn cn cn cn cn cn CO cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . • . . . . . . . . . . . . • . • . .□ o o o
un cn cn CM cnCD cn in CM in
cn cn in in in incn cn cn cn cn cn
. . . . . .□ a
<c— V cn CM incn r— CD «3" o
a cn a cn cn acn CO cn co CO cn
• • • • • •
cn CO cn CD ix cn CM CM cn CO ix r - CM ^ — r xcn <3- CM CM V - □ I X CM o co in CM r -in in LD in in in in >3* in in in in incn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
o■ • • •
a• * • •
□■ ■*
«3- cn CO cn rx cn t- CO r- cn «ct-o cn CO CM cn cn CO r— CM in cn cn a aa cn cn o □ CO CO cn □ a cn CO cn a ocn CO CO cn cn co co CO cn cn co CO CO cn cn
a• ■ • •
o• • • •
□
ix IX in cn co CO 3* *3" cn in IX cn 'v in CO ix <3* cn CDin cn CM CD in a CM a co cn CM CM CM CM a cn
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn COcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
t—\ . . . • . . . • . . . . . . . . . . . . .2 CD o o o o\
(N •C d CM rx in rx T- in in CD CD '3" cn r - <r- ix co1 CO CM CM cn cn 3" a V CO in in cn CM cn in CM '3" CM V
>—. in un in in in in in in in in in in *3* in <3" in in in in incn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
•r“ )C m . . . . . . . . . . . . . . . . . . . .o a a a CD
2 ’\
PI ■ CD V CD CM CD «3* CD cn CM in 'r* in COC d cn un IX in a in CM CD r— in cn CD □ in in IX>_> o o cn o O CD CD a a a CD cn cn cn a CD □ cn a cn aw cn cn CO cn cn cn cn cn cn cn cn CO CO CO cn cn cn CO cn CO cnv-- > . . . . . . . . . . . . . . . . . . . . .CO o a a □ □
CM cn *3- un o CM cn 3* in a CM cn *3” in CD CM CO '3_ in ac~ r~ r—
Ci(0 O (_>
«—> bO > cn o <—i CDLD •H u c n T“ in —
CD 0 •H 2> 0 in JO i—> > > > i-O "D JO •P i—i JO JO JOv—t o •H i—i >_< <—< v—>
cn 3 0UJ cn in o cn cn CO
> inCL LU >> JO JO c JO > > >e: m jo •p o a JO JO JO< M CM •H •H 0CO CO CM *_! 3 •P 0 CM cn CO
- 213 -
o E-P □
TO•P 0c 00 Pc <4-oa H -e □oo 0
0rH 00 PD bO
•a 0•H TOif )0 <— >
P Y0 £JO+J cr
CLO JO
4-5> i •Ho 20p CD oo •Ha 4-50 OJ
JO0 •H
JO P4-5 -P
0bO •HC . TO
•HC n•H£0 0X 4-5LU •H
CO
a
LU_ICD<
in 3" CO cn CM cn T- CD cn ■3* in CO X 3" CM X cn COco in 3" CM a CO CM O CO co IX in CO CM r— in 'T~ V- 'T~
CD cn cn cn cn cn cn cn cn cn CO cn cn cn cn cn cn cn cn cn cncn cn O) cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn■ • • ■ ■ • • ■ ■ • • 9 • 9 ■ • ■ 9 • ■ ■o o a o a
. CD T- V V in CM CO V V CM cn in X CO cn•<—) CO CO CO CD 3" cn cn V“ in CO <3* O <3- <3- V- 3" CO CD 3*•H in rx in in in in CD <3* m in in X CD in in in CD 3" 3* in in
2 \ /c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn• • • • • • • • • • • • • • • • • • • • •
a o □ a o•>—>
(N "Hcrv-- 'w CO CO CM CD in •3* CO in in CM 3* CD CD CD in CO CO CM COw a CO in <3* 3 - cn IX CO CO CM CD r— CD o X X CO 3*1 a CO a a a o CM CO cn a CO CM a a a cn T- CO CO cn O2 cn cn cn cn cn cn cn CO CO cn cn cn cn cn cn co cn CO CO CO cnv—f ■ • ■ • ■ ■ ■ 9 ■ ■ 9 • 9 ■ ■ 9 ■ • • • 9
- X a o a a o
CM CO 3^ in □ CM CO *3* in a CM CO 3" in o CM CO 3* in ot- t-
COi0 □
<—> bfl > CO CO ,—i ain ■H P c II 'T- in r—
0 0 •H 20 0 JO <—> >1 >» >
JO 10 JO 4-5 i—i JO JO JOV__4 o •H i—1 '__ i %_* <_>
CO 3 0LU CO in CJ CO CO CO> 0Q. LU >1 JO JO C JO > CO >1 in > □
e : ix JO 4-5 o 0 JO CD JO 3" JO cn< M CM •H •H 0 II n iiCO CO CM w 3 4-5 0 CM Z CO 2 CO
- 214 "
TABLE
10.3
.12
Comp
arin
g the
ac
cura
cy
of the
test
stat
isti
c ag
ains
t its
im
prov
ed
stat
isti
c for
exam
inin
g the
first
main
effe
ct,
p
V CD CD in ix in CM CO co IX in rx CM CM in <3* IX'3* CM T~ CM a in CD CM o in CM a cn CD cn CM V 'T-
CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD COCD CD CD CD CD CD CD CD CD CD CD CO CD CD CD CD CD CD CD CD CD
• ■ ■ . . • • . • » . . . . . . . . . •o CD a CD o
CD CM CO CD CM CM CO CM <r~ V CO CO r - V COa CO in in v~ CO in CO in cn □ CD IX >3* CD o CM cn 3" CO
in CO in in in in in in in in in CO in in in CO in in in X"^ co ? CD CD CD CD CD CD CD CD CD CD CD CD CD cn CD CD CD CD CD CDIX £1 . • • • . • . . ■ • » . . . . . . . . . .PQ o o a a o
CO CD ■sf CO r— cn CD cn <3" CM CO v- IX CM <r-o CD CO cn CD o CD CO CO «3- cn <3- CO a in IX CD
a T— CD o □ a o r- CD a a CD CD a o CD CD CD a CDCD CD CO CD CD CD CD CD CD CD CD CD CD CD CD CO CD CD CD CO CO• . • . . • . . ■ . » . . . » . . . ■ . .o a o CD CD
cn r - in CO rx in in CO CM CM OD CO CD CM cn CO COi \ CO cn cn ix CM «3* CM CO in cn CM CO *3* CM cn CD
CD CD CD CD CD CD CD CD CD CD CD CD CD cn CD CD CD CD CD CD COCD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD
• ■ * . . • • . . . » . . . . . . . ■ . .□ O □ CD a
LD
C\J
O r- in T- CM CD CM CO <r~ CO IX V- CM <3* CD T“in CO cn CD cn CM CO CO CM in CO r— CD o cn CM CO IX CO CMV_/ LD rx CO CO in in (X CO LO CO in rx IX. CO CO in IX CD in in in
> CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD coc c ■ . • . . . . . . . . . . . . . . . . . .□ □ a □ a CD
•H4->(DDCJ ix CO rx 'Cf CM CO in rx CO CD CM CM < - cn CD V IX CM CM
UJ in cn in CO CO *3* CD CD CD *3* <3* IX IX CD CD CO CDCD cn T— □ cn CM O O a cn CM t:— V CD cn CM o a a
CO CD CD CD CD CD CD CD CD CD CD CD CD CD CD CO CD CD CD CD CD CDIX . • • . • • . . * . . . . . . . * . . . .a a o CD a
CM cn in o CM cn <3" in CD CM cn in CD CM cn *3* in CDV V
t_N«—> a ,—, CDin V in
> - > > >JD JD JD JD
\— > v_/ V_/
lli cn cn cn cn_iOL UJ > O > CD > in > os: rx JD cn JD CO JD >3* JD CD< H ii II ii IIcn cn CM 2 CM 2 cn 2 cn 2
_ 215 _
TABLE
10.3
.13
Comp
arin
g the
ac
cura
cy
of the
test
stat
isti
c ag
ains
t its
im
prov
edst
atis
tic
for
exam
inin
g the
second
main
effe
ct,
q
CN CO cn rx CO LO V- CN CO IX IX cn V V CN cn V“<3* CO a □ in CO t— CN CO *3" CN CN co CO CO a CN
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn CD cn CDcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . » . . • . . .a o o a CD
CO CO CO CN CO CO v- CN CO V“ CO CO CO CM CO *3*CN *3* CD CO □ in in <3- CD CD «3* in rx CO CO CN V-
cn CD CD in in in in CD in in in CD CO in in in in CO in in in)( cn cn cn cn cn cn cn CD cn cn cn cn cn cn cn cn cn cn cn cn cn
cn '■ . » . . . . . • . . . . . . . . . . . . .N o o □ a CDCCS
CO CO CN cn IX cn <3" IX CN '3" v - inv CO in «3* CN cn <3* (X CO co a in in <3- CO a CN •3-
□ o □ a □ r— □ o a o r— a □ CD <r* r" CD CDcn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. . . . . . . . . . . . . . . . . . . . .o □ CD o o
CO CO CO IX CO <r~ CN r - CO CN CO CN V- cnrx rx CN CN CO CO CO CO CO CN CO CO <3* CN CO IX CD CO co ’T-
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn___ cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnCD • . • . . . • . . • . . . . . m • . . . .a o □ □ CDCN
Or~
CD <3* CN in in in *3" CN rx CO V“ CN CO cncn o CN CD co IX CD in o CO cn 3" cn CO CO *3" in CO co
C rx rx CO in in rx rx CO CO CO rx CO CO in in IX CO CD in in° ? c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
•H C- . . . . • . . . . . . . . . . . . . . .-P CD a a a □(DDcrUJ
un in rx CO CO CO rx in CN in 3" V CO CO IXcn CO cn a CO rx LO CO CO CO CO CD IX IX IX IX CO cn CO
cn CD CO CN t- r— a CO CN a CO CN a CO CN V— CD CDIxl cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn CD. • . . . . . . . . . . . . . . . . . . .a CD CD CD CD
CN CO <3- un O CN CO in o CN CO *3* in o CN CO in CDV" T“
<_(i—. CD i—> CDun V- in
> > > >JD JG JD JDw w
LU CO CO CO CO_I CDn_ lu > □ >i o > in > □z: m JD CO JD CO JD JD cn< M II II ii nCD CD CN 2 CN 2 CO 2 CO 2
- 216 -
CDszp
+->cn pc G
•H CDCD <PbO q-(D (D
CDCr - O
N •Hp
G G•H CDP Pcn 0
*H PP cCD •H
Pcn CD
.Cp PcnCD bO
p C•H
CD Cj= •rHp E
CDq- Xo 0
> Po oCD q-pD GG •HG PCD 0
•HCD P
JD CDP P
0bflc "O
•H 0P >CD oQ. pe ao E
CJ •H
in . in . CO CD <3* V - V CM IM cn ID [ \ IM *3* CO V - ID ID CDCO CO V CO cn ID < r- CD CO CM cn CM V - CO CM CO CD CD
cn cn cn cn cn CO cn cn cn cn cn cn cn CO cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
• • ■ a a a a a a a a a a a a a a a a a a
o a o CD CD
un 'T~ cn CD CO ID «3- CO CO V* C D *3“ CO T— fM fN CDo V - v~ cn cn CD CM < 3- CM IM LD CD <3- CM CD CO IN. ID CO
L D CD co un LD <3- ID ID ID ID ID ID ID ID ID ID ID ID ID ID IDa ) cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnV - *■ . • • a a a a a a a a a a a a a a a a a a
M a a CD o CDcq
! \ CO IM ID CD cn r - IM IM CM [N. ID cn LD cn ID V - IN.cn V CM ■sf- CO CM cn ID on CD CO cn CO CD CO CO ID cn
o o V CD O o o a CD CD cn CD cn cn cn CD cn a CD cncn cn cn cn cn cn cn cn cn cn cn co cn co CO CO cn CO cn cn cn
• a a a a a a a a a a a a a a a a a a a a
a CD o CD □
CD cn r — r - V [N. CM CM LD LD C N cn CM cnN LD CO ID o CO CD *3" CO CM IN . C N >3- «3* CM ID CD •3- CO r -
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cncn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnr • • a a a a a a a a a a a a a a a a a a a
o □ CD □ CD
CM
ar r V “ CM IN. V* ID (N. CM CO CO IM CM CO CN CM V"w r \ r - IN . IM CM CD CO CM CD 'Cl CM CO cn v- <3- CD CD CD cn CO
m rN [N. LD ID ID CO CD CD CD io CO CD LD CD LD rN CD CD ID IDc ^ cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cno 1 • a a a a a a a a a a a a a a a a a a a
•H o a CD CD CDP00cr
UJ CM T- CD IM <r~ CD CO T- V - CM *3* IM CO CO CM CM cn <r~a CO un CO CD CO CO *3* CO CD CO CO ID CO CO IN . CM <3- «3- L Da a <3- CO r ~ a CO CM v CO CM o CO CM O
^ r - cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cnM • • a a a m a a a a a a a a a a a a a a a
a CD CD CD CD
CM CO ID a CM CO *3- ID o CM CO *3* ID CD CM CO <3" ID ar~ T- 'r *
/—\<—> cn ,—1 CDin ID
> > > >JD JD JD JD'—’ *— > s_/ s_/
LU CO CO CO CO_i cnCL LU > CD > o > ) ID > CDsz n JD CO JD CD JD *3* JD cn< M II ii II iicn cn CM z : CM 2 CO 2 CO 2
- 217 -
10.4 Randomised Complete Block and Two-way Classification with Interaction
Designs for Small k
As discussed in Sections 9.7 and 9.8 only the associated component chi-squared
approximation may be used as a test of difference, since the corresponding total and
residual components, for small concentration parameter, do not form chi-squared
distributions. For the randomised complete block and two-way analysis designs with
small k the models may be seen as given in (10.2.1) and (10.2.4) but the total and
residual terms may not be assumed to be chi-squared distributed. From (10.2.1),
(10.2.4) and Section 9.2 the test statistic to examine the null hypothesis that there is
no difference between the p treatments gives:
which has a chi-squared distribution with 2 (p -l) degrees of freedom.
Similarly, the test statistic Z 1 2 will be produced to test the null hypothesis that there
is no difference between the q blocks within the randomised complete block, or q
treatments within the two-way design:
which has a chi-squared distribution with 2 (q -l) degrees of freedom. Testing for
differences between interaction terms within the two-way analysis gives test statistic
( 1 0 .4 . 1 )
( 1 0 .4 . 2 )
As for large concentration parameter, test statistics (10.4.1), (10.4.2) and (10.4.3)
indicate whether or not . the mean directions are equal, they do not allow for
discrimination amongst single mean directions.
10.5 Accuracy of the Associated Chi-Squared Approximations for Small k
The accuracy of the test statistics (10.4.1), (10.4.2) and (10.4.3) are determined by
simulation methods. The Monte Carlo samples from a von Mises distribution are
generated by the computer method described in Appendix B. As for the testing of
large k, 10,000 sets of samples of various size were drawn with varying concentration
parameter values. Before the tests are applied the equality of concentration
parameter is checked using test statistic (4.5.2). Ranges of small k are given as its
accuracy cannot be assumed for such small values. The test statistics are examined
with equivalent sample sizes and components within the two-way classification, since
the main effects for both the randomised block and two-way designs are similar.
Table 10.5.1 gives the first two moments of the chi-squared approximations for the
two main effects within the two-way design. The simulated mean value for each
component is seen to be a very good fit to the mean of the chi-squared
approximation. The variances are also good fits, increasing in accuracy as the size
of sample and k increase. Table 10.5.2 shows the accuracy of the first two
moments for the interaction term to the expected values. As for the main effects
the simulated means and variances are comparable to their associated chi-squared
approximations notably increasing in accuracy as N and predominantly k increase.
Tables 10.5.3 and 10.5.4 compare the accuracy of the test statistics, Z 11t Z 12 and
Z 13 to their corresponding chi-squared distributions in the upper 10 percent
significant levels. The goodness of fit for the tests with concentration parameter less
- 219 -
than 1 are seen to be very good for all the test statistics, and as for all previous
goodness of fit examinations, the accuracy is seen to increase as the size of sample
increases.
Tables 10.5.5 and 10.5.6 take the test statistics out of their proven range and
examine their accuracy for concentration parameter values of 1.25 and 1.75. The
test statistic for both main effects and interaction terms are tested. The tables
clearly indicate that the test statistics show very good fits to the upper 10 percent
significance levels and are reliable tests for experimental designs with concentration
parameters in the range 1 to 2.
- 220 -
TABL
E 10
.5.1
Comparing
the mean and
variance of
the
x2 approximations with
Cp-1)
and
(q-1)
degr
ees
of fr
eedo
m to
their
expe
cted
va
lues
, for
smal
l.
(N ■ CDCL 0v—/ c /—\ t—\ /--* /—* t—\ /—. /—v
1 (0 CD LD CM LD cn ^3- v -r—> •H CD rN •3" LD 3" CO O CD
• p CO CM rN ID a CM CDID . • • • • • • .
• > CO *3* CO IN CO CO INV_' w *_i X_> \_* V__ ! v—> V__>
CM •cr>—' <3" LD cn cn rN CD ow C CO fN □ CD 'C~ CM\_/ (D cn cn o CD O o □ CD/—* CD • . • . . . . .CM e : M— CM «3* *3* >3*
v_;
\CM
*3* *3" CO COt— \
cn II II II ii
> CM 0 CM 0 3* 0 ^ r 0jd O a 0 a'_! T3 II c TJ II c "0 II c ■a n c
0 (D 0 0 0 0 0 0ILI ' i ’ CO -P C •H -P C •H -P C •H -p c •H_J O (0 P O 0 p O 0 P a 0 0CL LU > 0 0 CO 0 0 0 0 0 0 0 0 0e : INI JO Q. rr- > CL ZZ > CL zz > CL e : >< 1—1 X X X Xco CO CM LU LU LU LU
cr
fM *Hcr
i
CM
CD0c<D•HP0>
<3- CO CO LD LD LD
CD in CD CD□ cn •3- CO
<3- CD IN . IN IN . CO
I N CO
<3- CM CO COo o<3* *3*
CDCM ^ LD CO
LD ID O CD O
CO *3*
CD v - < r- CO O coCO IN
CO CO □ IN O CD
<3- co
ID a IDCD ID a ID CD. . . . . . . .
a r— a CD CDtr-i i i 1 1 i 1 1
a ID a LD CDID CDLD. . . . . . . .
a CD CDO a O CDCD
CO CO co t_i CO<—i a ■—> aID ii r- ii LD n n> '3' 0 > *3* 0 > <sf" 0 > <3* 0JD o JD 0 JD 0 JD 0
<—< TD 11 c v__t TD II c >—• TD II c w TD II c0 0 0 0 0 0 0 0LU CO •P C •H CO -P C •H CO -P c •H CO -P C •H_! a 0 P 0 0 P 0 0 P 0 0 PCL LU > 0 0 0 > 0 0 0 > 0 0 0 >1 0 0 0e: INI JD CLe: > JD CLe: > JD a. e: > JD CLe: >< H X X X XCOCO CM LU CM LU CO LU CO LU
- 2 2 1 -
TABLE
10.5.2
Comparing
the mean and
variance
of the x2
approximation
with
Cp-lHq-1)
degrees
of freedom
to their
expected values in
cn cm m o •• in
o cm
^ v -co cn
cm cn cm cn <3- co
cn cm co cn in □
CM *H
co in cn cn
CM t- cn o cn o
co in co <3- co ocn cn co
CM
in o in o in □
□ in o ino o □ o □ o o o
co coCO COo oin in
co CO QJ
cn cn •p cn p cnQ_ LU
EZ M < Mcn cn
n cl e: cl JO □L
CM cn cn
- 2 2 2 -
TABLE
10.5
.3
Examining
the
accuracy
of the
x2 approximations for
the
two
main
effect components,
for
small
kCO(X CO CDrx
O cn CDCD 'T~ a o,—i cn CDCO COcn ODcn cn cn2 cn cn cn cn cn cn cn cn cn. . . • • • • • •• o a □ a CD
N ■DCl
□ CO COCO CNv-2 • cn cn cn cn CNCD V” in
/1X1 LD un in un un in< cn cn cn cn cn cn cn cn cn
'r~3 . ■ • • • • • • •n • o a a a CDDCw
a CDcn co CD cnCOCN cn o cn COcoo cn o cn CDcn cn acn COcn COcn cn CO COcn\ . • • • • • • • •CN] a a a CD a
cn COrx COCN IXcn «r* CDrx cn cn
S cn cn cn cn CO cn cn COco\ cn cn cn cn cn cn cn CDcn• . . . . . . . . .
• a CDa CD CD ori •q:i
•H COV- COv- ix. CNcnCO a cn cn CD□LD LD LD un un <sf <3* in, i cn cn cn cn cn cn cn cn cn, i . . . . . . . .a #H o □ a CD amLJW\—//—*N COCN ix CO CNIXr- cn in un cn CDt-v - a O □ a cn CDa CDCD'—> cn cn cn cn CO cn cn cn cn\ • . • • • • • . •CN □ o a CD CD
cn CD un a in CD un CD. . • . . . . .□ CDr- CD CD v -i 1 1 1 1 1 1 1a LD CDun a un O in
. . . . . . . .a □ CDo CDa CDCD
C•H
jC JZ4->-P• H •Hs 3 □C cn cn ,—> «—iCxOC II CD <—. CD
CD -H O 2 V un Vcn •H>1 CD+J ,-N >i > >JD TJ (0 >—I JD JD JD\_/ > rH v—/ V_/ s_>
cn 0 0LU cn 0 O cn cn cn_I > 0CLLU > JDJD JZ > CD > tn > CDz: M JD Q 0 JD CO JD JD cn< M CN 0 II n ncn CD CN w un 0 CN 2 cn 2 cn 2
223
TABLE
10.5.4
Examining
the accuracy
of the
x
2 approximation
for the
interaction
component
for
small
k
Ncr
cr
v.N *Hcr
• | - 3•Hz
\
01 *Hcr*—tww
\CN
COCD
y un 0 CD
oCD
un r \ cn co cn co cn co
co rN cn coCD CO CD CD
CN CO CO CO CD CO CD CO
O
COCN CN CD CD CO CO
COun cn in in CD co
cd
■e* IN CO CDun <3- cn cd
co coCO INun <3-CO CD
CD
CO 3" v-un unCD CD
CD
907
90
53
900
6 <3-COCDCO CD
COCO 9
07
2
CNCOCO. . • . . . ■ .
□ □ CD CD
in a un CD in CD un CD. . • . . . . .□ CD a a v"O in CD 1
in □1
inio 1
un. . • . . . . .□ a CDa CDa CDa
tu_JCL LU Z INI < H CO CO
C•H
JC JD•p •P*rH •H2 3c 0 CDM C CO CO in o
in -h o II0 •H z >
> 0 -P /—1 > JD >ID TD 1—1 JD >_> JDW > i-H 1__ V_l
CO P 0CO 0 o CO CO CO
> 0> JD JD J Z > O > in > CD
JD Q o JD CO JD JD CDCN 0 II it II
CN ^ in 0 CN z CO z CO Z
- 224 -
TABLE
10.5.5
Examining
the accuracy
of the
x
2 approximations for
the
two main
effect components,
for
k in
the
range
1 to
2
IS. 'T- co CM•—> CO CD CD MT“ m un rs IS2 : CD CO CO CO CO CO CO CO CO
CD CD CD CD CD CD CD CD CD• • . . . . . . • •• o □ a □ o
(N •c ri
■ CO rs CO un *3* CD CDun CO CD rs CO V“ CO OJ Ln 1* M" 'd - CO un
• 5 CD CD CD CD CD CD CD CD CD• • • . . • • .
CM • o □ O o ac r'—’
CMCO r - IS. CM v - '3 -
1 ^ r un a un CD o CDo CD CD o CD CO CD CD oCD CO co CD CO CO CO CO CD
• • • ■ ■ • . . •CM o o o a a
CD M" is . CO CM CO CO>—* r — IS CO a rs un V- un-2L CD CD CO CO CO CO CO CD CON CD CD CD CD CD CD CD CD CD
• • ■ • . . . . • •• a □ a o a
CM •c ri
•H V- CO T“ VLn CO CO CD IS O un
un *sf- <3* un• CD CD CD CD CD CD CD CD CD■ • . . . . . . . .
CM "H □ a o o ac rw
01a , CM CO CD CD IS COi CD is . un CD CD IS CD
o CO CO CO CO CO CD CD CD'—> CD CO CO CO CO CO CO CO CO\ ■ ■ • • • . . . •CM o a a o o
un in un un un in un unCM rs CM is CM IS. CM IS
• ■ • • . • . •v~ V- V V V V-
C•H
JZ JD-P -P•H •Hs 3
C cn a (_1<-> ClO c CO a ,—> aun -h o n in r -
cn •H 2> 0 -P <—. > >> >
JD TD in i—i JD JD JD> i—i \—> v_/
CO p 0LU CO 0 o CO CO CO__I > cnQ_ LU > jd jd JZ > a > un > ae : M n o u JD CD JD «cf JD CD< M CM 0 n it itcn cn CM w un 0 CM 2 CO 2 CO 2
_ 225 _
TABLE
10.5
.6
Examining
the accuracy
of the
x2 approximations for
the
interaction
component, for
k in
the
range
1 to
2
)
Ncr
cr
cncn
CD
(M *rHcr
incno
•H
\■|— 3•H
cr
\CN
Ocn
LU IQ_ UJz: m< Mcn cn
in cn cn i \ co co cn cn
co <r- o cn in <3- cn cn
CNcn un r\ co cn cn
CD CNco ^r co co cn cn
in cn cn co co co cn cn
co co cn <3- co <3- cn cn
□
CO rs. cocn cn
IN.o cnin •vj-cn cn o
IN. <3* r~ CO IN. COV CN in Cn CO CN CN r~O CD cn cn cn cn CD acn cn CO CO CO CO cn cn
• • • . . . • .CD a CD CD
in in m in in in in inCN [N. CN IN. CN IN. CN IN.
• • • ■ • • . •V- V V - r - v " V- * “ V-
C•H
JZ JZ-P 4->•H •H2 2
c cn CD/—\ b fl c CO a <—i Oin •H o II V in
cn •H 2> m -P (—1 > > >
JD TD ID 1—1 JD JD JDw > 1—I V_/ V_! V_>
CO t-. 0CO 0 o CO CO CO
> 0JD JD JZ > CD > in > O
JD O a JD CO JD 'T* JD cnCN 0 II II ii
CN w in 0 CN 2 CO 2 CO 2
- 226 -
10.6 Summary
This Chapter has investigated the adequacy of the generalised approach for the
randomised complete block and two-way classification designs, with particular
reference to their accuracy compared to the associated chi-squared and F
approximations. The randomised complete block and two-way classification designs
have been used to illustrate the validity of the approach for larger more complex
design situations.
Section 10.3 has show that for large k (k>2), and with the inclusion of the
correction factor /3, the new test statistics are reliable and show excellent
approximations to their associated F distributions. For small k only the main factors
may be tested against their corresponding chi-squared distributions, nevertheless, the
tests show excellent fits for a concentration parameter range of 0 to 1.
Of most interest is the examination of the test statistics in the concentration
parameter range 1 to 2. Ideally the test statistics for large k would be desirable
since measures of total and residual variation may be found and utilised. However,
the test statistics for small k show very good distribution fits and indicate that these
tests should be used for all experimental situations where the concentration parameter
is in the range 0 to 2.
- 227 -
U t lA T li iK 11
ANALYSIS OF VARIANCE EXAMPLES FOR CIRCULAR STATISTICS
11.1 Introduction
Having discounted the possibility of extending the previous one-way analysis of
variance approach to larger designs, we will now proceed to test the new approaches
and their validity with real data sets. The designs range from the simple one-way
design to the Graeco-Latin square and split plot designs. This section not only
demonstrates the application of the new procedures but also indicates how they may
be used to analyses other designs not discussed in this thesis.
11.2 One-way Analysis
Example 11.2.1 for Large k
For a one-way classification analysis with large k an example given by Gadsden and
Kanji (1983) will be used. The examination concerns the orientation of particles in
clay strata observed from photographs for various magnifications. The data, given in
Table 11.2.1, has been reproduced from Gadsden and Kanji (1983).
- 228 -
Table 11.2.1 Raw Data From Photographs
Magnification:100 8 2 , 7 1 , 8 5 ,8 9 , 7 8 , 7 7 , 7 4 , 7 1 . 6 8 , 8 3 , 7 2 ,7 3 , 8 1 , 6 5 , 6 2 , 9 0 ,
9 2 ,8 0 ,7 7 ,9 3 , 7 5 , 8 0 , 6 9 , 7 4 , 7 7 , 7 5 , 7 1 ,8 2 , 8 4 , 7 9 , 7 8 , 8 1 , 8 9 , 7 9 , 8 2 , 8 1 , 8 5 , 7 6 , 7 1 , 8 0 , 9 4 , 6 8 , 7 2 ,7 0 , 5 9 , 8 0 , 8 6 , 9 8 , 82,73
200 7 5 , 7 4 , 7 1 ,6 3 , 8 3 , 7 4 , 8 2 , 7 8 , 8 7 , 8 7 , 8 2 ,7 1 , 6 0 , 6 6 , 6 3 , 8 5 ,8 1 , 7 8 , 8 0 , 8 9 , 8 2 , 8 2 , 9 2 , 8 0 , 8 1 , 7 4 , 9 0 ,7 8 , 7 3 , 7 2 , 8 0 , 5 9 , 6 4 , 7 8 , 7 3 , 7 0 , 7 9 , 7 9 , 7 7 , 8 1 , 7 2 , 7 6 , 6 9 ,7 3 , 7 5 , 8 4 , 8 1 , 5 1 ,76,88
400 7 0 ,7 6 ,7 9 ,8 6 , 7 7 , 8 6 , 7 7 , 9 0 , 8 8 , 8 2 , 8 4 ,7 0 , 8 7 , 6 1 , 7 1 , 8 9 ,7 2 , 9 0 , 7 4 ,8 8 , 8 2 , 6 8 , 8 3 , 7 5 , 9 0 , 7 9 , 8 9 ,7 8 , 7 4 , 7 3 , 7 1 , 8 0 , 8 3 , 8 9 , 6 8 , 8 1 , 4 7 , 8 8 , 6 9 , 7 6 , 7 1 , 6 7 , 7 6 ,9 0 , 8 4 , 7 0 , 8 0 , 7 7 ,93,89
1200 7 3 , 9 0 , 7 2 ,9 1 , 7 3 , 7 9 , 8 2 , 8 7 , 7 8 , 8 3 , 7 4 ,8 2 , 8 5 , 7 5 , 6 7 , 7 2 ,7 8 , 8 8 , 8 9 ,7 1 , 7 3 , 7 7 , 9 0 , 8 2 , 8 0 , 8 1 , 8 9 ,8 7 , 7 8 , 7 3 , 7 8 , 8 6 ,7 3 . 8 4 . 6 8 .7 5 . 7 0 . 8 9 . 5 4 . 8 0 . 9 0 . 8 8 . 8 1 .8 2 . 8 8 . 8 2 . 7 5 . 7 9 , 83,82
400x1.3 8 8 , 6 9 , 6 4 , 7 8 , 7 1 , 6 8 , 5 4 , 8 0 , 7 3 , 7 2 , 6 5 ,7 3 , 9 3 , 8 4 , 8 0 , 4 9 ,7 8 . 8 2 . 9 5 . 6 9 . 8 7 . 8 3 . 5 2 . 7 9 . 8 5 . 6 7 . 8 2 .8 4 . 8 7 . 8 3 . 8 8 . 7 9 ,8 3 , 7 7 , 7 8 , 8 9 , 7 5 , 7 2 , 8 8 , 7 8 , 6 2 , 6 8 , 8 9 ,7 4 , 7 1 , 7 3 , 8 4 , 5 6 , 77,71
For the photograph magnifications we wish to test the null hypothesis
H0 • ^ o , i “ ^0 , 2 ” .............“ ^o , q
against the alternative hypothesis that at least one of the equalities does not hold.
Prior to testing the hypothesis it is necessary to test the assumptions that (a) the
samples are drawn from a von Mises population and (b) the concentration parameter
k has the same value in each sample. Each of the sample populations have been
tested by Watsons U 2 statistic (1961), using the critical values supplied by Stephens
(1964), to show von Mises distributed data sets. Similarly, the homogeneity of the
concentration parameters have been tested and validated via test statistic (4.5.5)
(U 3= 5.584 distributed as x |)*
- 229 -
The one-way analysis components of variation are given by:
R2 r q R2 . •J
R2 r qv R2 . •Jk N ------— - k y - —LL + k n - y
N LL j - i N . i N L
L J - i N . i
and the modified test for the null hypothesis will provide an F-ratio:
F ( q - l ) ( N - q ) - P
(N-q)
r -
\*T
c—
R2 .•J
N . i
R2 1
N
( q - Dr q C H - J
L j - i
CM •
•f*.
a.
(11.2 .2)
The analysis of variance for Gadsden and Kanji's data is given in Table 11.2.2.
Table 11.2.2 Analysis of Variance Table
Source o f V a r ia t io n d . f . Measure o f V a r ia t io n
Between Photographs q - i H r j 1 r ?.
j - i k j J N
W i th in Photographs N-qS ("r2 .1
n - y , j• 1 n ij “ i
To ta l N - lR2
N ------—N
Stat is t ics ij - 1
R2228.1931
N JJ
R2
N227.6244
q=5 N . 1 N. 2 N.q - 50 N - 250
- 230 -
which gives the following ANOVA table:
Table 11.2.3 Analysis of Variance Table:
Source o f V a r i a t i o n d . f . Measureo f
V a r i a t ion
MeanMV
F
Between Photographs 4 0.5687 0.142171.5974
W i th in Photographs 245 21.8069 0.089
Tota l 249 22.3756
1 1 1 ,k - 11.02 - - 1 ------------------------------ or 0 - 1.01959
f3 5k 10k2
The value of k is found via approximation (3.3.11) for large k. The modified F'
value is 0 x F 4>245 = 1.628; and as the table value of F 4>245(0.05) = 2.37, the
result is not significant and we can conclude (like Gadsden and Kanji) that there is
no observed significant difference between the orientation of the clay particles under
differing photographic magnifications.
Example 11.2.2 for Small k
An example of the test procedure for small k is given by using Mardia's (1972)
example on wind directions in degrees at Gorleston, England, at 11 hr - 12hr on
Sundays in 1968 classified according to the four seasons. The data has been
reproduced from Mardia (1972).
- 231 -
Table 11.2.4 Wind Directions in Degrees at Gorleston on Sundays in 1968
According to the Four Seasons
Season Wind D i rect io n s in Degrees
W inter 5 0 ,1 2 0 ,1 9 0 ,2 1 0 ,2 2 0 ,2 5 0 ,2 60 ,2 90 ,29 0 ,3 2 0 ,3 2 0 ,3 40
Spr ing 0 ,2 0 ,4 0 ,6 0 ,1 6 0 ,1 7 0 ,2 0 0 ,2 2 0 ,2 7 0 ,2 9 0 ,3 4 0 ,3 5 0
Summer 1 0 ,1 0 ,2 0 ,2 0 ,3 0 ,3 0 ,4 0 ,1 5 0 ,1 5 0 ,1 5 0 ,1 7 0 ,1 9 0 ,2 9 0
Autumn 3 0 ,7 0 ,1 1 0 ,1 7 0 ,1 8 0 ,1 9 0 ,2 4 0 ,2 5 0 ,2 6 0 ,2 6 0 ,2 9 0 ,3 5 0
Do the wind directions for the four seasons differ significantly within the given data
set? In this case the test statistic to be used is given by;
2r q
y R2 . ‘ J
R2
- P2 L• ja l N- j . N . ^
( 1 1 . 2 . 3 )
The associated circular statistics are given in Table 11.2.5
Table 11.2.5 Statistics for the Wind Directions
Season N. j R. j ^ o , jA
R. j
W inter 12 5.1185 272* 0 .94
Spr ing 12 2.1321 330* 0 .36
Summer 13 3.8680 57 ’ 0 .62
Aut umn 12 3.1878 232* 0.55
Combined Sample 49 5.8771 292* 0 .24
iJ-l
R2 .
N JJ
4 .5 5 9 8R2
N0.7049
- 232 -
The values of k j are found via the 'simple' approximation (3.2.11)
R
M*0 “ 2 —
CN15
N ." . j .
Prior to examining any difference between the four seasons the assumption that the
concentration parameters are equal must be tested. Test statistic (4.5.2) is used to
test their homogeneity and was also used by Mardia (1972). A statistic value of
U , = 0.6013 (distributed as x§) indicates that the concentration parameters for the
wind directions may be regarded as homogeneous.
Applying the test statistic leads to
iJ - l
RJ
N -j .
I , ( & )
l 0W
R2
N
and
3.8549
1 - p 2 2.0288
Hence
2 r sy R2 .
• JR2
- P2 LLj-i N.j. N
7.82
Table value x l (0-05) = 12.59
Therefore the wind direction for the four seasons are seen not to be significantly
different (the same solution as given by Mardia (1972)).
- 233 -
11.3 Randomised Complete Block Design
Example 11.3.1 for Large k
This example is taken from the field of clinical psychology and is an example given
by Ramano (1977). A psychiatrist was aware that certain organic compounds in the
fluid surrounding the brain will, when purified and separately placed into solution
rotate the plane of a polarized light source. The psychiatrist was interested in
determining whether the optical activity of a specific compound was measurably
different for various degrees of schizophrenia. Five distinct levels of schizophrenic
behaviour were recognised and for each level 4 patients were selected. Measurements
were taken as to the extent to which each sample rotated the plane of polarized
light under specified conditions.
In the original example a one-way analysis of the difference between schizophrenic
levels of behaviour was examined. Here an added factor of blocking will be
introduced as a 'row' effect where we may assume a blocking by, for example, age
of patient.
In the example given by Ramano, although the angle of rotation had been measured
a standard arithmetic mean had been calculated and a 'linear' analysis of variance
carried out. However, although this is an incorrect approach to the analysis, the
estimated concentration parameters are shown to be very large and therefore the
Normal approximation to the von Mises distribution may be used. Table 11.3.1
reproduces the data given in Ramano (1977):
- 234 -
Table 11.3.1 Optical Activity Measurements of a Specific Compound Contained in
the Brain Fluid of Persons Classified According to Levels of
Schizophrenic Behaviour
Blocks1 2
Leve1s 3 4 5
AngularMean
1 11.51* 12.80* 14.98* 15.71* 20.45* 1 5 .0 9 ’
2 1 1 .74 ’ 12.49* 12 .90 ’ 15.42* 1 9 .4 2 ’ 14.39*
3 1 2 .07 ’ 12 .01 ’ 14.25* 15.77* 2 0 .2 5 ’ 14.87*
4 13.15* 13.97* 15.27* 15.07* 17.17* 14.92*
AngularMean 1 2 .1 2 ’ 12.82* 14 .35 ’ 15.49* 1 9 .32 ’
Testing the homogeneity of concentration parameters utilises test statistic (4.5.5). A
test statistic value of U 3 = 4.7477 (distributed as x |) f ° r the 5 levels and
U 3 = 2.5107 (distributed as x§) f° r 4 blocks, indicates that the concentration
parameters may be regarded as homogeneous for both factors.
The randomised complete block components of variation are given by:
R2 r pv R? R2 r qv R2 . • J
R2k N ------— - k y l . - —— + k y _ —Ll
N L■ i -1 Ni - . N L
L j - i N - j . N
f PN - I
R?l .q
- 1R2 .
•JR2
Li » l Ni . L
J - l NJ . N
and the modified test for the null hypothesis of no difference between the q levels
will provide an F-ratio:
- 235 -
F( q - 0 . ( p - 0 ( q - 0 “ ^
r[R’ j |
R2(p**i) (q - i ) I ------—
LiLj-i k . j J N
r p R L q h ] R2(q - i ) N - I - 1 + —
LtL i - 1 k J Lt
j - i k j J N
( i i
For the null hypothesis of no difference between the p blocks
F( p - i ) . ( p - 0 ( q - 0 “ ^
r pv R2( p - l ) ( q - l ) I ------—
LaLi-i [Ni.J N
r pRi . '
qv h ]R2
(P-D N - I - I + —L t
i -1 LNi JLa
j - i k j J N
(11
The analysis of variance for the optical activity measurements is given in
11.3.2.
Table 11.3.2 Analysis of Variance Table
Source o f V a r ia t ion
d . f . Measure o f V a r ia t io n
Due tosch izophren iclev e l
Due to Block
q-i
p - i
Res idual ( q - l ) ( p - l )
iJ - l
P
ii-1
P
N " Ii -1
R2• j
LN J J
Ri .
N i .
R?l
R2
N
R2
N
q■ I
R2 . •J
R2
Lj - l k j J N
Tota l N - 1 N -R2
N
. 3 . 2 )
. 3 . 3 )
Table
- 236 -
r
Ii= l
R?JL
Ni19.9957
R2
N19.9564
iJ- l
R2 . • J
lN.Jj19.9568 p = 4 q => 5 N = 20
which gives the following ANOVA table.
Table 11.3.3 Analysis of Variance Table
Source o f V a r i a t ion
d . f . Measure o f V a r i a t ion
MV
MeanMV
F
Due tosch izophren ic leve 1
4 0.039263 0.009816 30.311
Due to Block 3 0.000414 0.000138 0.426
Res idual 12 0.003886 0.0003288
To ta l 19 0.043563
A A
From concentration parameter approximation (3.3.11) k « 459, as k is so large the
correction factor (3 may be neglected (j3 = 1.00043). Hence F 4>12 = 30.311 and as
the table values of F 4>12(0.05) = 3.26, F 4 1 2 (0.01) = 5.41, the analysis shows that
the optical activity of the compound differs significantly for schizophrenic behaviour
recognised by the psychiatrist.
Testing the difference between the blocks gives F 3>1 2 = 0.426, as the table value of
F 3, 12(0*05) = 3.49, the analysis indicates no significant difference between the 4
blocks under investigation.
- 237 -
Example 11.3.2
This example of a randomised complete block analyses a hypothetical data set where
the concentration parameter is close to two. As was discussed in Chapter 6, a data
set of this nature may not be analysed via extension of the original techniques.
Table 11.3.4 shows the hypothetical data set which will be used to illustrate the
application and robustness of the new approach.
Table 11.3.4
Block1 2
Treatment 3 4 5 6
AngularMean
1 311' 299’ 338* 298’ 286’ 305* 305 .966 ’
2 326* 39’ 354* 47* 10* 354* 8.283*
3 10* 45* 309’ 319* 25* 339* 345.974*
4 55* 48 ’ 54* 17* 29* 69* 4 5 .4 4 0 ’
AngularMean 353 .9 8 ’ 7.657* 351.89* 349.795* 2 .2 5 6 ’ 353.053*
As with all the previous examples the homogeneity of the concentration parameters
must be tested prior to the analysis of variance. Test statistic values of
U 3 = 0.0821 (distributed as x§) f° r the 6 treatments, and U 3 = 2.7967 (distributed
as x Q f° r the 4 blocks, indicates that the concentration parameters may be regarded
as homogeneous for both factors.
- 238 -
Using the test statistic (11.3.2) and (11.3.3) and the Analysis of Variance Table
11.3.2 from the previous example the resulting statistics are obtained.
r
Ii -1
R?l .Ni
20.519145R2
N13.360056
iJ - l
R2R. jN
13.539216 pq - N - 24
which gives the following ANOVA table
Table 11.3.5 Analysis of Variance
Source o f V a r i a t i o n
d . f . Measure o f V a r i a t ion
MV
MeanMV
F
Due to treatments 3 0.178658 0.05955 0.2705
Due to Blocks 5 7.158587 1.43172 6.5035
Res idual 15 3.302197 0.22015
To ta l 23 10.639442
A
From concentration parameter approximation (3.2.7) k = 2.34, hence /? = 1.1159.
The modified F ' value for the testing of the differing blocks becomes 1.1159 x
6.5035 = 7.257 and as the table values of F 5>15(0.05) = 2.9 and F 5>15(0.01) =
4.56, the analysis shows that there is a significant difference between the blocks.
The modified F ' value for the testing of the differing treatments becomes 1.1159 x
0.2705 = 0.3019. The table value of F 3>1 5(0.05) = 3.29 indicates that there is no
significant difference between the observed values within the treatments. Figures
11.3.1 and 11.3.2 illustrate the differences between treatments and blocks, and help
to understand and confirm the results obtained- Figure 11.3.1 shows the large spread
between the 4 blocks, whilst Figure 11.3.2 shows little spread between the 6
treatments, both confirmed by the analysis of variance
Figure 11.3.1 Angular Mean Responses for the Block Effects
Figure 11.3.2 Angular Mean Responses for the Treatment Effects
- 240 -
11.4 Two-way Classification with Interaction
Example 11.4.1 for large k
Table 11.4.1, overleaf, gives a hypothetical example data set for the two-way
classification design with large k. The data may, for example, be representative of
the time of on-set of an illness with relation to new drugs and differing groups of
people.
The two-way analysis components of variation are given by:
R2M ’ ' * a U
r p y R?l . . R24- k
r *y R2 .•J*
R2
N z. i« l Ni . .
N zLj“ i N-j . . N
r p qV V R2 .
+ k n - y y 1J •La La
L i - l j - i Ni j . .
+ /c
p q
1 1■i~i j = i
R? . 1J
N
r
- Ii j i = l
R?i_
N* - 1J - l
R2 . • J
N J-J N
( 1 1 . 4 . 1 )
Testing the homogeneity of concentration parameters between cells produces a test
statistic of U 3 = 0.682 (distributed as x§) indicating the equality of concentration
parameters. Table 11.4.2 provides the analysis of variance statistics required.
- 241 -
TABL
E 11
.4.1
Tw
o-wa
y cl
assi
fica
tion
o 0 ©LO vO LO 1—1 ClCN CN (—1 Cl LO00 . 00 • r—1 •
LO to . LO CN • o r^CN t—4 o CN t—1 LO to to CNi—H to r—4 to CN to
H. II . It II . II II II II. II
o T z f la fiV
DC Z^ I® * o f z : |CD:
oi—< 0
Cl 00 • ■Cl CNo • o i“H vO CN00 o Cl • O o CN
. LO • LO to • rH •to CN Cl CN
it. II II II. II. II. n. II .ti
<•0
o f z fA
icrr
rj•i
z la?■»
o f•>
2
o o o o 0 o 0 o 0 0CN CN LO CN to rr Cl f—H LO
CN LO CN CN r“—4 CNto to to to
o 000 to
i-H t > • r—1 ClCl • Cl CO i“ 4 •00 o 00 Cl Cl O LO
. LO r - • LO • • —H i-H< ^r CN •sf o vO to
ii. II n II . II . II . II . ii IIaco <
Mo f z? la r 2 ? 10? oc
*<z ICD*
E-cj< 0 0 © 0 • 0 © O « a£L- Cl L0 t-H CN Cl LO Cl 00
00 LO vO r—1 LO LOCM CN CN CN CN to to to
•rH
0Cl
CN CN '■O vO vO 00Cl • vO • CN •00 O 00 Cl o
LO o . lO O . <—1 o• r to •rr to Cl toII. II II ii . IL II . II . II _ II ^
0< o f z f la? DC* Z * la ? DC z~ lay
© 0 o a o o o o 0 oi*H O vO LO —1 CN
00 o Cl O Cl CN Cl i-H CNCN to to CN to CN to CN to to
oCQ co"
CQ
DCo
u<U.
- 242 -
Table 11.4.2 Analysis of Variance Table
Source o fV aria t ion
d . f . Measure o f V a r ia t io n
In te r a c t ion AB
Res idual
To ta l
Due toq - i
qy R2
Factor A Lj-1 NJ , N
Due topy Ri . . ' R2
Factor B P 1 Li-1 Ni . .N
( p - l ) ( q - l )p q
I 1i - 1 j - 1
q
- 1J-i
[ f j .l P. y R?l . .
Nu . .L
i - 1 Ni . . .
R2 . •J
p q ( l - l )
N - 1
p q
■ - 1 1i - i J - i
R2N - —
N
R2 + - * - -
N
R2 .i j
tr
Ii - 1
RiNi
R2- 21.601414 - 17.935976
N
ij - 1
R2 . •J
N J-J
22.44784 p - 2 q - 3 / - 5
N — pq — 30
p q
i ii - 1 j - 1
*11 lNU
28 .550744
- 243 -
which gives the following ANOVA table:
Table 11.4.3 Analysis of Variance Table
Sources o f d . f . Measure o f Mean FV a r i a t ion V a r i a t ion
MVMV
Due to Factor A 2 4.51186 2.25593 37.35877
Due to Factor B 1 3.66544 3.66544 60.7005
In te ra c t ion AB 2 2.43747 1.218735 20.1825
Res idual 24 1.44925 0.060385
To ta l 29 12.06402
From concentration parameter approximation (3.3.10) k = 2.57, hence /3 = 1.1025.
The modified test for the null hypothesis of no significant difference between the q
levels of Factor A provides the F-ratio:
F ( q - l ) , p q ( Z - l ) " 0
p q ( Z - l )f ^
y R2 .* J*
R2LU=i N* J , N
( q - 1 )
p q
■ - I Ii - i j - i
Ru
iNi j
( 1 1 . 4 . 2 )
Therefore F ' 2>24 = 1.1025 x 37.35877 = 41.188 and as the table values of
F 2>24(0.05) = 3.4, F 2>24(0.01) = 5.61, the analysis indicates a very large significant
difference between the q levels of Factor A.
- 244 -
Similarly, F-ratios (11.4.3) and (11.4.4) provide the null hypothesis tests with regards
to the p levels of Factor B and the pq levels of interaction AB respectively.
p q ( f - l )9y R2
LU - i
N,* N* * * -
(P -1 ) » - 2 Ii - 1 j - 1
RUN i j
( 1 1 . 4 . 3 )
F(p - i ) (q - i )»pq(f - i )1
pq(/-Dp qV V [Ri i 1
pv Ri . /
qv [R2j . l
R2) )
1J • - ) - ) +L L
L i -1 j - 1 LNi j JL
i - 1 [N i . . Ju
j - 1 k j j N
( p r l ) ( q - l ) »-l Ii - 1 j - 1
Rfj
lNi j . J
( 1 1 . 4 . 4 )
F ' 1>24 = 1.1025 x 60.7005 = 66.922 and as the table values of F 1>24(0.05) = 4.26,
1 , 2 4 (0.01) = 7.82, the analysis shows there to be a very large significant difference
between the observed results for Factor B. Testing the difference between the
interaction terms AB shows a similarly large significant difference, F ' 2, 24 = 22.25.
Figures 11.4.1, 11.4.2 and 11.4.3 emphasize the observed differences found between
the q levels of Factor A, the p levels of Factor B, and the pq levels of interaction
respectively.
- 245 -
e...
Figure 11.4.1 Angular Differences between the q Levels of Factor A
6 ,e...
Figure 11.4.2 Angular Differences between the p levels of Factor B
Figures 11.4.1 and 11.4.2 illustrate the angular mean differences between the q and p
levels of Factors A and B, the solid lines represent the angular mean for each level,
while the dotted line indicates the associated range of values accounting for that
mean. The larger significant differences between levels within both factors are
clearly -seen.
- 246 -
( a ) D i f f e r e n c e between l e v e l s o f A0 and A1
F a c t o r A
(b) D i f f e r e n c e between l e v e l s o f A, and A2
F a c t o r 2a
( c ) D i f f e r e n c e between l e v e l s o f B0 and B,
F a c t o r B
Figures 11.4.3 Mean Responses Indicating Interaction
Figures (a)-(c) indicate the presence of interaction between the two Factors,
represented by the length of the line segments between the differing levels.
- 247 -
The hypothetical example above shows a design data set where all the differing
groupings of, cells, rows, columns and overall, produce a large value of concentration
parameter. The problems arise, in the majority of cases, where the concentration
parameters for the differing groupings do not remain large and equal. As the
complexity of designs grow the chance that all the cells, rows, columns etc have the
same large and equal concentration parameter becomes highly unlikely. Confusion
arises in the original 'simple' one-way analysis when the q samples all have large
equal k, but their combined sample gives a small k. Batschelet (1981) assumes that
the parameter of concentration has the same value in each population and that k is
found from the average of the sample resultant lengths, and therefore ignores the
combined overall sample. The test statistic derived by Watson and William (1956) is
based, however, on the combined overall sample concentration parameter k, and it is
this value that should be used. This problem limits the usage of the original test
statistic and was highlighted in the data set of Example 7.2.1 where a two-way
design was examined. The new generalised approach was constructed in such a
manner as not to breakdown under such conditions. The following example
re-analyses the data set of Example 7.2.1, this time using the new approach.
Example 11.4.2 for small k
In Example 7.2.1 the possible breakdown of the original extended techniques was
discussed, here this data set is re-analysed via the new approach. Using the
component statistics of (11.4.1) and the analysis of variance Table 11.4.2 the
following ANOVA Table is produced:
- 248 -
Table 11.4.4 Analysis of Variance Table
Source o f V a r ia t io n d . f . Measure o f V a r ia t io n
Due to Factor A 1 9.33114
Due to Factor B 1 8.83213
In te ra c t io n AB 1 0.02437
Res idual 16 1.7936
T o ta l 19 19.98129
r
2i - 1
q
2j - 1
Ri
R2 . • J
8.85089R2
N0.01871
N• J
9.34985 p - 2 q - 2
N - pqZ - 20
2 2 i - 1 j - 1
Ri j
Ni j .
18.2064
In Example 7.2.1 the sum of the two mean effects measure of variation was greater
than the total measure of variation and consequently the interaction term was
negative. The first, and most important, property shown in Table 11.4.4 is that the
component measures of variation for the new approach remain positive and sum to
the associated total measure of variation.
The concentration parameters of Table 7.2.1 may be shown to be equal within cells
and between factors, however, the overall k is very small. Chapter 10 has shown
the residual and total measures are not chi-squared distributed when k is small and
therefore the F-ratio cannot be computed, and examination of the main effects and
- 249 -
interaction must oe carried out using me associated cm-squareu test statistic ior smau
k.
Test Statistic
k - 0 .0612 t 2_ -2- - 2 .00187 where p - A( k )I, (k)
Between measure of variation for Factor A
18.68
This is distributed as x i(q -l)> from tables x i ^ - ^ ) = 5.991 x 2^ * 0 1 ) = 4.605,
indicating a significant difference between the observed responses of Factor A.
Between measure of variation of Factor B
Distributed as X 2 (p-1)> once aSa n indicating a significant difference between the
observed responses of Factor B.
Between measure of variation for interaction term AB
2r p q
y y C—
Py * ? . '
qy
[R f j ]
R24- # *
- P 2 L L Li-1 j - 1 N i j .
Li - 1 N i * .
ij-i N.i N
Distributed as x 2(p -l)(q -l)» indicating that no interaction exists between Factors A
and B.
- 25 0 -
This hypothetical data set was designed solely to emphasize the possible breakdown of
the original extended techniques. The main effects were purposely set with maximum
distance from each other to produce a large significant difference within both
factors.
11.5 The Latin or Graeco-Latin Square
Example 11.5.1
The object of the experiment was to ascertain whether a modified annealing
procedure (to heat and then cool slowly to prevent brittleness) could be introduced
into the production of light gauge domestic copper tube. The original experiment
measured the subsequent tensile strength of the tube (Davies (1963)), here the results
will be taken as the breaking angle of the tube. In deciding the form of the tests it
is necessary to consider possible causes of variation in the results, including variation
in the material itself and variations in temperature over the annealing furnace.
Material variations are studied by taking samples of eight tubes at random on each
of eight days spread over a period of three weeks, thus allowing normal process
variations to be covered adequately. The 64 tubes were held in the furnace in a jig
having eight horizontal rows and eight vertical columns of holes i.e. on 8 x 8
square. The construction of the furnace indicated that temperature variations, if
present, would be horizontal or vertical, and no appreciable interaction was expected
between rows and columns.
The absence of appreciable interaction enabled factors to be examined by means of a
Latin or Graeco-Latin Square. For this example the Graeco-Latin arrangement is
employed in the experiment, so that two factors each of eight treatments could be
allotted to the 64 cells in such a way that the two factors and row and column
effects could be determined independently. The two factors were:
- 251 -
1. Day of Manufacture
2. Number allotted to an individual tube within the sample of eight.
From the point of view of this example the second is a dummy factor, since the
numbering of the tubes did not correspond to any physical reality.
A separate square selected at random was used for each temperature, and the rows,
columns, and letters were themselves randomised. The results for a nominal
temperature of 300 *C are shown in Table 11.5.1, in which the rows and columns
represent positions in the furnace, the letters A to H the day of manufacture, and
the numbers 1 to 8 the designation within the sample. The figures in brackets are
the associated breaking angles.
The Latin or Graeco-Latin square components are given by:
R2M * *r p y R2 i . R2 r p y R2 .
• J - •R2
N l. i - 1 N i. N L
. i - 1 N J . . . N
To ta l measure Measure o f V a r ia t io n Measure o f V a r ia t io n o f v a r i a t io n due to row o f j i g due to column o f j i g
+ kR?.z. R2 r p y R2 . . . a R2
N z.a - l N. . . a N
Measure o f V a r ia t io n due to day o f
manufacture
Measure o f V a r i a t io n due to number
des ignat ion w i th in the sample
+ k
i - 1
R? i ..
- 1 J-1
R2
N - iZ—1
R?.z.'Pn R2 R2_ y . . . a + 3 ’ ’ * •L
a - l N . . . a N
Res idual Measure o f V a r ia t io n ( 1 1 . 5 . 1 )
Carrying out the analysis of variance we obtain Table 11.5.2.
- 2 5 2 -
Tes
ts
on an
neal
ing
light
ga
uge
dom
estic
co
pper
tu
bes
GO
•Hg pcS ctf <L> *H
S > 0
o
00
vO
LO
to
cE3
fHo
u
LOCNr -
vO
00
LO
r -U-
LO
CN CD
CN
00
00
LO
00w
’ S’
r
vOCQ
-S’
LOu
cnvo
’ S’X
vO
vO
COa
CN
to
03
oo
00c j
00
r -
CQ
vO
CNX
o03
to<
00
LO
"S’CO
’ S’
vO
LOCU
03
LO
VOU -.
ooLO
to00
CN<
"S’
00
w
00
00ci«
oo
LO
CO
vO
vO
’S’a
CN
03r—i'_'to
00
vd
vOCJ
LOCQ
0003
VO
LO
vO
00CQ
00
00
CNQ
vO
r-~
u
ovO
LOG-
tovO
vOw
03
LO
toCO
I''.
’ S-<
ooLO
CN00
t '-
LO
voCO
03
LO
"3-tu
’ S’
00
toCO
tooCN
LO<
CN
03
00
vo
00a
or -
CNCQ
’ 3’
CJ
vO
03
LOX
’ S’
t''.
toCQ
LO
1
vOa
03
LO
00CO
03
vo
r^<
ovO
Cl-
LO
vO
CNCO
to00
toCO
LOa
"S-
03
Tt\ vOCO X
vO
’S’CQ
LO
vO
CO
03
LO
CNCO
03
vo
00<
00
LO
f"co
CN•4
vO
’S’
00
"S’CO
’S’
vO<
LO
LOCO
00
vO
toco
CN
03
CNCO
’ S’
r^.
CQ
’ S’
r -
Q
vO
00
00X
to vo 00
toLOCN
CNvO"S’
CNvOto
LOCN00
LOCNto
LO
’ S’
CNO03
VO
CNvOvO
vO
OLO03
vO
Co
•HG P G cS 0 *H
2 > o a
oLO
LO00
00
00
LO
ovo
oLO00
oLO
oLOo
oo00
00to
LOCN
CQ CN
CNvOCO
o00
rH
0 «H
aQ
- 2 5 3 -
Table 11.5.2 Analysis of Variance of Data from Table 11.5.1
Source o f V a r ia t ion
d . f . Measure o f V a r ia t ion
MV
MeanMV
F
Betweenrows o fJ ig
7 1.657 x !0 “ 3 2 .3 6 7 x l0 “ 4 1 .5
Betweencolumnso f j i g
7 2.347 x l0 “ 3 3 .3 5 3 x l0 - 4 2 . 1 2 *
Between pos it ions 14 4.004 xlO-3 2 . 8 6 x l 0 “ 4 1 .8 1 *
Betweenlo ts( l e t t e r s )
7 1 .4 7 17 x l0 “ 2 2.102x10-3 1 3 .3 2 * *
Betweennumbers 7
- 42
9 . 83x l0“ 4
" 6.629x10-3
1 .4 0 4 x l0 " 4
- 1 .5 7 8 x l0 “ 4
Res iduals 35 5.646x10-3 1 .6 1 3 x l0 -4
T o ta l 63 2 .5 3 5 x l0 - 2
Py T? 2
Ri . . .L
i - 1 Ni ...Py R2
• j . .Lj - i
pyL
/= i
py R2 . . . aL
a - l N - - - a
63.976307
63.976997
63.989367
63.975633
R2
- 63.97465N
N - 64
p - 8
- 254 -
rne concentration parameter approximation is so large me correction iacior p may
be neglected ( 0 = 1 )
* = significant at the 1 0 % level * * = highly significant (1 % level)
As expected, the mean measure of variation of the factor representing number
designation within the sample does not differ significantly from the mean measure of
variation. The number factor may therefore be combined with that for the residual
to provide a new estimate of error with 42 degrees of freedom.
Variation between columns (i.e. across the furnace) and total variation among
positions (variation over all positions in the furnace) may be judged significantly
different at the 10 percent level. Clearly the major factor causing variation in the
breaking angle was variation in the lots of tubes; position in the furnace is of little
or no importance.
F 7 t 4 2 ( 0 * 1 0 ) = 1 . 8 7
F 7>42(0.05) = 2.25
F 7, 4 2 (0 .0 1 ) = 3.12
F i 4 , 4 2 ( 0 * 1 0 ) = I - 6 6
F 14|42(0.05) = 1.92
F i 4 ,4 2 (0*01) = 2.52
- 255 -
11.6 The Split Plot Design
Although the theory behind a design such as the split plot has not been derived and
fully checked, the new generalised approach is robust enough (particularly for large
k) to analyse any experimental design if the components are adapted appropriately
and the assumptions are checked sufficiently.
Example 11.6.1
In an experiment on the preparation of chocolate cakes three recipes for preparing
the mixture were compared. Recipes I and I I differed in that the chocolate was
added at 40 *C and 60 *C, respectively, while Recipe m contained extra sugar. For
each recipe enough mixture was made for six cakes. These 18 cakes were then
placed in an oven which was then heated slowly. When the temperature had
reached 175*C three cakes, one from each recipe, were selected at random for
removal, another three at 185*C, and so on until the last three cakes were removed
at 225 *C. In this manner the recipes are representative of the "whole-unit"
treatments, while the baking temperatures are representative of the "sub-unit"
treatments. There were 7 replications, one replication was completed before starting
the next, so that differences among replicates represented a time difference.
A number of measurements were made on the cakes. Table 11.6.1 presents the
measurements of the breaking angle. One half of the cake was held fixed, while the
other half was pivoted about the middle until breakage occured.
- 256 -
The split components of variation are given by:
N -R 2
N
r
Ii - 1
R?i_
N i
R 2
N+ k i
■J-1
R 2 . •J
N J-J
R2
N
+ k
' III
y R?.z' R 2L.Z—1 N..z, N
+ k
r p q R ? . P [r 2 q R 2 2 R2 1
1 1 L i - l j - l
i j .
N i j ,- 1
i - 1
1 . .
N i . .- 1
j - i
• J-
NJ ,
+N
+ k
p m
+ k
U - i z - i
q m
1 1 J - 1 Z - l
H r 1 - !N i.Z
R ’ j *
i - 1
q
• ij - i
R2l
N i
R 2
N
+ k
p m
- 1 1i = l Z = l
R i . Z
N i A
q m
1 1 j - i z - i
R 2
m
- 1Z-l
R 2
N
m
- IZ -l
R 2
N
Pn + y
R?l . .q
+ IR 2 .
m
+ IR?.z
p q
- 1 I
R? . i j -
i i - i [Ni . . J j - i k j . J Z-l N. . 1 i - i j - i LN i j J
R 2
N
R 2
N
• j Z
LN - j z j
R 2
N( 11 . 6 . 1)
Carrying out the analysis of variance we obtain Table 11.6.2.
- 257 -
Table 11.6.1 Examination of Breaking Angles
Rep1ic a t ions
175* 185*
Temperature
195’ 205’ 215* 225’
AngularMean
1 42 46 47 39 53 42
2 47 29 35 47 57 45
3 32 32 37 43 45 45
RecipeI 4 26 32 35 24 39 26 3 5 .3 7 ’
5 28 30 31 37 41 47
6 24 2 2 2 2 29 35 26
7 26 23 25 27 33 35
1 39 46 51 49 55 42
2 35 46 47 39 52 61
3 34 30 42 35 42 35
RecipeI I
4 25 26 28 46 37 37 3 6 .0 2 ’
5 31 30 29 35 40 36
6 24 29 29 29 24 35
7 2 2 25 26 26 29 36
1 46 44 45 46 48 63
2 43 43 43 46 47 58
3 33 24 40 37 41 38
RecipeI I I 4 38 41 38 30 36 35 3 6 .1 4 ’
5 2 1 25 31 35 33 23
6 24 33 30 30 37 35
7 2 0 2 1 31 24 30 33
Angular Mean 31.42* 3 2 .2 2 ’ 3 5 .3 3 ’ 3 5 .8 5 ’ 4 0 .6 6 ’ 3 9 .6 2 ’ 35.84*
- 258 -
Table 11.6.1 Continued
R e p l ica t ion 1 2 3 4 5 6 7
Angular Mean 46.82* 45.55* 36.95* 33.28* 3 2 .3 8 ’ 2 8 .7 2 ’ 27.33*
Table 11.6.2 Analysis of Variance of Data from Table 11.6.1
Source o f V a r ia t ion
d . f . Measure o f V a r ia t ion
MV
MeanMV
F
BetweenRecipes 2 0.00207 0.001035 0.081
Between R e p l ica t ions
In te ra c t ion
6 1.92835 0.3213916 2 5 .1 6 9 *
R ec ipe /R ep l ica t ion
1 2 0.15323 0.0127691
BetweenTemperatures
In te ra c t ion
5 0.44041 0.088082 1 3 .4 7 *
Rec ipes/Temperatures
30 0.17493 0.005831 0 .677
Res idual 60 0.51635 0.00860583
To ta l 125 3.28073
- 259 -
Ii = l
q
ij - i
m
iZ=1
R2
N
R L . 122.72134N l . .
R2 . • J 123.15968
P 2K..z
N . .Z- 124.64762
122.71927 p - 3
* = Highly significant { \% level)
P q
1 1i - 1 j - 1
p m
1 1i=*l Z=T
q m
1 1 j - 1 Z - l
R? . i j
R?.Z
123.22714
N i . Z124.80292
N • JJJ
125.26296
q = 6 m = 7
N — pqm *= 126
The concentration parameter approximation is large and therefore the correction
factor may be neglected ((3 = 1)
From tables , 2(0.05) = 3.89
F s, , 2(°-05) = 3.00 Fb, 1 2 (0 -0 1 ) = 4.82
F s. , o(°*05) = 3.33 F s, ,o(0 -0 1 ) = 5.64oCO ,B0 (°*05) = 1.65
The analysis of variance Table 11.6.2, indicates that there was no significant
difference observed between the recipes. This is shown more clearly from
examination of the angular mean breaking angles associated with the three recipes,
given in Table 11.6.1, where little difference may be noted. Variation between
replicates and temperatures, however, are significantly large and are the dominating
factors causing variation in the breaking angle. The angular mean breaking angle is
seen to increase as the temperature of the oven increases and may be an
understandable effect. The angular mean breaking angle of the replications, however,
- 260 -
show an even clearer decrease in value as the experiments are carried out. This
heavy dependence on time is less understandable and may indicate, for example, that
another factor or condition may be affecting the oven state.
11.7 Summary
This chapter has shown how the new generalised approach may be applied to real
situations where directional data values are measured. The effect of large and small
concentration parameters have been emphasized together with the requirement to test
for the homogeneity of concentration parameters. The Graeco-latin square and split
plot examples have helped to show the suitability of the approach for many
experimental design situations and indicate the method by which further test statistics
may be built.
- 261 -
CHAPTER 12
SYNOPSIS OF RESULTS AND CONCLUSIONS
The aims of this study have been to extend the knowledge of the present methods of
analysis of directional data and to develop suitable analysis of variance techniques for
differing experimental design models. These objectives have been achieved, although
it has not been possible to produce a single unified approach for both large and
small concentration parameter. However, separate techniques were produced under a
generalised approach which has been shown to be applicable to many experimental
design problems. To obtain this approach the work has followed several steps
individually discussed within the thesis.
Before discussing the development of the new techniques an understanding of the
development, constructions and distributional form of the original methods was
required. Within the first four chapters a simple yet informative review of the many
approximations for the concentration parameter k , both large and small, was given.
This work included the plotting of the residuals and relative residuals for each
approximation to indicate the accuracy and range of application for the statistics.
Several were shown to be inappropriate despite their complex form. The analysis
culminated in a summary table of the 'best' and 'best simple' approximations for
both large and small k (Table 3.5.1) and indicated the required approximations for
use with the techniques to be developed.
Chapter 5 discussed a generalised linear modelling approach, analogous to the normal
theory of linear regression, in order to estimate the individual parameter values for
application to the maximum likelihood method. The work showed how the
observations may be 'added together' under the constraint that the sum of the sines
of the factor effects equals zero. Using this fact parameter estimates may be found
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for the one-way analysis of variance giving further understanding to the underlying
structure of the data. The approach, under the above constraint, was found to
produce the original one-way analysis of variance technique developed by Watson and
Williams. When applied to larger experimental design problems, however, the
optimisation of the circular constrained simultaneous equations could not be found,
without very good initial estimates, due to numerous local maxima. Further work in
this particular area, using improved techniques for convergence and ever increasing
improvements in computing methods, is worthy of investigation.
Although it would have been useful to have produced a computer optimisation
program to solve the constrained equations, the constant need for a computer
program as a general method for analysing circular experimental designs, would be
rather restrictive. Therefore, a simple construction of an analysis of variance was
still required. Chapter 6 investigated the possibility of extending the original
approach, with large k , for other designs such as the nested or hierarchical, the
randomised complete block and the two-way design with interaction. The methods
were seen to extend for these designs (k > 2 ) and with good chi-squared
approximations. However the assumption of equal and large k must hold, not only
for the cell, column and row observations, but for the overall sample. This, under
larger and larger designs i.e. with an increasing number of factors, is extremely
restrictive. For example, in situations where the individual row and column factors
have large concentration parameters and the mean directions differ, the overall
concentration parameter may be small. Under these conditions within a two-way
design the 'sum of squares' for the factor effects may add to a value greater than
the total 'sum of squares'. A full investigation followed in Chapter 7 where the
one-way analysis was reconstructed via a regression approach using basic vector
analysis to examine the make-up of the individual components. This brought to light
the possible lack of independence between the model components and the non-zero
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existence of a cross-product term. This questioned not only the use of the original
techniques for larger designs but the adequacy of the original one-way analysis.
The development of the new approach needed to overcome the faults of the original
techniques but be relatively simple and, if possible, be capable of generalising across
all designs. The vector approach used to investigate the original techniques was used
again to construct the new test statistics. The method utilizes the resultant lengths
associated with each sample mean direction, in order that when sample means are
combined the overall mean direction is still obtained. The vector approach
minimised the chord distance between mean directions to construct the test statistics.
Nevertheless, this is not only testing the difference between mean directions but the
associated concentration parameters as well. Hence, a separate examination of the
individual concentration parameters must also be carried out for a valid test of the
mean directions to be feasible.
The beauty of the construction of components in this manner is its simplicity and, in
comparison to the original techniques the independence of the individual components
and the zero cross product terms produced. The interpretation and calculation of
interaction is discussed and the generalised nature of the technique is illustrated in
the construction and proof of the interaction component.
As with Stephens (1969) and Upton (1970) an attempt was made to evaluate
numerically the exact distribution of the old and new test statistics. However, even
for the simplest single sample test statistics the numerical integration involved was
found to be very tedious.
R does not have a simply stated density function, and indeed a direct computation of
the significance points of R/N or R 2/N is not simple.
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The majority of the tests are complicated by involving R v R 2 and R, and, since
these statistics are not for the most part independent of one another, either one or
another of the conditional distributions needs to be employed. These distributions are
exceedingly complicated as are the relevant bounds for integration, and no results of
any use were obtained by attempting to integrate numerically.
Alternative investigations were carried out using the power series expansion of p to
examine the moments of the test statistics to compare with their associated
chi-squared distributions. This showed good approximations to the first moment but
worsening accuracy to higher moments, depending on the size of k. In a similar
manner to the adjustment advocated by Stephens, an important factor is produced to
increase the test statistics accuracy.
The simulation techniques, discussed in Appendix B, have been successful in obtaining
the characteristics of the von Mises distribution and hence the various test statistics.
Elaborate examination of the components for varying designs, concentration parameter
and sample size have been carried out together with tests of the statistics power and
robustness in comparison to any available alternative techniques. Compared to the
alternatives in the one-way analysis the new technique is seen as slightly less accurate
and powerful, although it does possess desirable properties as previously described.
For small concentration parameter the new approach, as with alternative tests, can
only examine the between measure of variation, seen to be chi-squared distributed.
No account can therefore be taken of the residual variation.
Following further justification for the new approach via examination of the
randomised complete block and two-way analysis designs, the test statistics were
successfully applied to differing problems with varying concentration parameter.
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APPENDIX A
INDEX OF NOTATION
A (k) or p : ratio of I^A:) to I 0 (fc)
C : sum of cosines of angles
C : mean of cosines of angles
F (0 ) : distribution function
f( 0 ) : circular density
H 0, H , : null and alternative hypothesis
Ip(fc) : modified Bessel function of the first kind and order p
i : subscript ranging from 1 ,2 , ....... ,p
j : subscript ranging from 1 , 2 ........ ,q
I : subscript ranging from 1 ,2 , ....... ,m
m : number of levels for factor 3
N : size of sample
p : number of levels for factor 1
q : number of levels for factor 2
R : the resultant length
Rj : R for the ith sample
r : mean resultant length of sample
S : sum of sines of angles
S : mean of sines of angles
VM ( p 0,k) : von Mises distribution with mean direction p 0 and
concentration parameter k
(3 : improvement factor for the new approach
7 : Stephens improvement factor
k : parameter of concentration
- I -
population mean direction
standard deviation
circular random variable
overall mean direction of sample
General Tabular Notation
FACTOR A
A 0 A 1 A 2
* 1 1 1 * 1 2 1 * 1 3 1
* 1 1 2 R 1 1 * 1 2 2 R 1 2 * 1 3 2 R 1 3 R 1 .
B o * 1 1 3
* 1 1 4
N n * 1 2 3
* 1 2 4
N 1 2 * 1 3 3
* 1 3 4
^ 1 3 N 1 .
F a c t o r B
* 2 1 1 * 2 2 1 * 2 3 1
* 2 1 2 R 2 j * 2 2 2 R 2 2 * 2 3 2
COCMPi
R 2 .
B i * 2 1 3 N 2 i * 2 2 3 N 2 2 * 2 3 3 N 2 3 N 2 .
* 2 1 4 * 2 2 4 * 2 3 4
R 1 R . 2
CO R
N . 1 N . 2 CO N
Mo
a
6
or 0
- II -
wherethe resultant length of cell observations in row i( i= l , 2 p) and the column j ( j= l ,2 , .,q)
the resultant length of all observations in row i( i= l,2 , p)
the resultant length of all observations in column j( j= l , 2 q)
the resultant length of all observations in the sample
the total number of observations in the sample
- I l l -
APPENDIX B
THE SIMULATION AND ACCURACY OF NUMERICAL RESULTS
All the numerical results presented in this thesis are the product of simulations; all
simulations carried out had a minimum of 10,000 samples. For standard simulation a
random number z is generated in the range 0 to 1, for a distribution function F (0 )
the number z corresponds to an observation 0 from this distribution which is the
solution of the equation
F (0 ) = z
For many distributions this equation can be inverted directly to obtain
0 = F“ ’ (z )
but this is not possible for the von Mises distribution. In addition generation of a
pseudo-random observation from VM(0,fc) cannot be obtained by a simple
transformation of VM(0,1) to VM(0,fc) and an alternative procedure was used.
The approach used was to fit a probability density function as an envelope around
the von Mises distribution to give an acceptance - rejection method which is both
simple to program and fast for all values of the concentration parameter. Initially
the simplest p.d.f. used was the Uniform function, simple but very slow. Best and
Fisher (1979) produced an algorithm to simulate samples from the von Mises
distribution using the wrapped Cauchy density (Equation 1.3.5) as the p.d.f. for the
envelope. Let f(x) be the p.d.f. of a random variable x which is to be sampled.
Let Y be a random variable with p.d.f. proportional to g(x), an upper envelope for
f(x) (i.e. g(x)^f(x)) and let U be a U(0,1) random variable. If (y,u) is a realization
of (Y,U), y is accepted as a realization of x if f(y)/g(y)>u. The distribution of the
accepted values is then exactly the required distribution. Once the observations from
the distribution specified by the null hypothesis are generated 1 0 , 0 0 0 sets of samples
- IV -
of various size and various concentration parameters may be grouped and analysed as
required. The homogeneity of concentration parameters was tested prior to all
analyses.
- V -
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