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The Assumptions ofANOVA
Dennis Monday
Gary Klein
Sunmi Lee
May 10, 2005

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Major Assumptions of Analysis of
Variance The Assumptions
Independence
Normally distributed
Homogeneity of variances
Our Purpose Examine these assumptions
Provide various tests for these assumptions Theory
Sample SAS code (SAS, Version 8.2)
Consequences when these assumptions are not met
Remedial measures

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Normality
Why normal? ANOVA is anAnalysis of Variance
Analysis of two variances, more specifically, the ratio of
two variances Statistical inference is based on the F distribution
which is given by the ratio of two chisquareddistributions
No surprise that each variance in the ANOVA ratio comefrom a parent normal distribution
Calculations can always be derived no matterwhat the distribution is. Calculations arealgebraic properties separating sums of squares.Normality is only needed for statistical inference.

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Normality
Tests Wide variety of tests we can perform to test if the
data follows a normal distribution.
Mardia (1980) provides an extensive list for both
the univariate and multivariate cases,categorizing them into two types Properties of normal distribution, more specifically, the
first four moments of the normal distribution
ShapiroWilks W (compares the ratio of the standard
deviation to the variance multiplied by a constant to one) Goodnessoffit tests,
KolmogorovSmirnov D
Cramervon Mises W2
AndersonDarling A2

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Normality
Tests
procunivariate data=temp normal plot;
var expvar;
run;
procunivariate data=temp normal plot;
var normvar;
run;
Tests for Normality
Test Statistic p Value
ShapiroWilk W 0.731203 Pr < W D WSq ASq D >0.1500
Cramervon Mises WSq 0.03225 Pr > WSq >0.2500
AndersonDarling ASq 0.224264 Pr > ASq >0.2500
Normal Probability Plot2.3+ ++ *
 ++* +** +** **** *** **+ ** *** **+ ***
0.1+ *** ** *** *** ** +*** +** +** **** ++ +*
2.1+*+++++++++++++
2 1 0 +1 +2
Stem Leaf # Boxplot22 1 1 20 7 1 18 90 2 16 047 3 14 6779 4 12 469002 6 10 2368 4 8 005546 6 ++6 228880077 9  4 5233446 7  2 3458447 7 **0 366904459 9  + 0 52871 5  2 884318651 9  4 98619 5 ++6 60 2 8 98557220 8 10 963 3 12 584 3 14 853 3 16 0 1 18 4 1 20 8 1 
++++Multiply Stem.Leaf by 10**1
Normal Probability Plot
8.25+
 *


 *

 *
 +
4.25+ ** ++++
 ** +++
 *+++
 +++*
 ++****
 ++++ **
 ++++*****
 ++******
0.25+* * ******************
+++++++++++
Stem Leaf # Boxplot8 0 1 *7766 1 1 *55 2 1 *4 5 1 04 4 1 03 588 3 03 3 1 02 59 2 2 00112234 8 1 56688 5 1 00011122223444 14 +++0 55555566667777778999999 23 **0 000011111111111112222222233333334444444 39 +++++++++

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Consequences of NonNormality
Ftest is very robust against nonnormal data,especially in a fixedeffects model
Large sample size will approximate normality by
Central Limit Theorem (recommended samplesize > 50)
Simulations have shown unequal sample sizesbetween treatment groups magnify any departurefrom normality
A large deviation from normality leads tohypothesis test conclusions that are too liberaland a decrease in power and efficiency

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Remedial Measures for Non
Normality Data transformation
Be aware  transformations may lead to afundamental change in the relationship between
the dependent and the independent variable andis not always recommended.
Dont use the standard Ftest. Modified Ftests
Adjust the degrees of freedom
Rank Ftest (capitalizes the Ftests robustness)
Randomization test on the Fratio
Other nonparametric test if distribution is unknown
Make up our own test using a likelihood ratio ifdistribution is known

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Independence
Independent observations No correlation between error terms
No correlation between independent variables and error
Positively correlated data inflates
standard error The estimation of the treatment means are more
accurate than the standard error shows.

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Independence Tests
If we have some notion of how the data wascollected, we can check if there exists any
autocorrelation.
The DurbinWatson statistic looks at the
correlation of each value and the value before it
Data must be sorted in correct order for meaningful
results
For example, samples collected at the same time wouldbe ordered by time if we suspect results could depend
on time

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Independence Tests
procglmdata=temp;class trt;
model y = trt / p;
output out=out_ds r=resid_var;
run;
quit;
data out_ds;
set out_ds;
time = _n_;run;
procgplot data=out_ds;
plot resid_var * time;
run;
quit;
procglmdata=temp;class trt;
model y = trt / p;
output out=out_ds r=resid_var;
run;
quit;
data out_ds;
set out_ds;
time = _n_;run;
procgplot data=out_ds;
plot resid_var * time;
run;
quit;
First Order Autocorrelation
0.00479029
DurbinWatson D
1.96904290
First Order Autocorrelation
0.90931
DurbinWatson D
0.12405

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Remedial Measures for Dependent
Data First defense against dependent data is proper
study design and randomization Designs could be implemented that takes correlation
into account, e.g., crossover design Look for environmental factors unaccounted for
Add covariates to the model if they are causingcorrelation, e.g., quantified learning curves
If no underlying factors can be found attributed to
the autocorrelation Use a different model, e.g., random effects model
Transform the independent variables using thecorrelation coefficient

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Homogeneity of Variances
Eisenhart (1947) describes the problem ofunequal variances as follows the ANOVA model is based on the proportion of the
mean squares of the factors and the residual meansquares
The residual mean square is the unbiased estimator of2, the variance of a single observation
The between treatment mean squares takes into accountnot only the differences between observations, 2,justlike the residual mean squares, but also the variance
between treatments If there was nonconstant variance among treatments,
we can replace the residual mean square with someoverall variance, a2, and a treatment variance, t2,which is some weighted version of a2
The neatness of ANOVA is lost

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Homogeneity of Variances
The omnibus (overall) Ftest is very robust
against heterogeneity of variances,
especially with fixed effects and equal
sample sizes.
Tests for treatment differences like ttests
and contrasts are severely affected,
resulting in inferences that may be tooliberal or conservative.

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Tests for Homogeneity of Variances
Levenes Test
computes a onewayanova on the absolute value (orsometimes the square) of the residuals, yiji with t1, Nt degrees of freedom
Considered robust to departures of normality, but tooconservative
BrownForsythe Test a slight modification of Levenes test, where the median is
substituted for the mean (Kuehl (2000) refers to it as theLevene (med) Test)
The Fmax Test
Proportion of the largest variance of the treatment groups
to the smallest and compares it to a critical value table Tabachnik and Fidell (2001) use the Fmax ratio more as a
rule of thumb rather than using a table of critical values.
Fmax ratio is no greater than 10
Sample sizes of groups are approximately equal (ratio ofsmallest to largest is no greater than 4)
No matter how the Fmax test is used, normality must beassumed.

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Tests for Homogeneity of Variances
procglmdata=temp;
class trt;
model y = trt;
means trt / hovtest=levene hovtest=bf;
run;
quit;
procglmdata=temp;
class trt;
model y = trt;
means trt / hovtest=levene hovtest=bf;
run;
quit;
Homogeneous Variances
The GLM Procedure
Levene's Test for Homogeneity of Y Variance
ANOVA of Squared Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
TRT 1 10.2533 10.2533 0.60 0.4389
Error 98 1663.5 16.9747
Brown and Forsythe's Test for Homogeneity of Y Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
TRT 1 0.7087 0.7087 0.56 0.4570
Error 98 124.6 1.2710
Heterogenous Variances
The GLM Procedure
Levene's Test for Homogeneity of y Variance
ANOVA of Squared Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
trt 1 10459.1 10459.1 36.71

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Tests for Homogeneity of Variances
SAS (as far as I know) does not have a procedure
to obtain Fmax (but easy to calculate)
More importantly:
VARIANCE TESTS ARE ONLY FOR ONEWAY
ANOVA
WARNING: Homogeneity of variance testing and Welch's
ANOVA are only available for unweighted oneway
models.

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Tests for Homogeneity of Variances
(Randomized Complete Block Design and/or
Factorial Design)
In a CRD, the variance of each treatment
group is checked for homogeneity
In factorial/RCBD, each cells varianceshould be checked
H0: ij
2= ij
2, For all i,j where i i, j j

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Tests for Homogeneity of Variances
(Randomized Complete Block Design and/or
Factorial Design)
Approach 1 Code each row/column to its own
group
Run HOVTESTS as before
Approach 2 Recall Levenes Test and Brown
Forsythe Test are ANOVAs based on
residuals
Find residual for each observation
Run ANOVA
data newgroup;
set oldgroup;
if block = 1 and treat = 1 then newgroup = 1;
if block = 1 and treat = 2 then newgroup = 2;
if block = 2 and treat = 1 then newgroup = 3;
if block = 2 and treat = 2 then newgroup = 4;
if block = 3 and treat = 1 then newgroup = 5;
if block = 3 and treat = 2 then newgroup = 6;
run;
procglm data=newgroup;
class newgroup;
model y = newgroup;
means newgroup / hovtest=levene hovtest=bf;
run;
quit;
procsort data=oldgroup; by treat block; run;
procmeans data=oldgroup noprint; by treat block;
var y;
output out=stats mean=mean median=median;
run;
data newgroup;
merge oldgroup stats;
by treat block;
resid = abs(mean  y);
if block = 1 and treat = 1 then newgroup = 1;
run;
procglm data=newgroup;
class newgroup;
model resid = newgroup;
run; quit;

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Tests for Homogeneity of Variances
(RepeatedMeasures Design) Recall the repeatedmeasures setup:
Treatment
a1 a2 a3
s1 s1 s1
s2 s2 s2
s3 s3 s3
s4 s4 s4

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Tests for Homogeneity of Variances
(RepeatedMeasures Design) As there is only one score per cell, the variance
of each cell cannot be computed. Instead, fourassumptions need to be tested/satisfied
Compound Symmetry Homogeneity of variance in each column
a12= a2
2 =a32
Homogeneity of covariance between columns
a1a2 =a2a3
= a3a1
No A x S Interaction (Additivity) Sphericity
Variance of difference scores between pairs are equal
Ya1Ya2 = Ya1Ya3
= Ya2Ya3

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Tests for Homogeneity of Variances
(RepeatedMeasures Design) Usually, testing sphericity will suffice
Sphericity can be tested using the Mauchly test inSAS
procglm data=temp;
class sub;
model a1 a2 a3 = sub / nouni;
repeated as 3 (123) polynomial / summary printe;
run; quit;
Sphericity Tests
Mauchly's
Variables DF Criterion ChiSquare Pr > ChiSq
Transformed Variates 2 Det = 0 6.01 .056
Orthogonal Components 2 Det = 0 6.03 .062

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Tests for Homogeneity of Variances
(LatinSquares/SplitPlot Design) If there is only one score per cell, homogeneity of
variances needs to be shown for the marginals of
each column and each row
Each factor for a latinsquare
Whole plots and subplots for splitplot
If there are repititions, homogeneity is to be
shown within each cell like RCBD
If there are repeatedmeasures, follow guidelinesfor sphericity, compound symmetry and additivity
as well

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Remedial Measures for
Heterogeneous Variances Studies that do not involve repeated measures
If normality is not violated, a weighted ANOVA is suggested(e.g., Welchs ANOVA)
If normality is violated, the data transformation necessary to
normalize data will usually stabilize variances as well If variances are still not homogeneous, nonANOVA tests
might be your option
Studies with repeated measures For violations of sphericity
modify the degrees of freedom have been suggested. GreenhouseGeisser
Huynh and Feldt
Only do specific comparisons (sphericity does not apply sinceonly two groups sphericity implies more than two)
MANOVA
Use an MLE procedure to specify variancecovariance matrix

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Other Concerns
Outliers and influential points
Data should always be checked for influential
points that might bias statistical inference Use scatterplots of residuals
Statistical tests using regression to detect outliers
DFBETAS
Cooks D

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References Casella, G. and Berger, R. (2002). Statistical Inference. United States: Duxbury.
Cochran, W. G. (1947). Some Consequences When the Assumptions for the Analysis ofVariances are not Satisfied. Biometrics. Vol. 3, 2238.
Eisenhart, C. (1947). The Assumptions Underlying the Analysis of Variance. Biometrics.Vol. 3, 121.
Ito, P. K. (1980). Robustness of ANOVA and MANOVA Test Procedures. Handbook ofStatistics 1: Analysis of Variance (P. R. Krishnaiah, ed.), 199236. Amsterdam: NorthHolland.
Kaskey, G., et al. (1980). Transformations to Normality. Handbook of Statistics 1: Analysisof Variance (P. R. Krishnaiah, ed.), 321341. Amsterdam: NorthHolland.
Kuehl, R. (2000). Design of Experiments: Statistical Principles of Research Design andAnalysis, 2nd edition. United States: Duxbury.
Kutner, M. H., et al. (2005). Applied Linear Statistical Models, 5th edition. New York:McGrawHill.
Mardia, K. V. (1980). Tests of Univariate and Multivariate Normality. Handbook of Statistics1: Analysis of Variance (P. R. Krishnaiah, ed.), 279320. Amsterdam: NorthHolland.
Tabachnik, B. and Fidell, L. (2001). ComputerAssisted Research Design and Analysis.Boston: Allyn & Bacon.