Chapter 13 1

Chapter 13

Assumptions Underlying Parametric Statistical Techniques

Chapter 13 2

Parametric Statistics

We have been studying parametric statistics.

They include estimations of mu and sigma, correlation, t tests and F tests.

Chapter 13 3

Five Assumptions

two research assumptions;

two assumptions about the type of the distributions in the samples,

and one assumption about the kind of numbering system that we are using.

To validly use parametric statistics, we make

Chapter 13 4

Research Assumptions

Subjects have to be randomly selected from the population.

Experimental error is randomly distributed across samples in the design.

(We will not discuss these any further).

Chapter 13 5

Distribution Assumptions

The distribution of sample means fit a normal curve.

Homogeneity of variance (using FMAX).

Chapter 13 6

Assumptions about Numbering Schemes

The measures we take are on an interval scale.

(Other numbering scales, such as ordinal and nominal, are non-parametric).

Chapter 13 7

Violating the Assumptions

If any of these assumptions are violated, we cannot use parametric statistics.

We must use less-powerful, non-parametric statistics.

Chapter 13 8

Chapter 13 9

Sample Means

An assumption we need to make is that the distribution of sample means is normally distributed.

This is not as extreme an assumption as it might seem.

We will follow an example from Chapter 4 to demonstrate.

Chapter 13 10

Example: Start with a tiny population N=5

The scores in this population form a perfectly rectangular distribution.

Mu = 5.00Sigma = 2.83We are going to list all the possible

samples of size 2 (n=2)First see the population, then the list

of samples

Chapter 13 11

Figure 4.5: Scores of the 5 research participants in this population. Frequency 3 2 1 x x x x x 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

Chapter 13 12

Figure 4.6: Means of all possible 25 samples (n=2) from this population Frequency 5 X 4 X X X 3 X X X X X 2 X X X X X X X 1 X X X X X X X X X Score 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

Chapter 13 13

Table 4.10: List of all 25 possible samples (n=2) of scores from the tiny population of 5 scores shown in Table 4.9 and Figure 4.5 I Sample Scores X Sample Scores X Summary statistics (all samples, n=2) AA 1,1 1.00 DA 7,1 4.00 X = 125.00 AB 1,3 2.00 DB 7,3 5.00 N = 25 AC 1,5 3.00 DC 7,5 6.00 mu = 5.00 AD 1,7 4.00 DD 7,7 7.00 SS = 100.00 AE 1,9 5.00 DE 7,9 8.00 BA 3,1 2.00 EA 9,1 5.00 BB 3,3 3.00 EB 9,3 6.00 BC 3,5 4.00 EC 9,5 7.00 BD 3,7 5.00 ED 9,7 8.00 BE 3,9 6.00 EE 9,9 9.00 CA 5,1 3.00 CB 5,3 4.00 CC 5,5 5.00 CD 5,7 6.00 CE 5,9 7.00

Chapter 13 14

Normal Curve for Sample Means Conclusion

Even if we have a small population (5),… with a rectangular distribution,… and a small sample size (2),… which yields a small number of possible

samples (52= 25)

… the sample means tend to fall in a normal distribution.

This assumption is seldom violated.This assumption is robust.

Chapter 13 15

Figure 4.6: Means of all possible 25 samples (n=2) from this population Frequency 5 X 4 X X X 3 X X X X X 2 X X X X X X X 1 X X X X X X X X X Score 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

Chapter 13 16

Violating the Normal Curve Assumption

Normal curves are symmetric are bell-shaped

have a single peak

Non-normal curves have skew have kurtosis-

platykutic or leptokurtic are polymodal

Distributions can vary from normal in many ways.

Chapter 13 17

Symmetry

Frequency

score

The left side is the same shape as the right side.

Chapter 13 18

Skewed

NORMAL

Skewed RightSkewed Left

Chapter 13 19

Bell-shaped

Frequency

score

Area under the curve occurs in a prescribed manner,as listed in the Z table.

1 sigma ~ 34%; 2 sigma ~ 48%; On each side of the mean

Chapter 13 20

Kurtosis

NORMAL

Leptokurtic Platykurtic

Chapter 13 21

One mode

Frequency

score

There is only one mode and it equals the medianand the mean.

Chapter 13 22

Polymodality

NORMAL

Bimodal Trimodal

Chapter 13 23

Violation of normally distributed sample means

If the distribution of sample means is… skewed,… or has kurtosis,… or more than one mode,

… then we cannot use parametric statistics.

Chapter 13 24

Chapter 13 25

For F Ratios and t Tests

We assume that the distribution of scores around each sample mean is similar.

The distributions within each group all estimate the same thing, that is, sigma2.

The mean squares within each group should be the same in each group.

For F ratios and t tests, this is called homogeneity of variance.

Chapter 13 26

For Correlation

For correlation, the scores must vary roughly the same amount around the entire length of the regression line.

This is called homoscedasticity.

Chapter 13 27

Homoscedasticity3

-3

2

1

0

-1

-2

3 -3 2 1 0 -1 -2

Chapter 13 28

Non-Homoscedasticity3

-3

2

1

0

-1

-2

3 -3 2 1 0 -1 -2

Chapter 13 29

Homogeneity of Variance

In mathematical terms, homogeneity of variance means that the mean squares for each group are about the same.

221 sigmaMSMSMSMS WK

MSW is a consistent estimate of sigma2. The more degrees of freedom for MSW, the closer it tends to come to sigma2.

Chapter 13 30

We assume the mean square is your best estimate of sigma2

Since MSW has more df than MS1 or MS2 or MSK, it should be a better estimate of sigma2. But that only works when the mean squares in all the groups are fairly good estimates of sigma2.We use the FMAX test to check if the group with the smallest mean squares is “too different” from the group with the largest mean squares for the combined mean square (MSW) to be a good estimate of sigma2.

Chapter 13 31

FMAX

If FMAX is significant, then the Mean Squares differ too much from each other to combine into a single estimate.

(Usually it means that the variance in one of the groups has virtually disappeared because of a floor or ceiling effect. When that happens, adding that groups sum of squares and df into the mix produces an underestimate of sigma2.

When that happens, it becomes too easy to make a Type 1 error.

We say that “The assumption of homogeneity of variance is violated.”

And we cannot use parametric statistics!

Chapter 13 32

7Wdf43.3WMS

Divide by df(nG-1) to get MS for each group.

7Wdf21.0WMS

Sum the deviations.

00.24WSS 48.1WSS

Book Example - no homogeneity

1.11.21.31.41.51.61.71.8

12355664

X#S2.12.22.32.42.52.62.72.8

98999989

9.004.001.001.001.004.004.000.00

21XX

Square thedeviations.

.06

.56

.06

.06

.06

.06

.56

.06

22XX -3.00-2.00-1.001.001.002.002.000.00

1XX

Calculate thedeviations.

.25-.75.25.25.25.25

-.75.25

2XX X#S4.004.004.004.004.004.004.004.00

1X

00.41 X

Within each groupCalculate

the means.

8.758.758.758.758.758.758.758.75

2X

75.82 X

Chapter 13 33

FMAX

In FMAX, the “MAX” part refers to the largest ratio that can be obtained by comparing the estimated variances from 2 experimental groups.

MinimumG

MaximumGMAX MS

MSF

The significance of FMAX is checked in an FMAX table.

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

K = number of variances

dfFMAX

alpha = .01.

nG(larger) - 1Default = larger df.

The number of groupsin the experiment.

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

k = number of variances

dfFMAX

The critical values.

Chapter 13 36

Book Example

43.31 MS 21.02 MS

33.16

21.0

43.3

MinimumG

MaximumGMAX MS

MSF

7MAXdf

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

k = number of variances

dfFMAX

FMAX = 16.33 > 8.89FMAX exceeds the critical value.

We cannot use parametric statistics.

Chapter 13 38

Examples

Number Subjects Critical valueDesign of Means in larger NG of FMAX

2X4 8 21 5.3 2X2 ? 16 ? 3X3 ? 11 ? 2X3 ? 9 ?

496

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

K = number of variances

dfFMAX

Chapter 13 40

CPE 14.2.1

Number Subjects Critical valueDesign of Means in larger NG of FMAX

2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 ? 2X3 6 9 ?

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

K = number of variances

dfFMAX

Chapter 13 42

CPE 14.2.1

Number Subjects Critical valueDesign of Means in larger NG of FMAX

2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 12.4 2X3 6 9 ?

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

K = number of variances

dfFMAX

Chapter 13 44

CPE 14.2.1

Number Subjects Critical valueDesign of Means in larger NG of FMAX

2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 9.5 2X3 6 9 14.5

Chapter 13 45

Example – other way

Design2X42X3 2X23X3

Numberof Means

8???

MSG

max18.226.334.218.0

MSG

min1.12.04.60.5

FMAX

16.5???

Subjects inlarger NG

1012217

dfFMAX

9???

p.01.01???

649

11206

13.27.4

36.0

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

k = number of variances

dfFMAX

FMAX(6,11) = 13.2

p.01

Chapter 13 47

Design2X42X3 2X23X3

Numberof Means

8???

MSG

max18.226.334.218.0

MSG

min1.12.04.60.5

FMAX

16.513.27.4

36.0

Subjects inlarger NG

1012217

dfFMAX

911206

p.01.01.01??

649

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

k = number of variances

dfFMAX

FMAX(4,20) = 7.4

p.01

Chapter 13 49

Design2X42X3 2X23X3

Numberof Means

8???

MSG

max18.226.334.218.0

MSG

min1.12.04.60.5

FMAX

16.513.27.4

36.0

Subjects inlarger NG

1012217

dfFMAX

911206

p.01.01.01.01?

649

2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106

5 14.9 22 28 33 38 42 46 50 54

6 11.1 15.5 19.1 22 25 27 30 32 34

7 8.89 12.1 14.5 16.5 18.4 20 22 23 24

8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9

9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3

10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9

12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9

15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5

20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6

30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0

60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6

K = number of variances

dfFMAX

FMAX(9,6) = 36.0

p.01

Chapter 13 51

Answers to examples

Design2X42X3 2X23X3

Numberof Means

8???

MSG

max18.226.334.218.0

MSG

min1.12.04.60.5

FMAX

16.513.27.4

36.0

Subjects inlarger NG

1012217

dfFMAX

911206

p.01.01.01.01.01

649

You cannot use theF test for any of these

experiments!

Chapter 13 52

Homogeneity of Variance Conclusions

If FMAX is significant, then the assumptionof homogeneity of variance has beenviolated.

If the assumption of homogeneity of varianceis violated, then we cannot use parametric statistics.

Chapter 13 53

Chapter 13 54

Assumption

Our last assumption that we must meet to use parametric statistics is that the measures in our experiment use an interval scale.

An interval scale is a set of numbers whose differences are equal at all points along the scale.

Chapter 13 55

Examples of Interval Scales

Integers - 1,2,3,4,…

Real numbers - 1.0, 1.1, 1.2, 1.3,…

Time - 1 minute, 2 minutes, 3 minutes, …

Distance - 1 foot, 2 feet, 3 feet, 4 feet, …

Chapter 13 56

Examples of Non-Interval Scales

Ordinal - ranks, such as first, second, third; high medium low; etc. The difference in time between first and

second can be very different from the time between second and third.

The median is the best measure of central tendency for ordinal data.

Chapter 13 57

Examples of Non-Interval Scales

Nominal - categories, such as, male, female; pass, fail.

There is not even an order for nominal data.

Categories should be mutually exclusive and exhaustive.

The best measure of central tendency is the mode.

Chapter 13 58

Comparing Scales

Interval scales have more information than ordinal scales, which in turn have more information than nominal scales.

The more information that is available, the more sensitive that a given statistical test can be.

Chapter 13 59

Book Example - test grades

Interval ScaleSCORES

9884777675626160

Ordinal ScaleRANKS

12345678

Nominal ScalePass/Fail

PPPPPFFF

Chapter 13 60

Book Example - test grades

Interval ScaleSCORES

9884777675626160

Ordinal ScaleRANKS

12345678

Nominal ScalePass/Fail

PPPPPFFF

Ordinal scales show the relative order of individual measures.

However, there is no information about how far apart individuals are.

Chapter 13 61

Book Example - test grades

Interval ScaleSCORES

9884777675626160

Ordinal ScaleRANKS

12345678

Nominal ScalePass/Fail

PPPPPFFF

Categories are mutually exclusive; you either pass or fail.

Categories are exhaustive; you can only pass or fail.

Chapter 13 62

Interval Scale Conclusion Parametric tests can only be performed on interval data. Non-parametric tests must be used on ordinal and nominal

data. Researchers prefer parametric tests because more

information is available, which makes it easier to find: Significant differences between experimental group

means or Significant correlations between two variables.

If any assumptions are violated, it is common practice to convert from the interval scale to another scale. Then you can use the weaker, non-parametric statistics.

There are non-parametric statistics that correspond to all of the parametric statistics that we have studied.

Chapter 13 63

Summary - Assumptions

Subjects are randomly selected from the population.

Experimental error is randomly distributed across samples in the design.

The distribution of sample means fit a normal curve.

There is homogeneity of variance demonstrated by using FMAX.

The measures we take are on an interval scale.

To use parametric statistics, it must be true that: