Example 30.10: Testing for Equal Group Variances

The GLM Procedure

Example 30.10: Testing for Equal Group Variances

This example demonstrates how you can test for equal group variances in a one-way design. The data come from the University of Pennsylvania Smell Identification Test (UPSIT), reported in O'Brien and Heft (1995). The study is undertaken to explore how age and gender are related to sense of smell. A total of 180 subjects 20 to 89 years old are exposed to 40 different odors: for each odor, subjects are asked to choose which of four words best describes the odor. The Freeman-Tukey modified arcsine transformation (Bishop, Feinberg, and Holland 1975) is applied to the proportion of correctly identified odors to arrive at an olfactory index. For the following analysis, subjects are divided into five age groups:

${\hv agegroup} & = & \{ 1 & {\rm if} & & & {\rm age} & \leq & 25 \ 2 & {\rm if}... ... 55 & \lt & {\rm age} & \leq & 70 \ 5 & {\rm if} & 70 & \lt & {\rm age} & & .$

The following statements create a data set named upsit, containing the age group and olfactory index for each subject.

   data upsit;
      input agegroup smell @@;
      datalines;
   1 1.381  1 1.322  1 1.162  1 1.275  1 1.381  1 1.275  1 1.322
   1 1.492  1 1.322  1 1.381  1 1.162  1 1.013  1 1.322  1 1.322
   1 1.275  1 1.492  1 1.322  1 1.322  1 1.492  1 1.322  1 1.381
   1 1.234  1 1.162  1 1.381  1 1.381  1 1.381  1 1.322  1 1.381
   1 1.322  1 1.381  1 1.275  1 1.492  1 1.275  1 1.322  1 1.275
   1 1.381  1 1.234  1 1.105
   2 1.234  2 1.234  2 1.381  2 1.322  2 1.492  2 1.234  2 1.381
   2 1.381  2 1.492  2 1.492  2 1.275  2 1.492  2 1.381  2 1.492
   2 1.322  2 1.275  2 1.275  2 1.275  2 1.322  2 1.492  2 1.381
   2 1.322  2 1.492  2 1.196  2 1.322  2 1.275  2 1.234  2 1.322
   2 1.098  2 1.322  2 1.381  2 1.275  2 1.492  2 1.492  2 1.381
   2 1.196
   3 1.381  3 1.381  3 1.492  3 1.492  3 1.492  3 1.098  3 1.492
   3 1.381  3 1.234  3 1.234  3 1.129  3 1.069  3 1.234  3 1.322
   3 1.275  3 1.230  3 1.234  3 1.234  3 1.322  3 1.322  3 1.381
   4 1.322  4 1.381  4 1.381  4 1.322  4 1.234  4 1.234  4 1.234
   4 1.381  4 1.322  4 1.275  4 1.275  4 1.492  4 1.234  4 1.098
   4 1.322  4 1.129  4 0.687  4 1.322  4 1.322  4 1.234  4 1.129
   4 1.492  4 0.810  4 1.234  4 1.381  4 1.040  4 1.381  4 1.381
   4 1.129  4 1.492  4 1.129  4 1.098  4 1.275  4 1.322  4 1.234
   4 1.196  4 1.234  4 0.585  4 0.785  4 1.275  4 1.322  4 0.712
   4 0.810
   5 1.322  5 1.234  5 1.381  5 1.275  5 1.275  5 1.322  5 1.162
   5 0.909  5 0.502  5 1.234  5 1.322  5 1.196  5 0.859  5 1.196
   5 1.381  5 1.322  5 1.234  5 1.275  5 1.162  5 1.162  5 0.585
   5 1.013  5 0.960  5 0.662  5 1.129  5 0.531  5 1.162  5 0.737
   5 1.098  5 1.162  5 1.040  5 0.558  5 0.960  5 1.098  5 0.884
   5 1.162  5 1.098  5 0.859  5 1.275  5 1.162  5 0.785  5 0.859
   ;

Older people are more at risk for problems with their sense of smell, and this should be reflected in significant differences in the mean of the olfactory index across the different age groups. However, many older people also have an excellent sense of smell, which implies that the older age groups should have greater variability. In order to test this hypothesis and to compute a one-way ANOVA for the olfactory index that is robust to the possibility of unequal group variances, you can use the HOVTEST and WELCH options in the MEANS statement for the GLM procedure, as shown in the following code.

   proc glm data=upsit;
      class agegroup;
      model smell = agegroup;
      means agegroup / hovtest welch;
   run;

Output 30.10.1, Output 30.10.2, and Output 30.10.3 display the usual ANOVA test for equal age group means, Levene's test for equal age group variances, and Welch's test for equal age group means, respectively. The hypotheses of age effects for mean and variance of the olfactory index are both confirmed.

Output 30.10.1: Usual ANOVA Test for Age Group Differences in Mean Olfactory Index

The GLM Procedure

Dependent Variable: smell

Source	DF	Type I SS	Mean Square	F Value	Pr > F
agegroup	4	2.13878141	0.53469535	16.65	<.0001

Output 30.10.2: Levene's Test for Age Group Differences in Olfactory Variability

The GLM Procedure

Levene's Test for Homogeneity of smell Variance ANOVA of Squared Deviations from Group Means
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
agegroup	4	0.0799	0.0200	6.35	<.0001
Error	175	0.5503	0.00314

Output 30.10.3: Welch's Test for Age Group Differences in Mean Olfactory Index

The GLM Procedure

Welch's ANOVA for smell
Source	DF	F Value	Pr > F
agegroup	4.0000	13.72	<.0001
Error	78.7489

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