The GLM Procedure

Example 45.10 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, Fienberg, 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:

$\begin{eqnarray*} \Variable{agegroup} & = & \left\{ \begin{array}{clcccccc} 1 & \textrm{if} & & & \textrm{age} & \leq & 25 \\ 2 & \textrm{if} & 25 & < & \textrm{age} & \leq & 40 \\ 3 & \textrm{if} & 40 & < & \textrm{age} & \leq & 55 \\ 4 & \textrm{if} & 55 & < & \textrm{age} & \leq & 70 \\ 5 & \textrm{if} & 70 & < & \textrm{age} & & \end{array}\right. \end{eqnarray*}$

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 statements.

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

Output 45.10.1, Output 45.10.2, and Output 45.10.3 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 45.10.1: Usual ANOVA Test for Age Group Differences in Mean Olfactory Index

The GLM Procedure

Dependent Variable: smell

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

Output 45.10.2: 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 45.10.3: Welch’s Test for Age Group Differences in Mean Olfactory Index

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

As discussed in Homogeneity of Variance in One-Way Models, Levene’s test or any other test for homogeneity of variance should not be used as a diagnostic for the assumption of equal group variances that underlies the usual analysis of variance. However, graphical diagnostics can be a useful informal tool for monitoring whether your data meet the assumptions of a GLM analysis. The following statements perform a one-way ANOVA as before, but with ODS Graphics enabled. In addition to the box plot that is produced by default, the PLOTS= DIAGNOSTICS option requests a panel of summary diagnostics for the fit. These additional plots are shown in Output 45.10.4 and Output 45.10.5.

ods graphics on;
proc glm data=upsit plot=diagnostics;
   class agegroup;
   model smell = agegroup;
run;
ods graphics off;

Output 45.10.4: Box Plot of Olfactory Index by Age Group

Output 45.10.5: Diagnostics for One-Way ANOVA of Olfactory Index by Age Group

Output 45.10.4 clearly shows different degrees of variability for olfactory index within different age groups, with the variability generally rising with age. Likewise, several of the plots in the diagnostics panel shown in Output 45.10.5 indicate a relationship between olfactory variability and mean olfactory index. Also, note that the plot of Cook’s D statistic indicates that observations in the higher, more variable age groups are overly influential on the analysis of group means. The overall inference from these plots is that an assumption of equal group variances is probably untenable and that the analysis of the group means should thus take this into account.