Statistical Methods for Psychology

334 Chapter 11 Simple Analysis of Variance

Control CogBeav

Treatment

Family

–10

0

10

20

Weight Gain

Figure 11.3 Weight gain in Everitt’s three groups

Source df SS MS F

Treatments 2 614.644 307.322 5.422*

Error 69 3910.742 56.677

Total 71 4525.386

* p, .05

From the summary table you can see that there is a significant effect due to treatment.

The presence of this effect is clear in Figure 11.3, where the control group showed no ap-

preciable weight gain, whereas the other two groups showed substantial gain. We do not

yet know whether the Cognitive-behavior group and the Family therapy group were signif-

icantly different, nor whether they both differed from the Control group, but we will re-

serve that problem until the next chapter.

11.8 Violations of Assumptions

As we have seen, the analysis of variance is based on the assumptions of normality and

homogeneity of variance. In practice, however, the analysis of variance is a robust sta-

tistical procedure, and the assumptions frequently can be violated with relatively minor

effects. This is especially true for the normality assumption. For studies dealing with

this problem, see Box (1953, 1954a, 1954b), Boneau (1960), Bradley (1964), and

Grissom (2000). The latter reference is somewhat more pessimistic than the others, but

there is still reason to believe that normality is not a crucial assumption and that the

homogeneity of variance assumption can be violated without terrible consequences, es-

pecially when we focus on the overall null hypothesis rather than on specific group

comparisons.

In general, if the populations can be assumed to be symmetric, or at least similar in

shape (e.g., all negatively skewed), and if the largest variance is no more than four times

the smallest, the analysis of variance is most likely to be valid. It is important to note, how-

ever, that heterogeneity of variance and unequal sample sizes do not mix. If you have reason

to anticipate unequal variances, make every effort to keep your sample sizes as equal as

possible. This is a serious issue, and people tend to forget that noticeably unequal sample

sizes make the test appreciably less robust to heterogeneity of variance.

The summary table for this analysis follows.