Statistical Methods for Psychology

The output here looks like what we computed. You would get the same general results

if you had selected Analyze/General Linear Model/Univariatefrom the menus, although

the summary table would contain additional lines of information that I won’t discuss until

the end of this chapter.

11.7 Unequal Sample Sizes

Most experiments are originally designed with the idea of collecting the same number of

observations in each treatment. (Such designs are generally known as balanced designs.)

Frequently, however, things do not work out that way. Subjects fail to arrive for testing, or

are eliminated because they fail to follow instructions. Animals occasionally become ill

during an experiment from causes that have nothing to do with the treatment. I still recall

an example first seen in graduate school in which an animal was eliminated from the study

for repeatedly biting the experimenter (Sgro & Weinstock, 1963). Moreover, studies con-

ducted on intact groups, such as school classes, have to contend with the fact that such

groups nearly always vary in size.

If the sample sizes are not equal, the analysis discussed earlier needs to be modified. For

the case of one independent variable, however, this modification is relatively minor.

(A much more complete discussion of the treatment of missing data for a variety of analysis

of variance and regression designs can be found in Howell (2008), or, in slightly simpler

form, at http://www.uvm.edu/~dhowell/StatPages/More_Stuff/Missing_Data/Missing.html)

Earlier we defined

We were able to multiply the deviations by n, because nwas common to all treatments. If

the sample sizes differ, however, and we define as the number of subjects in the jth treatment

, we can rewrite the expression as

which, when all are equal, reduces to the original equation. This expression shows us

that with unequal ns, the deviation of each treatment mean from the grand mean is

weighted by the sample size. Thus, the larger the size of one sample relative to the others,

the more it will contribute to , all other things being equal.

Effective Therapies for Anorexia

The following example is taken from a study by Everitt that compared the effects of two

therapy conditions and a control condition on weight gain in anorexic girls. The data are

reported in Hand et al., 1994. Everitt used a control condition that received no intervention,

a cognitive-behavioral treatment condition, and a family therapy condition. The dependent

variable analyzed here was the gain in weight over a fixed period of time. The data are

given in Table 11.5 and plotted in Figure 11.3. Although there is some tendency for the

Cognitive-behavior therapy group to be bimodal, that tendency is probably not sufficient to

distort our results. (A nonparametric test [see Chapter 18] that is not influenced by that bi-

modality produces similar results.)

The computation of the analysis of variance follows, and you can see that the change

required by the presence of unequal sample sizes is minor. I should hasten to point out that

unequal sample sizes will not be so easily dismissed when we come to more complex designs,

but there is no particular difficulty with the one-way design.

SStreat

nj

SStreat= a 3 nj(Xj 2 X..)^24

Aanj=NB

nj

SStreat=na(Xj 2 X..)^2

332 Chapter 11 Simple Analysis of Variance

balanced designs