Statistical Methods for Psychology

My sample sizes were unequal, but not seriously so. When we have quite unequal

sample sizes, and they are unavoidable, the smaller group should be as large as possible

relative to the larger group. You should never throw away subjects to make sample sizes

equal. This is just throwing away power.^2

8.5 Power Calculations for Matched-Sample t

When we want to test the difference between two matched samples, the problem becomes

a bit more difficult and an additional parameter must be considered. For this reason, the

analysis of power for this case is frequently impractical. However, the general solution to

the problem illustrates an important principle of experimental design, and thus justifies

close examination.

With a matched-sample ttest we definedas

where represents the expected difference in the means of the two populations of

observations (the expected mean of the difference scores). The problem arises because

is the standard deviation not of the populations of and , but of difference

scores drawn from these populations. Although we might be able to make an intelligent

guess at or , we probably have no idea about.

All is not lost, however; it is possible to calculate on the basis of a few assump-

tions. The variance sum law (discussed in Chapter 7, p. 204) gives the variance for a sum

or difference of two variables. Specifically,

If we make the general assumption of homogeneity of variance , for the

difference of two variables we have

where (rho) is the correlation in the population between and and can take on values

between 1 and 2 1. It is positive for almost all situations in which we are likely to want a

matched-sample t.

Assuming for the moment that we can estimate , the rest of the procedure is the same

as that for the one-sample t. We define

and

We then estimate as , and refer the value of dto the tables.

As an example, assume that I want to use the Aronson study of stereotype threat in

class, but this time I want to run it as a matched-sample design. I have 30 male subjects

sX 12 X 2 s 1 2(12r)

d=d 2 n

d=

m 1 2m 2

sX 12 X 2

r

r X 1 X 2

sX 12 X 2 =s 2 2(12r)

s^2 X 12 X 2 = 2 s^222 rs^2 = 2 s^2 (12r)

s^2 X 1 =s^2 X 2 =s^2

s^2 X 16 X 2 =s^2 X 1 1s^2 X 262 rsX 1 sX 2

sX 12 X 2

sX 1 sX 2 sX 12 X 2

sX 12 X 2 X 1 X 2

m 1 2m 2

d=

m 1 2m 2

sX 12 X 2

236 Chapter 8 Power

(^2) McClelland (1997) has provided a strong argument that when we have more than two groups and the independ-
ent variable is ordinal, power may be maximized by assigning disproportionately large numbers of subjects to the
extreme levels of the independent variable.