Statistical Methods for Psychology

Notice in Table 13.10 that when we have a nested design with a random variable nested

within a fixed variable our Fstatistic is going to be computed differently. We can test the

effect of Therapist(Gender) by dividing MST(G)by MSerror, but when we want to test Gender

we must divide MSGby MST(G). The resulting Fs are shown in Table 13.11, where I have

labeled the error terms to indicate how the Fs were constructed.

Notice that the Gender effect has the same sum of squares that it had in the original

study, but the Fis quite different because Therapist(Gender) served as the error term and

there was considerable variability among therapists. Notice also that SSTherapist(Gender)is

equal to the sum of SSConditionand SSAge 3 Conditionin the first example, although I prefer to

think of it as the sum of the two simple effects.)

Having a random factor such as Therapist often creates a problem. We really set out to

study Gender differences, and that is what we most care about. We don’t really care much

about therapist differences because we know that they will be there. But the fact that Ther-

apist is a random effect, which it should be, dramatically altered our test on Gender. The F

went from nearly 30 to nearly 1.0. This is a clear case where the design of the study has a

dramatic effect on power, even with the same values for the data. Maxwell and Delaney

(2004) make the point that in designs with random factors, power depends on both the

number of subjects (here, clients) and the number of levels of the random variable (here,

therapists). Generally the number of levels of the random variable is far more important.

Summary

I have presented three experimental designs. The crossed design with fixed factors is the

workhorse for most traditional experimental studies. The nested design with a random fac-

tor is an important design in much research in education and more applied areas of psy-

chology. The crossed design with a random factor occurs occasionally but is not as

common. In general when you have crossed effects they are most often fixed, and when

you have nested effects the nested factor is most often random. This helps to explain why

Section 13.8 Expected Mean Squares and Alternative Designs 437

Table 13.10 Expected mean squares for nested designs

Fixed Random Mixed

A fixed A random A fixed

Source Bfixed Brandom Brandom

A

B(A)

Error

s^2 e 1 ns^2 ab 1 nbu^2 a

s^2 e 1 ns^2 b

s^2 e

s^2 e 1 ns^2 b 1 nbs^2 a

s^2 e 1 ns^2 b

s^2 e

s^2 e 1 nbu^2 a

s^2 e 1 nau^2 b

s^2 e

Table 13.11 Tests for a nested design with a random nested factor

Source df SS MS F

Gender 1 240.25 240.250 1.127

Error 1 8 1705.24 213.155

Therapist(Gender) 8 1705.24 213.155 26.56*

Error 2 90 722.300 8.026

Total 99 2667.79

*p,.05