Statistical Methods for Psychology

Nested Designs

Now let’s modify our basic study again while retaining the same values of the dependent

variable so that we can compare results. Suppose that your clinical psychology program is

genuinely interested in whether female students are better therapists than male students. To

run the study the department will randomly sample 10 graduate students, split them into

two groups based on Gender, and have each of them work with 10 clients and produce a

measure of treatment effectiveness. In this case Gender is certainly a fixed variable because

every replication would involve Male and Female therapists. However, Therapist is best

studied as a random factor because therapists were sampled at random and we would want

to generalize to male and female therapists in general, not just to the particular therapists

we studied. Therapist is also a nestedfactor because you can’t cross Gender with Therapist—

Mary will never serve as a male therapist and Bob will never serve as a female therapist.

Over many replications of the study the variability in Fwill depend on random error

(MSerror) and also on the therapists who happen to be used. This variability must be taken

into account when we compute our Fstatistics.^3

The study as I have described it looks like our original example, but it really is not. In this

study therapists are nestedwithin gender. (Remember that in the first example each Condi-

tion (adjective, etc.) was paired with each Age, but that is not the case here.) The fact that we

have a nested design is going to turn out to be very important in how we analyze the data. For

one thing we cannot compute an interaction. We obviously cannot ask if the differences

between Barbara, Lynda, Stephanie, Susan, and Joan look different when they are males than

when they are females. There are going to be differences among the five females, and there

are going to be differences among the five males, but this will not represent an interaction.

In running this analysis we can still compute a difference due to Gender, and for these

data this will be the same as the effect of Case is the previous example. However, when we

come to Therapist we can only compute differences due to therapists within females, and

differences due to therapist within males. These are really just the simple effects of Therapist

at each Gender. We will denote this as “Therapist within Gender” and write it as Thera-

pist(Gender). As I noted earlier, we cannot compute an interaction term for this design, so

that will not appear in the summary table. Finally we are still going to have the same source

of random error as in our previous example, which, in this case, is a measure of variability

of client scores within each of the Gender/Therapist cells.

For a nested design our model will be written as

Notice that this model has a term for the grand mean a term for differences between

genders and a term for differences among therapists, but with subscripts indicating

that Therapist was nested within Gender There is no interaction because none can

be computed, and there is a traditional error term (eijk).

Calculation for Nested Designs

The calculations for nested designs are straightforward, though they differ a bit from what

you are used to seeing. We calculate the sum of squares for Gender the same way we

always would—sum the squared deviations for each gender and multiply by the number of

observations for each gender. For the nested effect we simply calculate the simple effect of

therapist for each gender and then sum the simple effects. For the error term we just calculate

the sum of squares error for each Therapist/Gender cell and sum those. The calculations

are shown in the Table 13.9. However before we can calculate the Fvalues for this design

(bj(i)).

(ai),

(m),

Xijk=m1ai1bj(i) 1 eijk

Section 13.8 Expected Mean Squares and Alternative Designs 435

(^3) It is possible to design a study in which a nested variable is a fixed variable, but that rarely happens in the
behavioral sciences and I will not discuss that design except to show the expected mean squares in a table.