Statistical Methods for Psychology

anticipated tolerance effect, and looking at the Different group, you see that it is much more

like the Control group than it is like the Same group. This is the result that King predicted.

Very little needs to be said about the actual calculations in Table 14.4b, since they are

really no different from the usual calculations of main and interaction effects. Whether a

factor is a between-subjects or within-subjects factor has no bearing on the calculation of

its sum of squares, although it does affect its placement in the summary table and the ulti-

mate calculation of the corresponding F.

In the summary table in Table 14.4c, the source column reflects the design of the ex-

periment, with first partitioned into and. Each of these sums

of squares is further subdivided. The double asterisks next to the three terms show we cal-

culate these by subtraction ( , , and ), based on the

fact that sums of squares are additive and the whole must be equal to the sum of its parts.

This simplifies our work considerably. Thus

These last two terms will become error terms for the analysis.

The degrees of freedom are obtained in a relatively straightforward manner. For each

of the main effects, the number of degrees of freedom is equal to the number of levels

of the variable minus 1. Thus, for Subjects there are 24 2 1 523 df, for Groups there are

32 1 52 df, and for Intervals there are 6 2 1 55 df. As for all interactions, the dffor I 3 G

is equal to the product of the dffor the component terms. Thus,.

The easiest way to obtain the remaining degrees of freedom is by subtraction, just as we

did with the corresponding sums of squares.

These dfcan also be obtained directly by considering what these terms represent. Within

each subject, we have 6 21 55 df. With 24 subjects, this amounts to

Within each level of the Groups factor, we have 8 21 57 dfbetween subjects, and with three

Groups we have. I 3 Ss w/in groups is really an interaction term, and

as such its dfis simply the product of and.

Skipping over the mean squares, which are merely the sums of squares divided by their

degrees of freedom, we come to F. From the column of Fit is apparent that, as we anticipated,

Groups and Intervals are significant. The interaction is also significant, reflecting, in part, the

fact that the Different group was at first intermediate between the Same and the Control group,

but that by the second 5-minute interval it had come down to be equal to the Control group.

This finding can be explained by a theory of conditioned tolerance. The really interesting find-

ing is that, at least for the later intervals, simply injecting an animal in an environment different

from the one in which it had been receiving the drug was sufficient to overcome the tolerance

that had developed. These animals respond almost exactly as do animals that had never experi-

enced midazolam. We will return to the comparison of Groups at individual Intervals later.

Assumptions

For the Fratios actually to follow the Fdistribution, we must invoke the usual assumptions

of normality, homogeneity of variance, and sphericity of. For the between-subjectsterm(s),

this means that we must assume that the variance of subject means within any one level of

gN

dfI dfSs w/in groups 5 (5)(21)= 105

(7)(3)= 21 dfw/in groups

(5)(24) 5120 dfw/in subj.

dfI 3 Ss w/in groups=dfw/in subj 2 dfintervals 2 dfIG

dfSs w/in groups=dfbetween subj 2 dfgroups

dfw/in subj=dftotal 2 dfbetween subj

dfIG=(6–1)(3–1)= 10

SSI 3 Ss w/in groups=SSw/in subj 2 SSintervals 2 SSIG

SSSs w/in groups=SSbetween subj 2 SSgroups

SSw/in subj=SStotal 2 SSbetween subj

SSw/in subj SSSs w/in groups SSI 3 Ss w/in groups

SStotal SSbetween subj SSw/in subj

Section 14.7 One Between-Subjects Variable and One Within-Subjects Variable 475