Statistical Methods for Psychology

14.1 The Structural Model

First, some theory to keep me happy. Two structural models could underlie the analysis of

data like those shown in Table 14.1. The simplest model is

where

the grand mean

a constant associated with the ith person or subject, representing how much

that person differs from the average person

a constant associated with the jth treatment, representing how much that

treatment mean differs from the average treatment mean

the experimental error associated with the ith subject under the jth treatment

The variables are assumed to be independently and normally distributed around zero

within each treatment. Their variances, and , are assumed to be homogeneous across

treatments. (In presenting expected means square, I am using the notation developed in the

preceding chapters. The error term and subject factor are considered to be random, so

those variances are presented as and. (Subjects are always treated as random.) How-

ever, the treatment factor is generally a fixed factor, so its variation is denoted as ) With these

assumptions it is possible to derive the expected mean squares shown in Model I of Table 14.2.

An alternative and probably more realistic model is given by

Here we have added a Subject 3 Treatment interaction term to the model, which allows

different subjects to change differently over treatments. The assumptions of the first model

will continue to hold, and we will also assume the to be distributed around zero inde-

pendently of the other elements of the model. This second model gives rise to the expected

mean squares shown in Model II of Table 14.2.

The discussion of these two models and their expected mean squares may look as if it is

designed to bury the solution to a practical problem (comparing a set of means) under a

mountain of statistical theory. However, it is important to an explanation of how we will run

our analyses and where our tests come from. You’ll need to bear with me only a little longer.

14.2 FRatios

The expected mean squares in Table 14.2 indicate that the model we adopt influences the F

ratios we employ. If we are willing to assume that there is no Subject 3 Treatment interac-

tion, we can form the following ratios:

E(MSbetween subj)

E(MSerror)

=

s^2 e 1 ks^2 p

s^2 e

ptij

Xij=m1pi1tj1ptij 1 eij

u^2 t

s^2 p s^2 e

s^2 p s^2 e

pi and eij

eij=

tj=

pi=

m=

Xij=m1pi1tj 1 eij

464 Chapter 14 Repeated-Measures Designs

Table 14.2 Expected mean squares for simple repeated-measures designs

Model I Model II

Source E(MS) Source E(MS)

Subjects Subjects

Treatments Treatments

Error s^2 e Error se^2 1s^2 pt

s^2 e 1 nut^2 s^2 e1spt^21 nut^2

s^2 e 1 ks^2 p s^2 e 1 ksp^2

Xij=m1pi1tj 1 eij Xij=m1pi1tj1ptij 1 eij