Statistical Analysis for Education and Psychology Researchers

8.8 One-way ANOVA F-test (related)

When to Use

Many research designs in psychology involve subjects in repeated measurements, that is
the same (or matched) subjects participate in each of the experimental conditions.
Observations or measurements are therefore correlated and treatment effects are analysed
using a repeated measures analysis of variance. Another name for this analysis is within
subjects ANOVA because comparison of treatment effects is within subjects.
In a related ANOVA differences in scores attributable to individuals, (subjects) can be
treated as a separate source of variance because the same subjects take part in each of the
treatment conditions. This source of variance is called Subjects variance, SS(subj). The
other variance components, we would have in a One-way ANOVA are a source of
variance attributable to between treatment conditions, SS(bet), and error variance which
represents differences among subjects within each of the treatment conditions, SS(error).
Recall that in a One-way unrelated ANOVA we only have two sources of variance:
between treatments and within individuals which is the error variance.
In a related ANOVA the F-test of significance is usually constructed on the ratio,
MS(bet)/MS(error), the error term is reduced in comparison to what it would be in an
unrelated analysis because variance accounted for by subjects has been partitioned
separately. In most repeated measurement analyses Subjects are treated as a random
effect in the statistical model, the treatment effect is usually considered to be fixed. The
distinction between fixed and random effects is of importance when there is more than
one factor; this is discussed in section 8.9. In a related ANOVA the MS(error) is not a pure
error term because part of the variation within subjects is attributable to the different
treatments, and part is due to individual differences. An F-test of significance for
Subjects is not therefore valid unless we are willing to assume that there is no interaction
between subjects and treatments. Ordinarily this is not a problem because the researcher
is interested in differences among treatment means and not differences among Subjects.
The assumptions for related ANOVA are the same as for the unrelated analysis with
the additional requirements of homogeneity of covariance among population error terms
for the different treatments and independence of errors for the different treatment
conditions. This assumption is of little concern in a practical setting but the interested
reader should consult Winer (1962), Chapter 4.

Example from the Literature

In an experiment designed to investigate the effects of visual interference on visuospatial
working memory (Toms, Morris and Foley, 1994) twelve subjects (university
undergraduates) were presented with two tasks—one spatial imagery and the other
verbal. Subjects performed each task under four conditions (eyes shut; looking at blank
screen; looking at a white square; and looking at a changing pattern) the order of
conditions was counterbalanced using a Latin square design (for explanation of this
design see Winer, 1962). Each condition was presented in a block of four trials. The
response variable score was the mean number of correctly recalled sentences per trial in

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