Statistical Analysis for Education and Psychology Researchers

Repeated measurement designs are also frequently analysed using ANOVA
techniques. Repeated measures or observations are treated as a factor in the analysis with
measurements on one variable at different occasions corresponding to levels of the factor.
The same subjects are involved in repeated measures. It is also possible to have mixed or
‘split-plot’ designs (a term derived from the initial development of the technique with
agricultural experiments) where there are two factors, one of which is a repeated
measurement where the same subjects are used and on the other where different subjects
are used (between-subjects factor). This chapter serves only as an introduction to the
analysis of experimental designs. There are many more complex designs requiring
sophisticated analytic strategies. The interested reader is referred to Mead (1992) for a
comprehensive guide to the principles of experimental design and analysis.

ANOVA and the General Linear Model

Consideration of analysis of variance from the point of view of an underlying general
linear model means that its relationship to regression can easily be seen; more
importantly this approach will form a foundation for the use of more sophisticated
techniques such as multivariate analysis of variance (MANOVA—this is used when there
are multiple response variables rather than a single response variable as is the case with
univariate ANOVA), factor analysis and discriminant analysis. The underlying general
linear model helps integrate ANOVA and regression which are often treated as
independent analytic strategies. In fact, ANOVA is a special case of multiple linear
regression. This common framework also helps the researcher see why ANOVA and
regression share many of the same underlying assumptions. Most proprietary computer
programmes for statistical analysis present data in a form consistent with the underlying
general linear model, and unless you understand commonalities and differences between
ANOVA and regression, you will be reduced to learning the meaning of computer output
by rote rather than understanding and reporting with insight.
Consider the linear model for the one-factor vocabulary teaching experiment
introduced in Chapter 3. It is represented here as an illustration of the general form of the
ANOVA statistical model for a one-factor design,
yij=μ+αi+εij
Statistical model for one-
factor ANOVA—8.20

This general linear model describes the observed vocabulary score for the jth individual
pupil from the ith treatment, yij, as the sum of three separate components: i) a response
common to all pupils in the target population of interest, μ, hence the term mean
response. This represents the average score of all pupils in the experiment; ii) a deviation
from the mean response for a particular treatment group, αi. In this experiment there are
three treatments, so we have α 1 corresponding to all pupils who receive silent reading, α 2
corresponding to the storytelling only condition, and α 3 corresponding to the storytelling
enhanced by pictures condition; iii) a unique deviation from the average treatment
response for a particular jth pupil in the ith treatment, εij. This is called the error term and
in ANOVA is estimated as the deviation of the observed score from the appropriate
treatment cell mean.

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