Statistical Analysis for Education and Psychology Researchers

As an illustration of how the statistical model apportions effects of the independent
variables treatment effects, consider a pupil in the silent reading condition who scores 16
on the vocabulary test. This score can be decomposed into the three components: i) the
population mean score is 9; ii) the difference between the population mean score and the
treatment mean for all pupils in the silent reading condition, say a later treatment mean of
12; and iii) the difference between the pupils score and the contribution of the mean
treatment effect. The three components of the pupils score are as follows:
16 = μ +α 1 +εij
Population mean score Treatment mean score Error residual score
9 12
16 = 9 + 12–9 + 16 − 12

One point of difference between ANOVA and regression is in the estimation of the error
term. In regression it is estimated as the difference between an observed and a predicted
score (based on the linear model) rather than as in ANOVA, the deviation between an
observed score and a cell mean. These different procedures can lead to different error
estimates and associated degrees of freedom. Interpretation may also be different and this
depends upon the assumptions the researcher makes about the relationship between the
independent variables and the response variable. Estes (1991) discusses these points with
illustrated examples.

Comparison of ANOVA and Regression Models

For the purpose of comparing the structural (statistical) models for ANOVA and
regression, a two-factor design is described so that the interaction term in the model can
be illustrated and interpretation of this effect discussed. Assume we modify the
vocabulary experiment and make it a two-factor fixed effects design, one factor is sex
with two levels male, female, and the other factor is treatment with two levels
storytelling, and storytelling enhanced by pictures. The investigator wants to see whether
there is an added effect of pictures and whether this is the same for both males and
females. Sex is clearly a fixed effect (not under the control of the researcher) and
treatment can be considered a fixed effect if we assume that the two treatments are not
chosen at random from a range of possible treatments and the treatment would be the
same in all replications of the experiment.
The statistical model for a Two-way fixed effect ANOVA can be written as,

Full

model

for

ANO

VA 2-

way—

8.21

yijk represents the vocabulary score of the kth pupil, in treatment condition ij, μ is the
population mean vocabulary score, αi is the population treatment effect for the

Inferences involving continuous data 311