Statistical Analysis for Education and Psychology Researchers

intervention (α 1 =storytelling, α 2 =storytelling+pictures), βj is the population effect for sex
(β 1 =male, β 2 =female), αβij is the interaction effect of treatments and εijk is the error term
for pupil k. The interaction term in the model represents the average effect on the pupils’
vocabulary score attributable to a particular combination of teaching method and sex.
The full 2-Way ANOVA statistical model can be rewritten in a regression format,
Yijk=μx 0 +α 2 x 2 +β 1 x 3 +β 2 x 4 +αβ 11 x 5 +αβ 12 x 6 +αβ 21 x 7 +αβ 22 x 8 +εijk

Each value of x will be either 0 or 1 depending upon the treatment combination. For
example, a pupil who was in treatment combination 2 for both factors (α=2 is
storytelling+pictures and β=2 is female) would have: x 0 set to 1 because the overall mean
always has an effect, x 2 , x 4 and x 8 would also be set to 1 because they represent the main
effects of storytelling+pictures, the main effect of being a female and the interaction
effect of being in the storytelling+pictures and female group. The other x’s would be set
to zero (in the regression framework x is the value of what is called an indicator variable)
indicating that the other treatment effects and combinations do not contribute to pupil k’s
score.
When comparing the ANOVA and regression statistical models a commonality which,
on reflection, should be clear is that the response variable is hypothesized to be a
weighted combination of independent variables, in regression these weights are called
regression coefficients and in ANOVA they are called treatment effects. Both models
are also linear in their parameters, that is the weighted parameters are assumed to be
additive. In ANOVA this is termed the ‘additivity’ of the model and in regression the
term linearity of the model is used.

Significance Tests and Estimation in ANOVA

As in regression analysis, the sums of squares derived from sample data are used to
estimate the various components of the ANOVA model. Sums of squares for the overall
model are partitioned into component sums of squares representing independent
variables, any interactions and error variance. Associated with each source of variance
are degrees of freedom, mean squares and F-statistics. These component sums of squares
and associated statistics are output in most statistical packages (although the terminology
might vary).
The general linear model approach to testing the significance of a linear model
(significant model effect) is to compare the fit of two statistical models, a full model
(sometimes called an effects model) and a reduced model (when there is no ‘treatment’
this is called a means only model). In a One-way ANOVA the full model, where the
factor has an effect, is:
yij=μ+αi+εij

The reduced model which is just the overall mean effect and underlying variation, is
yij=μi+εij

yij is the value for the jth observation for the mean treatment i plus underlying variation.

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