15.4: Factorial Designs

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Extending from the previous example, let us say that the effect of the special curriculum (treatment) relative to traditional curriculum (control) depends on the amount of instructional time (three or six hours per week). Now, we have a $2 \times 2$ factorial design, with the two factors being curriculum type (special versus traditional) and instructional type (three or six hours per week). Such a design not only helps us estimate the independent effect of each factor, called main effects, but also the joint effect of both factors, called the interaction effect. The generalised linear model for this two-way factorial design is designated as follows:

$y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{1} x_{2} + \varepsilon\,,$

where represents students’ post-treatment performance scores, $x_{1}$ is the treatment (special versus traditional curriculum), $x_{2}$ is instructional time (three or six hours per week). Note that both $x_{1}$ and $x_{2}$ are dummy variables, and although $x_{2}$ looks like a ratio-scale variable (3 or 6), it actually represents two categories in the factorial design. Regression coefficients $\beta_{1}$ and $\beta_{2}$ provide effect size estimates for the main effects and $\beta_{3}$ for the interaction effect. Alternatively, the same factorial model can be analysed using a two-way ANOVA analysis. Regression analysis involving multiple predictor variables is sometimes called multiple regression, which is different from multivariate regression that uses multiple outcome variables.

A note on interpreting interaction effects. If $\beta_{3}$ is significant, it implies that the effect of the treatment (curriculum type) on student performance depends on instructional time. In this case, we cannot meaningfully interpret the independent effect of the treatment ( $\beta_{1}$ ) or of instructional time ( $\beta_{2}$ ), because the two effects cannot be isolated from each other. Main effects are interpretable only when the interaction effect is non-significant.

Covariates can be included in factorial designs as new variables, with new regression coefficients—e.g., $\beta_{4}$ . Covariates can be measured using interval or ratio-scaled measures, even when the predictors of interest are designated as dummy variables. Interpretation of covariates also follows the same rules as that of any other predictor variable.