Two-factor normal linear model ($Y_{aj\ell }=\mu +{\alpha }_a+{\gamma }_j+{\delta }_{aj}+{\epsilon }_{aj\ell }$)

A two-factor normal linear model describes independent responses with factor-specific means and common normal error variance. The additive mean $\mu +{\alpha }_a+{\gamma }_j$ assumes no interaction term; the full mean $\mu +{\alpha }_a+{\gamma }_j+{\delta }_{aj}$ permits the first factor effect to vary with the second. Constraints or treatment coding identify the parameters. In a balanced two-by-five design the full model has ten free cell means.

To pool factor levels in a normal linear model, impose equality of their fitted means while retaining any needed interaction terms. Compare the resulting nested model with the full model using a nested-model F-test. For two cells each of size $m$, pooling their fitted means adds $(m/2)({\overline{Y}}_1-{\overline{Y}}_2{)}^2$ to the residual sum of squares. If pooling two pairs separately within two other-factor levels, there are four restrictions.