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In a complete two-factor design with equal cell replication, centred factor indicator columns for different factors have zero cross-products. Consequently the two main effects in an additive normal linear model are orthogonal, and their extra sums of squares do not depend on the order in which these main effects are entered. An omitted interaction term can still require a separate adequacy check.

A logarithmic count-component contrast $b$ multiplies the susceptible mean by $e^b$, while a zero-logit contrast $g$ multiplies the structural-zero odds by $e^g$. If the baseline zero predictor is $\eta$, the marginal mean ratio isThus the count ratio alone is not a population-average effect when a covariate changes both components. With interaction terms, the relevant count contrast must first include the interactions for the stated reference group.

A factorial design includes every combination of specified factor levels. Balanced replication permits separate estimation of marginal effects and interaction terms. The factor allocation, not the number of subsamples, determines the appropriate error term for each effect.

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.

An interaction contrast compares a treatment difference across levels of another factor. A nonzero difference of differences shows that an additive representation of those cell means is inadequate. Its standard error must use the error stratum in which that interaction term is randomized.

A statistical interaction means that the effect of one predictor depends on another on the chosen response scale. In a linear regression, an interaction term ${\beta }_{3}XZ$ changes the slope in $X$ to ${\beta }_1+{\beta }_{3}Z$. Whether interaction is present depends on the comparison scale; additive effects and multiplicative effects use different null contrasts. Comparing subgroup effects requires an interaction contrast, rather than contrasting their separate significance labels.