Interaction contrast ($({\mu }_{1a}-{\mu }_{1b})-({\mu }_{0a}-{\mu }_{0b})$)

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.

For binary predictors $A,B$, a logistic regression with linear predictor $a+bA+cB+\delta AB$ has conditional odds ratios $e^b$ at $B=0$ and $e^{b+\delta }$ at $B=1$. Their ratio is $e^{\delta }$. The same interaction can be computed from the four cell logit link values as $\delta ={\eta }_{11}-{\eta }_{10}-{\eta }_{01}+{\eta }_{00}$. Independent positive event and nonevent counts in each cell give an estimated variance for this contrast equal to the sum of all eight reciprocal counts. An interaction on the odds ratio scale is not an interaction on the absolute-risk scale.