Multivariate interactions in regression

A “multivariate interaction” in a regression model is a product of two independent variates (linear functions of the regressors) that is an additive component of the regression function E(Y |X). In many cases a substantial portion of the overall pairwise interaction structure in a regression function can be captured by a single multivariate interaction. Due to its parsimonious form, a multivariate interaction can be estimated directly, without the use of sequential variable selection techniques. Inference can be carried out using a straightforward likelihood ratio approach, which yields hypothesis tests of considerably greater power than a comparable regression test when the number of regressor variables is large. We investigate three methods for estimating multivariate interactions. The optimizations required in computing these estimates are neither orthogonalization nor eigenvalue problems, as is the case for other common regression and dimension reduction estimators. We describe an efficient algorithm, present results from a simulation study, and demonstrate the method using U.S. temperature field data and data from a small experiment.