Partial one-dimensional regression models

Many regression problems have one grouping variable or factor and several covariates, and the goal of the problem is to understand differences in the regressions between levels of the grouping variable. Standard approaches to this require fitting main effects and grouping variable by covariate interactions, potentially leading to complex models if the number of covariates is greater than one. Using the ideas of the dimension of a regression problem, and dimension reduction, we describe the use of one-dimensional models fit separately to each level of the grouping variable. These models are less general than the interaction models usually studied, but they can lead to very simple results and can also lead to simple and useful summary graphs.