Constrained Inverse Regression for Incorporating Prior Information

Inverse regression methods facilitate dimension-reduction analyses of high-dimensional data by extracting a small number of factors that are linear combinations of the original predictor variables. But the estimated factors may not lend themselves readily to interpretation consistent with prior information. Our approach to solving this problem is to first incorporate prior information via theory- or data-driven constraints on model parameters, and then apply the proposed method, constrained inverse regression (CIR), to extract factors that satisfy the constraints. We provide chi-squared and t tests to assess the significance of each factor and its estimated coefficients, and we also generalize CIR to other inverse regression methods in situations where both dimension reduction and factor interpretation are important. Finally, we investigate CIR's small-sample performance, test data-driven constraints, and present a marketing example to illustrate its use in discovering meaningful factors that influence the desirability of brand logos.