Sufficient dimension reduction with additional information

Abstract

Sufficient dimension reduction is widely applied to help model building between the response and covariate X. In some situations, we also collect additional covariate W that has better performance in predicting Y, but has a higher obtaining cost, than X. While constructing a predictive model for Y based on (X, W) is straightforward, this strategy is not applicable since W is not available for future observations in which the constructed model is to be applied. As a result, the aim of the study is to build a predictive model for Y based on X only, where the available data is (Y, X, W). A naive method is to conduct analysis using (Y, X) directly, but ignoring W can cause the problem of inefficiency. On the other hand, it is not trivial to utilize the information of W to infer (Y, X), either. In this article, we propose a two-stage dimension reduction method for (Y, X) that is able to utilize the information of W. In the breast cancer data, the risk score constructed from the two-stage method can well separate patients with different survival experiences. In the Pima data, the two-stage method requires fewer components to infer the diabetes status, while achieving higher classification accuracy than the conventional method.