Sufficient dimension reduction with additional information

Sufficient dimension reduction is widely applied to help model building between the response Y 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.