Sufficient dimension reduction with additional information

Sufficient dimension reduction is widely applied to help model building between the response [Formula: see text] and covariate [Formula: see text] In some situations, we also collect additional covariate [Formula: see text] that has better performance in predicting [Formula: see text], but has a higher obtaining cost, than [Formula: see text] While constructing a predictive model for [Formula: see text] based on [Formula: see text] is straightforward, this strategy is not applicable since [Formula: see text] 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 [Formula: see text] based on [Formula: see text] only, where the available data is [Formula: see text] A naive method is to conduct analysis using [Formula: see text] directly, but ignoring [Formula: see text] can cause the problem of inefficiency. On the other hand, it is not trivial to utilize the information of [Formula: see text] to infer [Formula: see text], either. In this article, we propose a two-stage dimension reduction method for [Formula: see text] that is able to utilize the information of [Formula: see text] 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.