  Sufficient dimension reduction is widely applied to help model building
between the response $Y$ and covariate $X$. While the target of interest is the
relationship between $(Y,X)$, in some applications we also collect additional
variable $W$ that is strongly correlated with $Y$. From a statistical point of
view, making inference about $(Y,X)$ without using $W$ will lose efficiency.
However, it is not trivial to incorporate the information of $W$ to infer
$(Y,X)$. In this article, we propose a two-stage dimension reduction method for
$(Y,X)$, that is able to utilize the additional information from $W$. The main
idea is to confine the searching space, by constructing an envelope subspace
for the target of interest. In the analysis of 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 conventional method.
