On the calculation of derived variables in the analysis of multivariate responses

The multivariate regression of a p - 1 vector Y of random variables on a q - 1 vector X of explanatory variables is considered. It is assumed that linear transformations of the components of Y can be the basis for useful interpretation whereas the components of X have strong individual identity. When p >= q a transformation is found to a new q - 1 vector of responses Y* such that in the multiple regression of, say, Y1* on X, only the coefficient of X1 is nonzero, i.e. such that Y1* is conditionally independent of X2, ..., Xq, given X1. Some associated inferential procedures are sketched. An illustrative example is described in which the resulting transformation has aided interpretation.