The dr package

The dr package for R for dimension reduction regression was first documented in Weisberg (2002). This is a revision of that article, to correspond to version 3.0.0 of dr for R added to CRAN (cran.r-project.org) in Fall 2007. Regression is the study of the dependence of a response variable y on a collection of p predictors collected in x. In dimension reduction regression, we seek to find a few linearly independent linear combinations β′ 1x, . . . , β ′ dx, such that all the information about the regression is contained in these d linear combinations. If d is very small, perhaps one or two, then the regression problem can be summarized using simple graphics; for example, for d = 1, the plot of y versus β′ 1x contains all the regression information. When d = 2, a 3D plot contains all the information. Formal estimation methods can additionally be used given the lower dimensional problem based on the few linear combinations. The primary goals of dimension reduction regression are determining the dimension d, and estimating β1, . . . , βd, or more precisely the subspace of < spanned by β1, . . . , βd. Several methods for these goals have been suggested in the literature and some of these are implemented in dr. Methods implemented in the package include sliced inverse regression or sir, sliced average variance estimates, or save, principal Hessian directions, or phd, and inverse regression estimation, or ire. The first three of these methods estimate the span of the βs by examining the inverse regression problem of x|y, rather than the forward regression problem of y|x, and all estimate the βs as the solution of an eigenvalue problem. The method ire is new in Version 3.0.0 and estimates a basis for the subspace of interest by minimizing a quadratic objective function. Determining d is approached as a testing problem. Marginal dimension tests are used for a hypothesis of the form d = d0 versus an alternative d > d0, without restriction on the subspaces involved other than their dimension. The first of these tests was proposed by Li (1991) in his paper on sir. Added to Version 3.0.0 are tests of coordinate hypotheses, proposed by Cook (2004), which effectively provides a test for elimination of predictors to reduce dimension, although the method is more general than this. These latter tests provide the basis for stepwise elimination of predictors. These tests are available in dr for methods sir, save, and ire.