Rank-Based Analysis of the Heteroscedastic Linear Model

Abstract

Heteroscedasticity often causes problems in the analysis of linear models. Frequently for such cases, scale is a function of the response. Here, for such models, a methodology is presented based on well-known rank-based procedures. The procedure is iterative. Using the residuals from an initial R estimate of the regression coefficients, scale is estimated by inverting a linear rank test for scale. This in turn leads to a weighted R estimate of the regression coefficients and then to a final estimate of scale. Asymptotic linearity results for these estimates are derived, from which their asymptotic distribution is obtained. The weighted R estimate has the same asymptotic distribution as the optimal (known scale) R estimate; hence it is efficiently robust. Consistent estimates of the standard errors of the R estimates of scale and the regression coefficients are determined. Based on these results, a complete inference for the regression coefficients and the scale parameter is realized, including a test for homogeneity. The procedure is illustrated by the analysis of a dose—response data set drawn from pharmaceutical science. Results of a simulation study that support the asymptotic theory are reported.

Key Words:

• Linear rank statistics
• Nonconstant variance
• R estimates
• R-scale estimates
• Robust
• Weighted estimates