A Study of Estimation in a Normal Heteroscedastic Regression Model with Type I Censored Data

Much interest has focused on the regression problem with censored data, which has usefulness In biological, pharmaceutical, industrial, survival, and failure time experiments. In these last three applications, the usual assumption is that data after a suitable transformation follow a normal, extreme value, or other distribution. Although usual methods of analysis have traditionally assumed constant dispersion for all responses, some recent interest has focused on censored regression models that accomodate heterogeneity of dispersion by modeling it as a function of location parameters, known variables, and possibly unknown additional parameters. The model can accomodate nonconstant dispersion or serve as the basis for formal assessment. For uncensored data, there is a large literature for heteroscedastic regression that has its roots in normal theory. One key issue is the choice between full normal maximum likelihood or generalized least squares methods which are inefficient but offer robustness properties and relative ease and stability of computation not enjoyed by the former method. In this article, a similar issue is considered for a normal heteroscedastic regressIOn model with Type I censored data, and an estimation method analogous to generalized least squares is developed that, although again inefficient, may be computed by an intuitive procedure with standard software and compares favorably to maximum likelihood. An interesting application of the normal heteroscedastic regression model is to the study of a detection limit problem for assay data. These data are approximately normal, heteroscedastic, and often such that variance is small relative to the range of mean response. Often, there is a detection limit such that the response can not be observed if sufficiently small, and it is common to ignore these censored observations in the analysis. "Small sigma" asymptotic theory used by in Davidian and Carroll (1987) and others is used to show that, theoretically, this practice may not be unreasonable, although to be safe accomdation of censoring may be wise. Furthermore, under this theory, deviations from the assumed normal distribution may also be tolerated.