Nonparametric Multivariate Inference on Shift Parameters

Abstract

Consider a study evaluating the prognostic value of prostate-specific antigen (PSA), in the presence of Gleason score, in differentiating between early and advanced prostate cancer. This data set features subjects divided into two groups (early versus advanced cancer), with one manifest variable (PSA), one covariate (Gleason score), and one stratification variable (hospital, taking three values). We present a nonparametric method for estimating a shift in median PSA score between the two groups, after adjusting for differences in Gleason score and stratifying on hospital. This method can also be extended to cases involving multivariate manifest variable. Our method uses estimating equations derived from an existing rank-based estimator of the area under the receiver operating characteristic curve (AUC). This existing AUC estimator is adjusted for stratification and for covariates. We use the adjusted AUC estimator to construct a family of tests by shifting manifest variables in one of the treatment groups by an arbitrary constant. The null hypothesis for these tests is that the AUC is half. We report the set of shift values for which the null hypothesis is not rejected as the resulting confidence region. Simulated data show performance consistent with the distributional approximations used by the proposed methodology. This methodology is applied to two examples. In the first example, the mean difference in PSA levels between advanced and nonadvanced prostate cancer patients is estimated, controlling for Gleason score. In the second example, to assess the degree to which age and baseline tumor size are prognostic factors for breast cancer recurrence, differences in age and tumor size between subjects with recurrent and nonrecurrent breast cancer, stratified on Tamoxifen treatment and adjusting for tumor grade, are estimated. The proposed methodology provides a nonparametric method for a statistic measuring adjusted AUC to be used to estimate shift between two manifest variables.