Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

Chun Li (), Yuqi Tian, Donglin Zeng and Bryan E. Shepherd
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Chun Li: Division of Biostatistics, Department of Population and Public Health Sciences, University of Southern California, Los Angeles, CA 90033, USA
Yuqi Tian: Department of Biostatistics, Vanderbilt University, Nashville, TN 37203, USA
Donglin Zeng: Department of Biostatistics, University of Michigan, Ann Arbor, MI 48109, USA
Bryan E. Shepherd: Department of Biostatistics, Vanderbilt University, Nashville, TN 37203, USA

Mathematics, 2023, vol. 11, issue 24, 1-21

Abstract: Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.

Keywords: cumulative probability model; semiparametric transformation model; uniform consistency; asymptotic distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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