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Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

Author

Listed:
  • 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)

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.

Suggested Citation

  • Chun Li & Yuqi Tian & Donglin Zeng & Bryan E. Shepherd, 2023. "Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes," Mathematics, MDPI, vol. 11(24), pages 1-21, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4896-:d:1295539
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    References listed on IDEAS

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    1. Torsten Hothorn & Lisa Möst & Peter Bühlmann, 2018. "Most Likely Transformations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 45(1), pages 110-134, March.
    2. Kani Chen, 2002. "Semiparametric analysis of transformation models with censored data," Biometrika, Biometrika Trust, vol. 89(3), pages 659-668, August.
    3. D. Zeng & D. Y. Lin, 2007. "Maximum likelihood estimation in semiparametric regression models with censored data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 507-564, September.
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