IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i24p4896-d1295539.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/24/4896/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/24/4896/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jin-Jian Hsieh & A. Adam Ding & Weijing Wang, 2011. "Regression Analysis for Recurrent Events Data under Dependent Censoring," Biometrics, The International Biometric Society, vol. 67(3), pages 719-729, September.
    2. Yanqing Sun & Rajeshwari Sundaram & Yichuan Zhao, 2009. "Empirical Likelihood Inference for the Cox Model with Time‐dependent Coefficients via Local Partial Likelihood," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 444-462, September.
    3. Chyong-Mei Chen & Pao-sheng Shen & Yi Liu, 2021. "On semiparametric transformation model with LTRC data: pseudo likelihood approach," Statistical Papers, Springer, vol. 62(1), pages 3-30, February.
    4. Donglin Zeng & Fei Gao & D. Y. Lin, 2017. "Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data," Biometrika, Biometrika Trust, vol. 104(3), pages 505-525.
    5. Jianbo Li & Minggao Gu & Tao Hu, 2012. "General partially linear varying-coefficient transformation models for ranking data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(7), pages 1475-1488, January.
    6. Xi Ning & Yinghao Pan & Yanqing Sun & Peter B. Gilbert, 2023. "A semiparametric Cox–Aalen transformation model with censored data," Biometrics, The International Biometric Society, vol. 79(4), pages 3111-3125, December.
    7. Pao-sheng Shen, 2011. "Semiparametric analysis of transformation models with left-truncated and right-censored data," Computational Statistics, Springer, vol. 26(3), pages 521-537, September.
    8. Lin, Huazhen & Peng, Heng, 2013. "Smoothed rank correlation of the linear transformation regression model," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 615-630.
    9. Lu Mao & D. Y. Lin, 2017. "Efficient estimation of semiparametric transformation models for the cumulative incidence of competing risks," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 573-587, March.
    10. Yilong Zhang & Xiaoxia Han & Yongzhao Shao, 2021. "The ROC of Cox proportional hazards cure models with application in cancer studies," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 27(2), pages 195-215, April.
    11. Pao-sheng Shen, 2012. "Analysis of left-truncated right-censored or doubly censored data with linear transformation models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 584-603, September.
    12. Pao-sheng Shen, 2014. "Semiparametric regression analysis for clustered doubly-censored data," Computational Statistics, Springer, vol. 29(3), pages 813-828, June.
    13. Lu Chen & Li Hsu & Kathleen Malone, 2009. "A Frailty-Model-Based Approach to Estimating the Age-Dependent Penetrance Function of Candidate Genes Using Population-Based Case-Control Study Designs: An Application to Data on the BRCA1 Gene," Biometrics, The International Biometric Society, vol. 65(4), pages 1105-1114, December.
    14. Jin Wang & Donglin Zeng & D. Y. Lin, 2022. "Semiparametric single-index models for optimal treatment regimens with censored outcomes," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(4), pages 744-763, October.
    15. Wang, Xuan & Wang, Qihua, 2015. "Semiparametric linear transformation model with differential measurement error and validation sampling," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 67-80.
    16. Huazhen Lin & Ling Zhou & Chunhong Li & Yi Li, 2014. "Semiparametric transformation models for semicompeting survival data," Biometrics, The International Biometric Society, vol. 70(3), pages 599-607, September.
    17. Changrong Yan & Dixin Zhang, 2013. "Sparse dimension reduction for survival data," Computational Statistics, Springer, vol. 28(4), pages 1835-1852, August.
    18. Pao-sheng Shen & Yi Liu, 2019. "Pseudo maximum likelihood estimation for the Cox model with doubly truncated data," Statistical Papers, Springer, vol. 60(4), pages 1207-1224, August.
    19. Chen, Songxi, 2012. "Estimation in semiparametric models with missing data," MPRA Paper 46216, University Library of Munich, Germany.
    20. Mingzhe Wu & Ming Zheng & Wen Yu & Ruofan Wu, 2018. "Estimation and variable selection for semiparametric transformation models under a more efficient cohort sampling design," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 570-596, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4896-:d:1295539. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.