1 / 27

A Proportional Odds Model with Time-varying Covariates

A Proportional Odds Model with Time-varying Covariates. Logistic Regression Model. Logistic regression model when outcome is binary How do we extend the logistic regression model for time-to-event outcome? It depends on how we view the time progression. 0. Time progression.

spencer
Download Presentation

A Proportional Odds Model with Time-varying Covariates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. A Proportional Odds Model with Time-varying Covariates

  2. Logistic Regression Model • Logistic regression model when outcome is binary • How do we extend the logistic regression model for time-to-event outcome? • It depends on how we view the time progression

  3. 0 Time progression Renewal time progression 0 0 Cumulative time progression Time Progression

  4. Extend Logistic Regression Model • Renewal time progression • Efron (1988, JASA) “Logistic-Regression, Survival Analysis and the Kaplan-Meier Curve” • Suppose time is counted by months • : # of patients at risk at the beginning of month • : # of patients who die during month • Assume that

  5. Extend Logistics Regression Model • Cox proportional hazards model • Interpretation of the regression parameter • Instantaneous hazards ratio • In terms of cumulative event rates

  6. Extend Logistics Regression Model • So, why this happens? • nonlinearity • The fundamental issue is how we deal with different denominators of summing fractions • What if we always count the cumulative events from time zero • Common denominator

  7. Proportional Odds Model • Logistic regression model • Proportional odds model with time-varying covariates at time : Yang & Prentice (1999, JASA)

  8. Proportional Odds Model • Yang-Prentice PO Model • Model closed under log-logisitic distributions • Interpretation of regression parameter • Without time-varying covariates • Special case of the transformation models when the error term follows standard logistic distribution with unspecified transformation • Rank estimation: Cheng, et al. (1995, Bmka) • NPMLE

  9. Proportional Odds Model • Transformation models with time-varying covariates • Kosorok, et al. (2004, Ann Stat) • is some frailty-induced Laplace transform • Zeng & Lin (2006, Bmka; 2007, JRSS-B) • is some known transformation, e.g., Box-Cox transformation • These models are not the Yang-Prentice models when the same error distributions/transformation would be chosen to obtain the proportional odds model without time-varying covariates

  10. Yang-Prentice Proportional Odds Model • Yang & Prentice (1999, JASA) • Inference procedures developed mostly without time-varying covariates • Time-varying covariates

  11. Estimation of Yang-Prentice PO Model • By way of integral equation for baseline odds function • Under Yang-Prentice PO model, individual hazard function is • Therefore, • Then we can solve it to get

  12. Estimation of Yang-Prentice PO Model • With time-varying covariates

  13. Estimation by Differential Equations • Consider • Let

  14. Estimating Equations for Baseline Function • Assume that we know

  15. Estimation of Baseline function • Then we solve to obtain a closed form solution for baseline odds function • Moreover • This shall lead to consistency and asymptotic normality of this baseline odds function estimator with true regression parameter

  16. Estimation of Regression Parameters • Estimating equations for regression parameters or • We can obtain all the necessary asymptotic properties of • Straightforward to extend to weighted estimation

  17. Consideration of Optimal Estimation • Hazard function under Yang-Prentice PO Model • A form of optimal weight function in weighted estimation is calculated as

  18. Simulation Studies • Simulation setup

  19. Data Analysis • VA Lung Cancer Clinical Trial (Prentice, 1973, Bmka) • Subgroup of 97 patients’ lung cancer survival with two covariates • Performance score • Tumor type • Bennett (1983, Stat Med) justified the PO model by a visual assessment of survival functions of dichotomized performance score • Most of the work analyzed this data without model checking. We include covariates and time interaction as time-varying covariates to serve this purpose

  20. Discussion • More thoughts on the PO model • Drug resistance or viral mutation • Weaning of breastfeeding in mother-to-child transmission • When-to-start design • Trial monitoring • Sequential methods

  21. More thoughts on Cox Model • Without time-varying covariates • Expressed in survival functions • Complementary log-log • Interpretation of rate ratio, c.f. odds ratio in the PO model

  22. An Infectious Disease Model • Assume constant probability of infection per contact • HIV infection: per sexual contact, per breastfeeding, per needle exchange, per blood transfusion • Probability of no infection after an average contacts • When average contact is associated with covariates by a log-linear model , and becomes the cumulative incidences over a period of time , it becomes a Cox model

  23. Cox Model with Time-varying Covariates • With time-varying covariates • c.f. the usual Cox model with time-varying covariates

  24. Generalized Linear Risk Model • With time-varying covariates • : functional operator link

More Related