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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.

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A Proportional Odds Model with Time-varying Covariates

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  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

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