1、A Proportional Odds Model with Time-varying Covariates,Logistic Regression Model,Logistic regression model when outcome is binaryHow do we extend the logistic regression model for time-to-event outcome? It depends on how we view the time progression,Time Progression,Extend Logistic Regression Model,
2、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,Extend Logistics Regression Model,Cox proportional haza
3、rds modelInterpretation of the regression parameter Instantaneous hazards ratio In terms of cumulative event rates,Extend Logistics Regression Model,So, why this happens? nonlinearity The fundamental issue is how we deal with different denominators of summing fractionsWhat if we always count the cum
4、ulative events from time zero Common denominator,Proportional Odds Model,Logistic regression modelProportional odds model with time-varying covariates at time : Yang & Prentice (1999, JASA),Proportional Odds Model,Yang-Prentice PO Model Model closed under log-logisitic distributions Interpretation o
5、f regression parameter Without time-varying covariates Special case of the transformation models when the error term follows standard logistic distribution with unspecified transformationRank estimation: Cheng, et al. (1995, Bmka) NPMLE,Proportional Odds Model,Transformation models with time-varying
6、 covariates Kosorok, et al. (2004, Ann Stat)is some frailty-induced Laplace transform Zeng 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
7、 odds model without time-varying covariates,Yang-Prentice Proportional Odds Model,Yang & Prentice (1999, JASA) Inference procedures developed mostly without time-varying covariates Time-varying covariates,Estimation of Yang-Prentice PO Model,By way of integral equation for baseline odds function Und
8、er Yang-Prentice PO model, individual hazard function isTherefore, Then we can solve it to get,Estimation of Yang-Prentice PO Model,With time-varying covariates,Estimation by Differential Equations,ConsiderLet,Estimating Equations for Baseline Function,Assume that we know,Estimation of Baseline func
9、tion,Then we solve to obtain a closed form solution for baseline odds functionMoreoverThis shall lead to consistency and asymptotic normality of this baseline odds function estimator with true regression parameter,Estimation of Regression Parameters,Estimating equations for regression parametersor W
10、e can obtain all the necessary asymptotic properties of Straightforward to extend to weighted estimation,Consideration of Optimal Estimation,Hazard function under Yang-Prentice PO ModelA form of optimal weight function in weighted estimation is calculated as,Simulation Studies,Simulation setup,Data
11、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
12、analyzed this data without model checking. We include covariates and time interaction as time-varying covariates to serve this purpose,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 monito
13、ring Sequential methods,More thoughts on Cox Model,Without time-varying covariatesExpressed in survival functionsComplementary log-log Interpretation of rate ratio, c.f. odds ratio in the PO model,An Infectious Disease Model,Assume constant probability of infection per contact HIV infection: per sex
14、ual contact, per breastfeeding, per needle exchange, per blood transfusion Probability of no infection after an average contactsWhen 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,Cox Model with Time-varying Covariates,With time-varying covariatesc.f. the usual Cox model with time-varying covariates,Generalized Linear Risk Model,With time-varying covariates: functional operator link,
