Transcript A Proportional Odds Model with Time

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
Time Progression
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Time progression
Renewal time progression
Cumulative time progression
Extend Logistic Regression Model
• Renewal time progression
– Efron (1988, JASA) “Logistic-Regression, Survival
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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 hazards model
• Interpretation 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 fractions
• What if we always count the cumulative
events from time zero
– Common denominator
Proportional Odds Model
• Logistic regression model
• Proportional 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 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
Proportional Odds Model
• Transformation models with time-varying covariates
– Kosorok, et al. (2004, Ann Stat)
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is some frailty-induced Laplace transform
– Zeng & Lin (2006, Bmka; 2007, JRSS-B)
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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
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
– Under Yang-Prentice PO model, individual hazard function is
– Therefore,
– Then we can solve it to get
Estimation of Yang-Prentice PO Model
• With time-varying covariates
Estimation by Differential Equations
• Consider
• Let
Estimating Equations for Baseline Function
• Assume that we know
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
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
Consideration of Optimal Estimation
• Hazard function under Yang-Prentice PO
Model
• A form of optimal weight function in
weighted estimation is calculated as
Simulation Studies
• Simulation setup
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
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
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
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
Cox Model with Time-varying Covariates
• With time-varying covariates
• c.f. the usual Cox model with time-varying
covariates
Generalized Linear Risk Model
• With time-varying covariates
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: functional operator link