Time Discrete Survival Models
j = P (Ti = j | Ti j, xi) = h(j + xi), where j is the discrete hazard, = (1, . . . , k ) are parameters characterizing the baseline hazard xi are time-independent covariates are regression coecients
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Proportional Hazards in Discrete Time
Assuming (t) = 0(t) exp(xi), and let Si denote the continuous event time Suppose that Ti = j if Si (aj1, aj ], for j = 1, . . . , k. Then, the discrete hazard is as follows: Pr(Ti = j | Ti j, xi) = 1 S(aj ) exp(i(aj )) =1 S(aj1) exp(i(aj1))
aj aj1
= 1 exp
0(t) exp(xi) dt
= 1 exp { exp(xi)(0(aj ) 0(aj1)} = 1 exp { exp(j + xi)}, where j = log(0(aj ) 0(aj1)) and 0(t) =
t 0 0 (s) ds.
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Thus, a Cox proportional hazards model can be t using a discretetime approximation by using a binary response GLM with a complementary log-log link In doing this, the discrete event time Ti must be coded as a Ti 1 vector of binary responses, yi = (0, ,0, i) The corresponding design matrix is then, Xi = (xi1, . . . , xi,Ti ) , where xij is a (k + p) 1 vector consisting of 0s in each of the rst k positions except for the jth which has a 1. The last p elements are xed at xi.
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Time-Varying Covariates
Often, in applications one or more of the predictors may vary over time. For example, suppose that we are interested in assessing the eect of air pollution levels on mortality. The air pollution levels vary from day to day.
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Reasonable model for discrete hazard of death in age group j? Pr(Ti = j | Ti = j, xi, zij ) = h(j + xi + zij ), where zij is the level of population for individual i at age j This model accommodates the time-varying covariate Are we making a restrictive assumption?
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In the previous model, we assumed that the eect of air pollution was constant at dierent ages. In fact, infants and the elderly are more susceptible to pollutioninduced mortality. How can we generalize the model, to account for this age-dependent susceptibility?
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Time-Varying Coecients
Pr(Ti = j | Ti = j, xi, zij ) = h(j + xi + zij j ), where we have now added a j subscript to the parameter characterizing the air pollution eect. Potentially, the eect of the time-independent predictors can also vary with time by allowing dierent s for the dierent age intervals Dimensionality rapidly become problematic - Order Restrictions?
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What about computation & inference from these models? Well, if were frequentist, we can just t the binary response GLM and proceed as before (maximum likelihood estimation, analysis of deviance, etc) If were Bayesian, we can potentially also proceed as in binary response GLMs - either using adaptive rejection sampling or (if probit) the Albert and Chib approach
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ContinuationRatioProbitModels
Pr(Ti j = | Ti = j, xij ) = (xij ), where we can potentially parameterize xij to allow a nonparametric baseline and time-varying coecients. Note that Ti {1, . . . , k}, with k potentially large Thus, we have a potentially large number of parameters, including the time-varying coecients
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By choosing a probit model, we can update the high dimensional vector jointly after augmenting the data with latent normal variables. Now, we have yi = (0, . . . , 0, i) as a Ti 1 outcome vector for subject i We introduce a zi = (zi1, . . . , zi,Ti ) vector of independent normal variables underlying yi yij = 1(zij > 0) and zij N (xij , 1), for j = 1, . . . , Ti. Gibbs sampler proceeds as before.
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What about the prior specication? We have a potentially high-dimensional vector of time-varying baseline parameters and coecients. Potentially, no individuals with the event in certain intervals. What type of information prior is reasonable?
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Focusing initially on the model with no time-varying coecients, we may want to do some smoothing Values of j and j are likely to be similar is j is close to j Autoregressive - Gaussian random walk prior: j N(j1, 1), where is a precision parameter controlling the degree of smoothing.
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Penalized Likelihood
The autoregressive prior essentially penalizes values of j that are far from the neighboring values From a frequentist perpective, we can use a similar idea by including a penalty term in the likelihood and then maximizing the resulting penalized likelihood. The penalty term can follow many forms, including an autoregressive normal density for the s
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Oft...