Specifics for Bayesian Analysis |
To request a Bayesian analysis, you specify the new BAYES statement in addition to the PROC PHREG statement and the MODEL statement. You include a CLASS statement if you have effects that involve categorical variables. The FREQ or WEIGHT statement can be included if you have a frequency or weight variable, respectively, in the input data. The STRATA statement can be used to carry out a stratified analysis for the Cox model, but it is not allowed in the piecewise constant baseline hazard model. Programming statements can be used to create time-dependent covariates for the Cox model, but they are not allowed in the piecewise constant baseline hazard model. However, you can use the counting process style of input to accommodate time-dependent covariates that are not continuously changing with time for the piecewise constant baseline hazard model and the Cox model as well. The HAZARDRATIO statement enables you to request a hazard ratio analysis based on the posterior samples. The ASSESS, CONTRAST, ID, OUTPUT, and TEST statements, if specified, are ignored. Also ignored are the COVM and COVS options in the PROC PHREG statement and the following options in the MODEL statement: BEST=, CORRB, COVB, DETAILS, HIERARCHY=, INCLUDE=, MAXSTEP=, NOFIT, PLCONV=, SELECTION=, SEQUENTIAL, SLENTRY=, and SLSTAY=.
Let be the observed data. Let be a partition of the time axis.
The hazard function for subject is
where
The baseline cumulative hazard function is
where
The log likelihood is given by
where .
Note that for , the full conditional for is log-concave only when , but the full conditionals for the ’s are always log-concave.
For a given , gives
Substituting these values into gives the profile log likelihood for
where . Since the constant does not depend on , it can be discarded from in the optimization.
The MLE of is obtained by maximizing
with respect to , and the MLE of is given by
For , let
The partial derivatives of are
The asymptotic covariance matrix for is obtained as the inverse of the information matrix given by
See Example 6.5.1 in Lawless (2003) for details.
By letting
you can build a prior correlation among the ’s by using a correlated prior , where .
The log likelihood is given by
Then the MLE of is given by
Note that the full conditionals for ’s and ’s are always log-concave.
The asymptotic covariance matrix for is obtained as the inverse of the information matrix formed by
Let be the observed data. Let be a partition of the time axis, where for all .
Replacing with
the formulation for the single failure time variable applies.
For a Cox model, the model parameters are the regression coefficients. For a piecewise exponential model, the model parameters consist of the regression coefficients and the hazards or log-hazards. The priors for the hazards and the priors for the regression coefficients are assumed to be independent, while you can have a joint multivariate normal prior for the log-hazards and the regression coefficients.
Let be the constant baseline hazards.
The joint prior density is given by
This prior is improper (nonintegrable), but the posterior distribution is proper as long as there is at least one event time in each of the constant hazard intervals.
The joint prior density is given by
This prior is improper (nonintegrable), but the posteriors are proper as long as there is at least one event time in each of the constant hazard intervals.
The gamma distribution has a pdf
where is the shape parameter and is the scale parameter. The mean is and the variance is .
Suppose for , has an independent prior. The joint prior density is given by
are correlated as follows:
The joint prior density is given by
Write .
The joint prior density is given by
Note that the uniform prior for the log-hazards is the same as the improper prior for the hazards.
Assume has a multivariate normal prior with mean vector and covariance matrix . The joint prior density is given by
Let be the vector of regression coefficients.
The joint prior density is given by
This prior is improper, but the posterior distributions for are proper.
Assume has a multivariate normal prior with mean vector and covariance matrix . The joint prior density is given by
Assume has a multivariate normal prior with mean vector and covariance matrix . The joint prior density is given by
Assume has a multivariate normal prior with mean vector and covariance matrix , where is the design matrix and is either a constant or it follows a gamma prior with density where and are the SHAPE= and ISCALE= parameters. Let be the rank of . The joint prior density with g being a constant c is given by
The joint prior density with g having a gamma prior is given by
Denote the observed data by .
where is the partial likelihood function with regression coefficients as parameters.
where is the likelihood function with hazards and regression coefficients as parameters.
where is the likelihood function with log-hazards and regression coefficients as parameters.
For the Gibbs sampler, PROC PHREG uses the ARMS (adaptive rejection Metropolis sampling) algorithm of Gilks, Best, and Tan (1995) to sample from the full conditionals. This is the default sampling scheme. Alternatively, you can requests the random walk Metropolis (RWM) algorithm to sample an entire parameter vector from the posterior distribution. For a general discussion of these algorithms, refer to section Markov Chain Monte Carlo Method.
You can output these posterior samples into a SAS data set by using the OUTPOST= option in the BAYES statement, or you can use the following SAS statement to output the posterior samples into the SAS data set Post:
ods output PosteriorSample=Post;
The output data set also includes the variables LogLike and LogPost, which represent the log of the likelihood and the log of the posterior log density, respectively.
Let be the parameter vector. For the Cox model, the ’s are the regression coefficients ’s, and for the piecewise constant baseline hazard model, the ’s consist of the baseline hazards ’s (or log baseline hazards ’s) and the regression coefficients ’s. Let be the likelihood function, where is the observed data. Note that for the Cox model, the likelihood contains the infinite-dimensional baseline hazard function, and the gamma process is perhaps the most commonly used prior process (Ibrahim, Chen, and Sinha; 2001). However, Sinha, Ibrahim, and Chen (2003) justify using the partial likelihood as the likelihood function for the Bayesian analysis. Let be the prior distribution. The posterior is proportional to the joint distribution .
The full conditional distribution of is proportional to the joint distribution; that is,
For example, the one-dimensional conditional distribution of , given , is computed as
Suppose you have a set of arbitrary starting values . Using the ARMS algorithm, an iteration of the Gibbs sampler consists of the following:
draw from
draw from
draw from
After one iteration, you have . After iterations, you have . Cumulatively, a chain of samples is obtained.
PROC PHREG uses a multivariate normal proposal distribution centered at . With an initial parameter vector , a new sample is obtained as follows:
sample from
calculate the quantity
sample from the uniform distribution
set if ; otherwise set
With taking the role of , the previous steps are repeated to generate the next sample . After iterations, a chain of samples is obtained.
When the BAYES statement is specified, PROC PHREG generates one Markov chain that contains the approximate posterior samples of the model parameters. Additional chains are produced when the Gelman-Rubin diagnostics are requested. Starting values (initial values) can be specified in the INITIAL= data set in the BAYES statement. If the INITIAL= option is not specified, PROC PHREG picks its own initial values for the chains based on the maximum likelihood estimates of and the prior information of .
Denote as the integral value of .
For the first chain that the summary statistics and diagnostics are based on, the initial values are
For subsequent chains, the starting values are picked in two different ways according to the total number of chains specified. If the total number of chains specified is less than or equal to 10, initial values of the th chain () are given by
with the plus sign for odd and minus sign for even . If the total number of chains is greater than 10, initial values are picked at random over a wide range of values. Let be a uniform random number between 0 and 1; the initial value for is given by
The ’s are the regression coefficients ’s, and in the piecewise exponential model, include the log-hazard parameters ’s. For the first chain that the summary statistics and regression diagnostics are based on, the initial values are
If the number of chains requested is less than or equal to 10, initial values for the th chain () are given by
with the plus sign for odd and minus sign for even . When there are more than 10 chains, the initial value for the is picked at random over the range ; that is,
where is a uniform random number between 0 and 1.
Denote the observed data by . Let be the vector of parameters of length . Let be the likelihood. The deviance information criterion (DIC) proposed in Spiegelhalter et al. (2002) is a Bayesian model assessment tool. Let Dev. Let and be the corresponding posterior means of and , respectively. The deviance information criterion is computed as
Also computed is
where is interpreted as the effective number of parameters.
Note that defined here does not have the standardizing term as in the section Deviance Information Criterion (DIC). Nevertheless, the DIC calculated here is still useful for variable selection.
Let be the parameter vector. For the Cox model, the ’s are the regression coefficients ’s; for the piecewise constant baseline hazard model, the ’s consist of the baseline hazards ’s (or log baseline hazards ’s) and the regression coefficients ’s. Let be the chain that represents the posterior distribution for .
Consider a quantity of interest that can be expressed as a function of the parameter vector . You can construct the posterior distribution of by evaluating the function for each in . The posterior chain for is Summary statistics such as mean, standard deviation, percentiles, and credible intervals are used to describe the posterior distribution of .
As shown in the section Hazard Ratios, a log-hazard ratio is a linear combination of the regression coefficients. Let be the vector of linear coefficients. The posterior sample for this hazard ratio is the set .
Let be a covariate vector of interest.
Let be the observed data. Define
Consider the th draw of . The baseline cumulative hazard function at time is given by
For the given covariate vector , the cumulative hazard function at time is
and the survival function at time is
Let be a partition of the time axis. Consider the th draw in , where consists of and . The baseline cumulative hazard function at time is
where
For the given covariate vector , the cumulative hazard function at time is
and the survival function at time is