SamplePosteriorDepGamma {BayesSurvival} R Documentation

## Draw samples from the posterior for the hazard, using the piecewise exponential (histogram) prior with dependent Gamma heights

### Description

The sampler is described in the Supplement to Castillo and Van der Pas (2020) and uses MCMC within Gibbs, with a Gamma proposal with shape parameter equal to the number of events in each interval plus some epsilon (to prevent proposals equal to zero if there are no events in an interval) and rate parameter equal to the parameter alpha (set by the user) divided by histogram height on the previous interval, plus the total amount of time all individuals were exposed during this interval. Most users of the package will not work with this function directly, but instead use the main function BayesSurv, in which this particular function is incorporated.

### Usage

```SamplePosteriorDepGamma(
failures,
exposures,
N = 1000,
alpha.dep = 1,
alpha0.dep = 1.5,
beta0.dep = 1
)
```

### Arguments

 `failures` A vector of length K (the total number of intervals), containing for each interval the number of individuals who had an event during that interval. `exposures` A vector of length K (the total number of intervals), containing for each interval the total amount of time all individuals together were under follow-up during that interval. `N` The number of draws to take. `alpha.dep` The main parameter α for the dependent Gamma prior, as described in the documentation for BayesSurv. It is recommended to take `alpha.dep` smaller than `alpha0.dep`. `alpha0.dep` The shape parameter for the Gamma prior on the histogram height for the first interval. It is recommended to take `alpha.dep` smaller than `alpha0.dep`. `beta0.dep` The rate parameter for the Gamma prior on the histogram height for the first interval.

### Details

The samples returned by this function are draws from the posterior for the hazard function. To obtain draws from the posterior for the cumulative hazard, one can use numerical integration. One way to achieve this is by first finding the values of the cumulative hazard at the end of each interval, e.g. by `t(apply(samples*time.max/K, 1, cumsum))`, where `samples` is the output from the present function and `time.max` and `K` are as described for BayesSurv, and then using `approxfun()` to linearly interpolate in between. To obtain posterior samples from the survival, one could then use SurvivalFromCumhaz.

### Value

 `samples` A N by K (the number of draws by the number of intervals) matrix, with each row containing a draw from the posterior for the hazard, based on a histogram prior with dependent Gamma heights.

### References

Castillo and Van der Pas (2020). Multiscale Bayesian survival analysis. <arXiv:2005.02889>.

### See Also

BayesSurv, which computes the posterior mean and credible bands for the cumulative hazard and survival functions, as well as the posterior mean for the hazard. Within BayesSurv, the present function is called.

[Package BayesSurvival version 0.2.0 Index]