hsaft {PanCanVarSel}

Function to implement the horseshoe shrinkage prior in Bayesian survival regression

Description

This function employs the algorithm provided by van der Pas et. al. (2016) for log normal Accelerated Failure Rate (AFT) model to fit survival regression. The censored observations are updated according to the data augmentation of approach of Tanner and Wong (1984).

Usage

hsaft(ct, X, method.tau = c("fixed", "truncatedCauchy", "halfCauchy"),
  tau = 1, method.sigma = c("fixed", "Jeffreys"), Sigma2 = 1,
  burn = 1000, nmc = 5000, thin = 1, alpha = 0.05)

Arguments

ct

Response, a n*2 matrix with first column as response and second column as right censored indicator, 1 is event time and 0 is right censored.

X

Matrix of covariates, dimension n*p.

method.tau

Method for handling \tau. Select "truncatedCauchy" for full Bayes with the Cauchy prior truncated to [1/p, 1], "halfCauchy" for full Bayes with the half-Cauchy prior, or "fixed" to use a fixed value (an empirical Bayes estimate, for example).

tau

Use this argument to pass the (estimated) value of \tau in case "fixed" is selected for method.tau. Not necessary when method.tau is equal to"halfCauchy" or "truncatedCauchy". The default (tau = 1) is not suitable for most purposes and should be replaced.

method.sigma

Select "Jeffreys" for full Bayes with Jeffrey's prior on the error variance \sigma^2, or "fixed" to use a fixed value (an empirical Bayes estimate, for example).

Sigma2

A fixed value for the error variance \sigma^2. Not necessary when method.sigma is equal to "Jeffreys". Use this argument to pass the (estimated) value of Sigma2 in case "fixed" is selected for method.sigma. The default (Sigma2 = 1) is not suitable for most purposes and should be replaced.

burn

Number of burn-in MCMC samples. Default is 1000.

nmc

Number of posterior draws to be saved. Default is 5000.

thin

Thinning parameter of the chain. Default is 1 (no thinning).

alpha

Level for the credible intervals. For example, alpha = 0.05 results in 95% credible intervals.

Details

The model is: t_i is response, c_i is censored time, t_i^* = \min_(t_i, c_i) is observed time, w_i is censored data, so w_i = \log t_i^* if t_i is event time and w_i = \log t_i^* if t_i is right censored \log t_i=X\beta+\epsilon, \epsilon \sim N(0,\sigma^2)

Value

SurvivalHat

Predictive survival probability.

LogTimeHat

Predictive log time.

BetaHat

Posterior mean of Beta, a p by 1 vector.

LeftCI

The left bounds of the credible intervals.

RightCI

The right bounds of the credible intervals.

BetaMedian

Posterior median of Beta, a p by 1 vector.

Sigma2Hat

Posterior mean of error variance \sigma^2. If method.sigma = "fixed" is used, this value will be equal to the user-selected value of Sigma2 passed to the function.

TauHat

Posterior mean of global scale parameter tau, a positive scalar. If method.tau = "fixed" is used, this value will be equal to the user-selected value of tau passed to the function.

BetaSamples

Posterior samples of Beta.

TauSamples

Posterior samples of tau.

Sigma2Samples

Posterior samples of Sigma2.

LikelihoodSamples

Posterior Samples of likelihood.

References

Stephanie van der Pas, James Scott, Antik Chakraborty and Anirban Bhattacharya (2016). horseshoe: Implementation of the Horseshoe Prior. R package version 0.1.0. https://CRAN.R-project.org/package=horseshoe

Arnab Kumar Maity, Anirban Bhattacharya, Bani K. Mallick, and Veerabhadran Baladandayuthapani (2017). Joint Bayesian Estimation and Variable Selection for TCPA Protein Expression Data

Examples

burnin <- 500   # number of burnin
nmc    <- 1000  # number of Markov Chain samples
y.sd   <- 1     # standard deviation of the data
p      <- 80    # number of covariates
n      <- 40    # number of samples
beta   <- as.vector(smoothmest::rdoublex(p))  # from double exponential distribution
x      <- mvtnorm::rmvnorm(n, mean = rep(0, p))  # from multivariate normal distribution
y.mu   <- x %*% beta  # mean of the data
y      <- as.numeric(stats::rnorm(n, mean = y.mu, sd = y.sd))  # from normal distribution
T      <- exp(y)   # AFT model
C      <- rgamma(n, shape = 1.75, scale = 3)  # censoring time
time   <- pmin(T, C)  # observed time is min of censored and true
status = time == T   # set to 1 if event is observed
ct     <- as.matrix(cbind(time = time, status = status))  # censored time

posterior.fit <- hsaft(ct, x, method.tau = "truncatedCauchy", method.sigma = "Jeffreys",
                       burn = burnin, nmc = nmc)
summary(posterior.fit$BetaHat)