afths: Horseshoe shrinkage prior in Bayesian survival regression

In intsurvbin: Survival and Binary Data Integration

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

afths(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, Xtest = NULL,
  cttest = NULL)

Arguments

ct	survival 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 τ. 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 τ 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 σ^2, or "fixed" to use a fixed value (an empirical Bayes estimate, for example).
Sigma2	A fixed value for the error variance σ^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.
Xtest	test design matrix.
cttest	test survival response.

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=Xβ+ε, ε \sim N(0,σ^2).

Value

SurvivalHat	Predictive survival probability
LogTimeHat	Predictive log time
Beta.sHat	Posterior mean of Beta, a p by 1 vector
LeftCI.s	The left bounds of the credible intervals
RightCI.s	The right bounds of the credible intervals
Beta.sMedian	Posterior median of Beta, a p by 1 vector
LambdaHat	Posterior samples of λ, a p*1 vector
Sigma2Hat	Posterior mean of error variance σ^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
Beta.sSamples	Posterior samples of β
TauSamples	Posterior samples of τ
Sigma2Samples	Posterior samples of Sigma2

References

Maity, A. K., Carroll, R. J., and Mallick, B. K. (2019) "Integration of Survival and Binary Data for Variable Selection and Prediction: A Bayesian Approach", Journal of the Royal Statistical Society: Series C (Applied Statistics).

Examples

burnin <- 500
nmc    <- 500
thin <- 1
y.sd   <- 1  # standard deviation of the response

p <- 100  # number of predictors
ntrain <- 100  # training size
ntest  <- 50   # test size
n <- ntest + ntrain  # sample size
q <- 10   # number of true predictos

beta.t <- c(sample(x = c(1, -1), size = q, replace = TRUE), rep(0, p - q))  # randomly assign sign

Sigma <- matrix(0.9, nrow = p, ncol = p)
for(j in 1:p)
{
Sigma[j, j] <- 1
}

x <- mvtnorm::rmvnorm(n, mean = rep(0, p), sigma = Sigma)    # correlated design matrix

tmean <- x %*% beta.t
yCorr <- 0.5
yCov <- matrix(c(1, yCorr, yCorr, 1), nrow = 2)


y <- mvtnorm::rmvnorm(n, sigma = yCov)
t <- y[, 1] + tmean
X <- scale(as.matrix(x))  # standarization

t <- as.numeric(as.matrix(c(t)))
T <- exp(t)   # AFT model
C <- rgamma(n, shape = 1.75, scale = 3)   # 42% 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


# Training set
cttrain <- ct[1:ntrain, ]
Xtrain  <- X[1:ntrain, ]

# Test set
cttest <- ct[(ntrain + 1):n, ]
Xtest  <- X[(ntrain + 1):n, ]

posterior.fit.aft <- afths(ct = cttrain, X = Xtrain, method.tau = "halfCauchy",
                             method.sigma = "Jeffreys", burn = burnin, nmc = nmc, thin = 1,
                             Xtest = Xtest, cttest = cttest)
                             
posterior.fit.aft$Beta.sHat

intsurvbin documentation built on Oct. 3, 2019, 9:02 a.m.