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 approach described in Maity et. al. (2019) and Maity et. al. (2020).

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
)

Arguments

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

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).

Maity, A. K., Bhattacharya, A., Mallick, B. K., & Baladandayuthapani, V. (2020). Bayesian data integration and variable selection for pan cancer survival prediction using protein expression data. Biometrics, 76(1), 316-325.

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

Enes Makalic and Daniel Schmidt (2016). High-Dimensional Bayesian Regularised Regression with the BayesReg Package arXiv:1611.06649

Examples

burnin <- 500
nmc    <- 1000
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)) 
x <- mvtnorm::rmvnorm(n, mean = rep(0, p), sigma = diag(p))    

tmean <- x %*% beta.t
y <- rnorm(n, mean = tmean, sd = y.sd)
X <- scale(as.matrix(x))  # standarization

T <- exp(y)   # 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 <- afths(ct = cttrain, X = Xtrain, method.tau = "halfCauchy",
                             method.sigma = "Jeffreys", burn = burnin, nmc = nmc, thin = 1,
                             Xtest = Xtest)
                             
posterior.fit$BetaHat

# Posterior processing to recover the true predictors
cluster <- kmeans(abs(posterior.fit$BetaHat), centers = 2)$cluster
cluster1 <- which(cluster == 1)
cluster2 <- which(cluster == 2)
min.cluster <- ifelse(length(cluster1) < length(cluster2), 1, 2)
which(cluster == min.cluster)  # this matches with the true variables

Horseshoe shrinkage prior in Bayesian linear regression

Description

This function employs the algorithm provided by van der Pas et. al. (2016) for linear model to fit Bayesian regression.

Usage

lmhs(
  y,
  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
)

Arguments

Details

The model is: y_i is response, y_i=X\beta+\epsilon, \epsilon \sim N(0,\sigma^2).

Value

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).

Maity, A. K., Bhattacharya, A., Mallick, B. K., & Baladandayuthapani, V. (2020). Bayesian data integration and variable selection for pan cancer survival prediction using protein expression data. Biometrics, 76(1), 316-325.

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

Enes Makalic and Daniel Schmidt (2016). High-Dimensional Bayesian Regularised Regression with the BayesReg Package arXiv:1611.06649

Examples

burnin <- 500
nmc    <- 1000
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))  
x <- mvtnorm::rmvnorm(n, mean = rep(0, p), sigma = diag(p))    

tmean <- x %*% beta.t
y <- rnorm(n, mean = tmean, sd = y.sd)
X <- scale(as.matrix(x))  # standarization

# Training set
ytrain <- y[1:ntrain]
Xtrain <- X[1:ntrain, ]

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

posterior.fit <- lmhs(y = ytrain, X = Xtrain, method.tau = "halfCauchy",
                       method.sigma = "Jeffreys", burn = burnin, nmc = nmc, thin = 1,
                       Xtest = Xtest)
                             
posterior.fit$BetaHat

# Posterior processing to recover the true predictors
cluster <- kmeans(abs(posterior.fit$BetaHat), centers = 2)$cluster
cluster1 <- which(cluster == 1)
cluster2 <- which(cluster == 2)
min.cluster <- ifelse(length(cluster1) < length(cluster2), 1, 2)
which(cluster == min.cluster)  # this matches with the true variables

Horseshoe shrinkage prior in Bayesian Logistic regression

Description

This function employs the algorithm provided by Makalic and Schmidt (2016) for binary logistic model to fit Bayesian logistic regression. The observations are updated according to the Polya-Gamma data augmentation of approach of Polson, Scott, and Windle (2014).

Usage

logiths(
  z,
  X,
  method.tau = c("fixed", "truncatedCauchy", "halfCauchy"),
  tau = 1,
  burn = 1000,
  nmc = 5000,
  thin = 1,
  alpha = 0.05,
  Xtest = NULL
)

Arguments

Details

The model is: z_i is response either 1 or 0, \log \Pr(z_i = 1) = logit^{-1}(X\beta).

Value

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

Enes Makalic and Daniel Schmidt (2016). High-Dimensional Bayesian Regularised Regression with the BayesReg Package arXiv:1611.06649

Polson, N.G., Scott, J.G. and Windle, J. (2014) The Bayesian Bridge. Journal of Royal Statistical Society, B, 76(4), 713-733.

Examples

burnin <- 100
nmc    <- 500
thin <- 1

 
p <- 100  # number of predictors
ntrain <- 250  # training size
ntest  <- 100   # 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))  
x <- mvtnorm::rmvnorm(n, mean = rep(0, p), sigma = diag(p))    

zmean <- x %*% beta.t
z <- rbinom(n, size = 1, prob = boot::inv.logit(zmean))
X <- scale(as.matrix(x))  # standarization


# Training set
ztrain <- z[1:ntrain]
Xtrain  <- X[1:ntrain, ]

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

posterior.fit <- logiths(z = ztrain, X = Xtrain, method.tau = "halfCauchy",
                         burn = burnin, nmc = nmc, thin = 1,
                         Xtest = Xtest)
                             
posterior.fit$BetaHat


# Posterior processing to recover the true predictors
cluster <- kmeans(abs(posterior.fit$BetaHat), centers = 2)$cluster
cluster1 <- which(cluster == 1)
cluster2 <- which(cluster == 2)
min.cluster <- ifelse(length(cluster1) < length(cluster2), 1, 2)
which(cluster == min.cluster)  # this matches with the true variables

Horseshoe shrinkage prior in Bayesian Probit regression

Description

This function employs the algorithm provided by Makalic and Schmidt (2016) for binary probit model to fit Bayesian probit regression. The observations are updated according to the data augmentation of approach of Albert and Chib (1993).

The model is: z_i is response either 1 or 0, \log \Pr(z_i = 1) = \Phi(X\beta), \Phi \sim N(0,\sigma^2).

Usage

probiths(
  z,
  X,
  method.tau = c("fixed", "truncatedCauchy", "halfCauchy"),
  tau = 1,
  burn = 1000,
  nmc = 5000,
  thin = 1,
  alpha = 0.05,
  Xtest = NULL
)

Arguments

Value

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

Enes Makalic and Daniel Schmidt (2016). High-Dimensional Bayesian Regularised Regression with the BayesReg Package arXiv:1611.06649

Albert, J. H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association, 88(422), 669-679.

Examples

burnin <- 100
nmc    <- 200
thin   <- 1
y.sd   <- 1  # statndard deviation of the response

p <- 200  # number of predictors
ntrain <- 250  # training size
ntest  <- 100   # 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))  
x <- mvtnorm::rmvnorm(n, mean = rep(0, p))    
zmean <- x %*% beta.t

y <- rnorm(n, mean = zmean, sd = y.sd)
z <- ifelse(y > 0, 1, 0)
X <- scale(as.matrix(x))  # standarization
z <- as.numeric(as.matrix(c(z)))

# Training set
ztrain <- z[1:ntrain]
Xtrain  <- X[1:ntrain, ]

# Test set
ztest <- z[(ntrain + 1):n]
Xtest <- X[(ntrain + 1):n, ]
 
posterior.fit <- probiths(z = ztrain, X = Xtrain, method.tau = "halfCauchy",
                          burn = burnin, nmc = nmc, thin = 1,
                          Xtest = Xtest)

posterior.fit$BetaHat

# Posterior processing to recover the significant predictors
cluster     <- kmeans(abs(posterior.fit$BetaHat), centers = 2)$cluster  # return cluster indices
cluster1    <- which(cluster == 1)
cluster2    <- which(cluster == 2)
min.cluster <- ifelse(length(cluster1) < length(cluster2), 1, 2)
which(cluster == min.cluster)  # this matches with the true variables