# This function extends the main function `hsaft` to create correlation among covariates. ```r hsaftallcorr(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, r, n.seq, pk) ``` ## 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. - `r`: number of groups. - `n.seq`: a vector of sample sizes for all groups. - `pk`: number of covariates in each group. ## Returns - **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. - **BGHat**: Posterior samples of b which is a part of the mean of $\beta$. - **BPHat**: Posterior samples of b which is the other part of the mean of $\beta$. - **LikelihoodSamples**: Posterior Samples of likelihood. ## Examples ```r # Examples for hsaftallcorr function burnin <- 50 # number of burnin nmc <- 100 # number of Markov Chain samples y.sd <- 1 # standard deviation of the data p <- 80 # number of covariates r <- 5 # number of groups p <- 80 # number of covariate in each group n1 <- 40 # sample size of 1st group n2 <- 50 # sample size of 2nd group n3 <- 70 # sample size of 3rd group n4 <- 100 # sample size of 4th group n5 <- 120 # sample size of 5th group n <- sum(c(n1, n2, n3, n4, n5)) # total sample size n.seq <- c(n1, n2, n3, n4, n5) Beta <- matrix(smoothmest::rdoublex(p * r), nrow = r, ncol = p, byrow = TRUE) # from double exponential distribution beta <- as.vector(t(Beta)) # vectorize Beta x1 <- mvtnorm::rmvnorm(n1, mean = rep(0, p)) x2 <- mvtnorm::rmvnorm(n2, mean = rep(0, p)) x3 <- mvtnorm::rmvnorm(n3, mean = rep(0, p)) x4 <- mvtnorm::rmvnorm(n4, mean = rep(0, p)) x5 <- mvtnorm::rmvnorm(n5, mean = rep(0, p)) # from multivariate normal distribution y.mu1 <- x1 %*% Beta[1, ] y.mu2 <- x2 %*% Beta[2, ] y.mu3 <- x3 %*% Beta[3, ] y.mu4 <- x4 %*% Beta[4, ] y.mu5 <- x5 %*% Beta[5, ] y1 <- stats::rnorm(n1, mean = y.mu1, sd = y.sd) y2 <- stats::rnorm(n2, mean = y.mu2, sd = y.sd) y3 <- stats::rnorm(n3, mean = y.mu3, sd = y.sd) y4 <- stats::rnorm(n4, mean = y.mu4, sd = y.sd) y5 <- stats::rnorm(n5, mean = y.mu5, sd = y.sd) y <- c(y1, y2, y3, y4, y5) x <- Matrix::bdiag(x1, x2, x3, x4, x5) X <- as.matrix(x) y <- as.numeric(as.matrix(y)) # 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 <- hsaftallcorr(ct, X, method.tau = "truncatedCauchy", method.sigma = "Jeffreys", burn = burnin, nmc = nmc, r = r, n.seq = n.seq, pk = p) summary(posterior.fit$BetaHat) ``` ## 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