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R/pic.R
In missingHE: Missing Outcome Data in Health Economic Evaluation
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pic
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pic
#' Predictive information criteria for Bayesian models fitted in \code{JAGS} using the funciton \code{\link{selection}}, \code{\link{selection_long}}, \code{\link{pattern}} or \code{\link{hurdle}}
#'
#' Efficient approximate leave-one-out cross validation (LOO), deviance information criterion (DIC) and widely applicable information criterion (WAIC) for Bayesian models,
#' calculated on the observed data.
#' @keywords loo waic dic JAGS
#' @param x A \code{missingHE} object containing the results of a Bayesian model fitted in cost-effectiveness analysis using the function \code{\link{selection}}, \code{\link{selection_long}},
#' \code{\link{pattern}} or \code{\link{hurdle}}.
#' @param criterion type of information criteria to be produced. Available choices are \code{'dic'} for the Deviance Information Criterion,
#' \code{'waic'} for the Widely Applicable Information Criterion, and \code{'looic'} for the Leave-One-Out Information Criterion.
#' @param module The modules with respect to which the information criteria should be computed. Available choices are \code{'total'} for the whole model,
#' \code{'e'} for the effectiveness variables only (\code{'u'} for longitudinal models), \code{'c'} for the cost variables only, and \code{'both'} for both outcome variables.
#' @return A named list containing different predictive information criteria results and quantities according to the value of \code{criterion}. In all cases, the measures are
#' computed on the observed data for the specific modules of the model selected in \code{module}.
#' \describe{
#' \item{d_bar}{Posterior mean deviance (only if \code{criterion} is \code{'dic'}).}
#' \item{pD}{Effective number of parameters calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}.
#' \item{dic}{Deviance Information Criterion calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}.
#' \item{d_hat}{Deviance evaluated at the posterior mean of the parameters and calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}
#' \item{elpd, elpd_se}{Expected log pointwise predictive density and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{p, p_se}{Effective number of parameters and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{looic, looic_se}{The leave-one-out information criterion and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'loo'}).}
#' \item{waic, waic_se}{The widely applicable information criterion and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'}).}
#' \item{pointwise}{A matrix containing the pointwise contributions of each of the above measures calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{pareto_k}{A vector containing the estimates of the shape parameter \eqn{k} for the generalised Pareto fit to the importance ratios for each leave-one-out distribution
#' calculated on the observed data for the model nodes indicated in \code{module} (only if \code{criterion} is \code{'loo'}).
#' See \code{\link[loo]{loo}} for details about interpreting \eqn{k}.}
#' \item{sum_dic}{DIC value calculated by summing up all model dic evaluated at each time point (only for longitudinal models). Similar estimates can
#' are obtained also for the other criteria, either \code{sum_waic} or \code{sum_looic}.}
#' \item{sum_pdic}{DIC value calculated by summing up all model effective number of parameter estimates based on dic evaluated at each time point (only for longitudinal models). Similar estimates can
#' are obtained also for the other criteria, either \code{sum_pwaic} or \code{sum_plooic}.}
#' }
#' @seealso \code{\link[R2jags]{jags}}, \code{\link[loo]{loo}}, \code{\link[loo]{waic}}
#' @author Andrea Gabrio
#' @importFrom stats var
#' @details The Deviance Information Criterion (DIC), Leave-One-Out Information Criterion (LOOIC) and the Widely Applicable Information Criterion (WAIC) are methods for estimating
#' out-of-sample predictive accuracy from a Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameters. If \code{x} contains the results from
#' a longitudinal model, all parameter names indexed by "e" should be instead indexed by "u". In addition, for longitudinal models information criteria results are displayed by time
#' and only a general approximation to the total value of the criteria and pD is given as the sum of the corresponding measures computed at each time point.
#' DIC is computationally simple to calculate but it is known to have some problems, arising in part from it not being fully Bayesian in that it is based on a point estimate.
#' LOOIC can be computationally expensive but can be easily approximated using importance weights that are smoothed by fitting a generalised Pareto distribution to the upper tail
#' of the distribution of the importance weights. For more details about the methods used to compute LOOIC see the PSIS-LOO section in \code{\link{loo-package}}.
#' WAIC is fully Bayesian and closely approximates Bayesian cross-validation. Unlike DIC, WAIC is invariant to parameterisation and also works for singular models.
#' In finite cases, WAIC and LOO give similar estimates, but for influential observations WAIC underestimates the effect of leaving out one observation.
#'
#' @references
#' Plummer, M. \emph{JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.} (2003).
#'
#' Vehtari, A. Gelman, A. Gabry, J. (2016a) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.
#' \emph{Statistics and Computing}. Advance online publication.
#'
#' Vehtari, A. Gelman, A. Gabry, J. (2016b) Pareto smoothed importance sampling. \emph{ArXiv} preprint.
#'
#' Gelman, A. Hwang, J. Vehtari, A. (2014) Understanding predictive information criteria for Bayesian models.
#' \emph{Statistics and Computing} 24, 997-1016.
#'
#' Watanable, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information
#' criterion in singular learning theory. \emph{Journal of Machine Learning Research} 11, 3571-3594.
#'
#' @export
#' @examples
#' # For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
#' # \code{\link{pattern}} or \code{\link{hurdle}}
#' #
#' #
pic <- function(x, criterion = "dic", module = "total") {
if(!isTRUE(requireNamespace("loo", quietly = TRUE))) {
stop("You need to install the R package 'loo'. Please run in your R terminal:\n install.packages('loo')")
}
if(!inherits(x, what = "missingHE")) {
stop("Only objects of class 'missingHE' can be used")
}
if(x$model_output$`model summary`$BUGSoutput$n.chains == 1){
stop("information criteria cannot be computed if n.chain=1")
}
if(is.character(criterion) == FALSE | is.character(module) == FALSE) {
stop("criterion and module must be character names")
}
criterion <- tolower(criterion)
module <- tolower(module)
if(!criterion %in% c("dic", "waic", "looic")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
if(x$data_format == "wide") {
if(!module %in% c("total", "e", "c", "both")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
m_e1 <- x$data_set$missing_effects$Control
loglik_e1 <- x$model_output$loglik$effects$control[, m_e1 == 0]
m_e2 <- x$data_set$missing_effects$Intervention
loglik_e2 <- x$model_output$loglik$effects$intervention[, m_e2 == 0]
m_c1 <- x$data_set$missing_costs$Control
loglik_c1 <- x$model_output$loglik$costs$control[, m_c1 == 0]
m_c2 <- x$data_set$missing_costs$Intervention
loglik_c2 <- x$model_output$loglik$costs$intervention[, m_c2 == 0]
if(x$model_output$type == "SELECTION" | x$model_output$type == "SELECTION_e" |
x$model_output$type == "SELECTION_c" | x$model_output$type == "SELECTION_ec" ) {
loglik_me1 <- x$model_output$loglik$`missing indicators effects`$control
loglik_me2 <- x$model_output$loglik$`missing indicators effects`$intervention
loglik_mc1 <- x$model_output$loglik$`missing indicators costs`$control
loglik_mc2 <- x$model_output$loglik$`missing indicators costs`$intervention
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_me <- rowSums(cbind(loglik_me1, loglik_me2))
loglik_mc <- rowSums(cbind(loglik_mc1, loglik_mc2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_me, loglik_mc))
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_me <- cbind(loglik_me1, loglik_me2)
loglik_mc <- cbind(loglik_mc1, loglik_mc2)
loglik_e_c <- cbind(loglik_e, loglik_c)
loglik_total <- cbind(loglik_e_c, loglik_me, loglik_mc)
}
}
if(x$model_output$type == "PATTERN" | x$model_output$type == "PATTERN_e" |
x$model_output$type == "PATTERN_c" | x$model_output$type == "PATTERN_ec" ) {
loglik_d1 <- x$model_output$loglik$`pattern indicators`$control
loglik_d2 <- x$model_output$loglik$`pattern indicators`$intervention
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_d <- rowSums(cbind(loglik_d1, loglik_d2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_d))
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_d <- cbind(loglik_d1, loglik_d2)
loglik_e_c <- cbind(loglik_e, loglik_c)
loglik_total <- cbind(loglik_e_c, loglik_d)
}
}
if(x$model_output$type == "HURDLE" | x$model_output$type == "HURDLE_e" |
x$model_output$type == "HURDLE_c" | x$model_output$type == "HURDLE_ec") {
if(x$model_output$type == "HURDLE_ec" | x$model_output$type == "HURDLE_e") {
m_de1 <- ifelse(is.na(x$data_set$structural_effects$Control) == TRUE, 1, 0)
loglik_e1 <- x$model_output$loglik$effects$control[, m_e1 == 0]
loglik_de1 <- x$model_output$loglik$`structural indicators effects`$control[, m_e1 == 0]
m_de2 <- ifelse(is.na(x$data_set$structural_effects$Intervention) == TRUE, 1, 0)
loglik_e2 <- x$model_output$loglik$effects$intervention[, m_e2 == 0]
loglik_de2 <- x$model_output$loglik$`structural indicators effects`$intervention[, m_e2 == 0]
}
if(x$model_output$type == "HURDLE_ec" | x$model_output$type == "HURDLE_c") {
m_dc1 <- ifelse(is.na(x$data_set$structural_costs$Control) == TRUE, 1, 0)
loglik_c1 <- x$model_output$loglik$costs$control[, m_c1 == 0]
loglik_dc1 <- x$model_output$loglik$`structural indicators costs`$control[, m_c1 == 0]
m_dc2 <- ifelse(is.na(x$data_set$structural_costs$Intervention) == TRUE, 1, 0)
loglik_c2 <- x$model_output$loglik$costs$intervention[, m_c2 == 0]
loglik_dc2 <- x$model_output$loglik$`structural indicators costs`$intervention[, m_c2 == 0]
}
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
if(x$model_output$type == "HURDLE_ec") {
loglik_de <- rowSums(cbind(loglik_de1, loglik_de2))
loglik_dc <- rowSums(cbind(loglik_dc1, loglik_dc2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_de, loglik_dc))
} else if(x$model_output$type == "HURDLE_e") {
loglik_de <- rowSums(cbind(loglik_de1, loglik_de2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_de))
} else if(x$model_output$type == "HURDLE_c") {
loglik_dc <- rowSums(cbind(loglik_dc1, loglik_dc2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_dc))
}
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_e_c <- cbind(loglik_e, loglik_c)
if(x$model_output$type == "HURDLE_ec") {
loglik_de <- cbind(loglik_de1, loglik_de2)
loglik_dc <- cbind(loglik_dc1, loglik_dc2)
loglik_total <- cbind(loglik_e_c, loglik_de, loglik_dc)
} else if(x$model_output$type == "HURDLE_e") {
loglik_de <- cbind(loglik_de1, loglik_de2)
loglik_total <- cbind(loglik_e_c, loglik_de)
} else if(x$model_output$type == "HURDLE_c") {
loglik_dc <- cbind(loglik_dc1, loglik_dc2)
loglik_total <- cbind(loglik_e_c, loglik_dc)
}
}
}
if(module == "total") {
loglik <- loglik_total
} else if(module == "e") {
loglik <- loglik_e
} else if(module == "c") {
loglik <- loglik_c
} else if(module == "both") {
loglik <- loglik_e_c
}
if(criterion == "dic") {
d_bar <- mean(-2 * loglik)
pD <- var(-2 * loglik) / 2
dic <- d_bar + pD
d_hat <- 2 * d_bar - dic
ic <- list("d_bar" = d_bar, "pD" = pD, "dic" = dic, "d_hat" = d_hat)
}
if(criterion == "waic") {
waic_l <- suppressWarnings(loo::waic(loglik))
elpd <- waic_l$estimates[1, 1]
elpd_se <- waic_l$estimates[1, 2]
p <- waic_l$estimates[2, 1]
p_se <- waic_l$estimates[2, 2]
waic <- waic_l$estimates[3, 1]
waic_se <- waic_l$estimates[3, 2]
pointwise <- waic_l$pointwise
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se,
"waic" = waic, "waic_se" = waic_se, "pointwise" = pointwise)
}
if(criterion == "looic") {
loo_l <- suppressWarnings(loo::loo(loglik))
elpd <- loo_l$estimates[1, 1]
elpd_se <- loo_l$estimates[1, 2]
p <- loo_l$estimates[2, 1]
p_se <- loo_l$estimates[2, 2]
looic <- loo_l$estimates[3, 1]
looic_se <- loo_l$estimates[3, 2]
pointwise <- loo_l$pointwise
pareto_k <- loo_l$pareto_k
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se,
"looic" = looic, "looic_se" = looic_se, "pointwise" = pointwise, "pareto_k" = pareto_k)
}
}
if(x$data_format == "long") {
if(!module %in% c("total", "u", "c", "both")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
m_u1 <- is.na(x$data_set$effects$Control)
m_u2 <- is.na(x$data_set$effects$Intervention)
m_c1 <- is.na(x$data_set$costs$Control)
m_c2 <- is.na(x$data_set$costs$Intervention)
loglik_u1 <- replicate(dim(m_u1)[2], list())
loglik_u2 <- replicate(dim(m_u2)[2], list())
loglik_c1 <- replicate(dim(m_c1)[2], list())
loglik_c2 <- replicate(dim(m_c2)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u1[[time]] <- x$model_output$loglik$effects$control[, m_u1[, time] == FALSE, time]
loglik_u2[[time]] <- x$model_output$loglik$effects$intervention[, m_u2[, time] == FALSE, time]
loglik_c1[[time]] <- x$model_output$loglik$costs$control[, m_c1[, time] == FALSE, time]
loglik_c2[[time]] <- x$model_output$loglik$costs$intervention[, m_c2[, time] == FALSE, time]
}
if(x$model_output$type == "SELECTION" | x$model_output$type == "SELECTION_u" |
x$model_output$type == "SELECTION_c" | x$model_output$type == "SELECTION_uc" ) {
loglik_mu1 <- replicate(dim(m_u1)[2], list())
loglik_mu2 <- replicate(dim(m_u2)[2], list())
loglik_mc1 <- replicate(dim(m_c1)[2], list())
loglik_mc2 <- replicate(dim(m_c2)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_mu1[[time]] <- x$model_output$loglik$`missing indicators effects`$control[, , time]
loglik_mu2[[time]] <- x$model_output$loglik$`missing indicators effects`$intervention[, , time]
loglik_mc1[[time]] <- x$model_output$loglik$`missing indicators costs`$control[, , time]
loglik_mc2[[time]] <- x$model_output$loglik$`missing indicators costs`$intervention[, , time]
}
if(criterion == "dic") {
loglik_u <- replicate(dim(m_u1)[2], list())
loglik_c <- replicate(dim(m_u1)[2], list())
loglik_mu <- replicate(dim(m_u1)[2], list())
loglik_mc <- replicate(dim(m_u1)[2], list())
loglik_u_c <- replicate(dim(m_u1)[2], list())
loglik_total <- replicate(dim(m_u1)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]))
loglik_c[[time]] <- c(rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]))
loglik_mu[[time]] <- c(rowSums(loglik_mu1[[time]]), rowSums(loglik_mu2[[time]]))
loglik_mc[[time]] <- c(rowSums(loglik_mc1[[time]]), rowSums(loglik_mc2[[time]]))
loglik_u_c[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]), rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]))
loglik_total[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]), rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]),
rowSums(loglik_mu1[[time]]), rowSums(loglik_mu2[[time]]), loglik_mc1[[time]], rowSums(loglik_mc2[[time]]))
}
} else if(criterion == "looic" | criterion == "waic"){
loglik_u <- replicate(dim(m_u1)[2], list())
loglik_c <- replicate(dim(m_u1)[2], list())
loglik_mu <- replicate(dim(m_u1)[2], list())
loglik_mc <- replicate(dim(m_u1)[2], list())
loglik_u_c <- replicate(dim(m_u1)[2], list())
loglik_total <- replicate(dim(m_u1)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]])
loglik_c[[time]] <- cbind(loglik_c1[[time]], loglik_c2[[time]])
loglik_mu[[time]] <- cbind(loglik_mu1[[time]], loglik_mu2[[time]])
loglik_mc[[time]] <- cbind(loglik_mc1[[time]], loglik_mc2[[time]])
loglik_u_c[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]], loglik_c1[[time]], loglik_c2[[time]])
loglik_total[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]], loglik_c1[[time]], loglik_c2[[time]],
loglik_mu1[[time]], loglik_mu2[[time]], loglik_mc1[[time]], loglik_mc2[[time]])
}
}
}
if(module == "total") {
loglik <- loglik_total
} else if(module == "u") {
loglik <- loglik_u
} else if(module == "c") {
loglik <- loglik_c
} else if(module == "both") {
loglik <- loglik_u_c
}
if(criterion == "dic") {
d_bar <- pD <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
d_bar[time] <- mean(-2 * loglik[[time]])
pD[time] <- var(-2 * loglik[[time]]) / 2
}
dic <- d_bar + pD
d_hat <- 2 * d_bar - dic
ic <- list("d_bar" = d_bar, "pD" = pD, "dic" = dic, "d_hat" = d_hat, "sum_dic" = sum(dic), "sum_pD" = sum(pD))
}
if(criterion == "waic") {
waic_l <- pointwise <- replicate(dim(m_u1)[2], list())
elpd <- elpd_se <- rep(NA, dim(m_u1)[2])
p <- p_se <- waic <- waic_se <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
waic_l[[time]] <- suppressWarnings(loo::waic(loglik[[time]]))
elpd[time] <- waic_l[[time]]$estimates[1, 1]
elpd_se[time] <- waic_l[[time]]$estimates[1, 2]
p[time] <- waic_l[[time]]$estimates[2, 1]
p_se[time] <- waic_l[[time]]$estimates[2, 2]
waic[time] <- waic_l[[time]]$estimates[3, 1]
waic_se[time] <- waic_l[[time]]$estimates[3, 2]
pointwise[[time]] <- waic_l[[time]]$pointwise
}
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se, "waic" = waic, "waic_se" = waic_se, "pointwise" = pointwise,
"sum_waic" = sum(waic), "sum_pwaic" = sum(p))
}
if(criterion == "looic") {
loo_l <- pointwise <- pareto_k <- replicate(dim(m_u1)[2], list())
elpd <- elpd_se <- rep(NA, dim(m_u1)[2])
p <- p_se <- looic <- looic_se <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
loo_l[[time]] <- suppressWarnings(loo::loo(loglik[[time]]))
elpd[time] <- loo_l[[time]]$estimates[1, 1]
elpd_se[time] <- loo_l[[time]]$estimates[1, 2]
p[time] <- loo_l[[time]]$estimates[2, 1]
p_se[time] <- loo_l[[time]]$estimates[2, 2]
looic[time] <- loo_l[[time]]$estimates[3, 1]
looic_se[time] <- loo_l[[time]]$estimates[3, 2]
pointwise[[time]] <- loo_l[[time]]$pointwise
pareto_k[[time]] <- loo_l[[time]]$pointwise
}
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se, "looic" = looic, "looic_se" = looic_se, "pointwise" = pointwise, "pareto_k" = pareto_k,
"sum_looic" = sum(looic), "sum_plooic" = sum(p))
}
}
return(ic)
}
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Defines functions pic
Documented in pic
#' Predictive information criteria for Bayesian models fitted in \code{JAGS} using the funciton \code{\link{selection}}, \code{\link{selection_long}}, \code{\link{pattern}} or \code{\link{hurdle}}
#'
#' Efficient approximate leave-one-out cross validation (LOO), deviance information criterion (DIC) and widely applicable information criterion (WAIC) for Bayesian models,
#' calculated on the observed data.
#' @keywords loo waic dic JAGS
#' @param x A \code{missingHE} object containing the results of a Bayesian model fitted in cost-effectiveness analysis using the function \code{\link{selection}}, \code{\link{selection_long}},
#' \code{\link{pattern}} or \code{\link{hurdle}}.
#' @param criterion type of information criteria to be produced. Available choices are \code{'dic'} for the Deviance Information Criterion,
#' \code{'waic'} for the Widely Applicable Information Criterion, and \code{'looic'} for the Leave-One-Out Information Criterion.
#' @param module The modules with respect to which the information criteria should be computed. Available choices are \code{'total'} for the whole model,
#' \code{'e'} for the effectiveness variables only (\code{'u'} for longitudinal models), \code{'c'} for the cost variables only, and \code{'both'} for both outcome variables.
#' @return A named list containing different predictive information criteria results and quantities according to the value of \code{criterion}. In all cases, the measures are
#' computed on the observed data for the specific modules of the model selected in \code{module}.
#' \describe{
#' \item{d_bar}{Posterior mean deviance (only if \code{criterion} is \code{'dic'}).}
#' \item{pD}{Effective number of parameters calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}.
#' \item{dic}{Deviance Information Criterion calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}.
#' \item{d_hat}{Deviance evaluated at the posterior mean of the parameters and calculated with the formula used by \code{JAGS} (only if \code{criterion} is \code{'dic'})}
#' \item{elpd, elpd_se}{Expected log pointwise predictive density and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{p, p_se}{Effective number of parameters and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{looic, looic_se}{The leave-one-out information criterion and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'loo'}).}
#' \item{waic, waic_se}{The widely applicable information criterion and standard error calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'}).}
#' \item{pointwise}{A matrix containing the pointwise contributions of each of the above measures calculated on the observed data for the model nodes indicated in \code{module}
#' (only if \code{criterion} is \code{'waic'} or \code{'loo'}).}
#' \item{pareto_k}{A vector containing the estimates of the shape parameter \eqn{k} for the generalised Pareto fit to the importance ratios for each leave-one-out distribution
#' calculated on the observed data for the model nodes indicated in \code{module} (only if \code{criterion} is \code{'loo'}).
#' See \code{\link[loo]{loo}} for details about interpreting \eqn{k}.}
#' \item{sum_dic}{DIC value calculated by summing up all model dic evaluated at each time point (only for longitudinal models). Similar estimates can
#' are obtained also for the other criteria, either \code{sum_waic} or \code{sum_looic}.}
#' \item{sum_pdic}{DIC value calculated by summing up all model effective number of parameter estimates based on dic evaluated at each time point (only for longitudinal models). Similar estimates can
#' are obtained also for the other criteria, either \code{sum_pwaic} or \code{sum_plooic}.}
#' }
#' @seealso \code{\link[R2jags]{jags}}, \code{\link[loo]{loo}}, \code{\link[loo]{waic}}
#' @author Andrea Gabrio
#' @importFrom stats var
#' @details The Deviance Information Criterion (DIC), Leave-One-Out Information Criterion (LOOIC) and the Widely Applicable Information Criterion (WAIC) are methods for estimating
#' out-of-sample predictive accuracy from a Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameters. If \code{x} contains the results from
#' a longitudinal model, all parameter names indexed by "e" should be instead indexed by "u". In addition, for longitudinal models information criteria results are displayed by time
#' and only a general approximation to the total value of the criteria and pD is given as the sum of the corresponding measures computed at each time point.
#' DIC is computationally simple to calculate but it is known to have some problems, arising in part from it not being fully Bayesian in that it is based on a point estimate.
#' LOOIC can be computationally expensive but can be easily approximated using importance weights that are smoothed by fitting a generalised Pareto distribution to the upper tail
#' of the distribution of the importance weights. For more details about the methods used to compute LOOIC see the PSIS-LOO section in \code{\link{loo-package}}.
#' WAIC is fully Bayesian and closely approximates Bayesian cross-validation. Unlike DIC, WAIC is invariant to parameterisation and also works for singular models.
#' In finite cases, WAIC and LOO give similar estimates, but for influential observations WAIC underestimates the effect of leaving out one observation.
#'
#' @references
#' Plummer, M. \emph{JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.} (2003).
#'
#' Vehtari, A. Gelman, A. Gabry, J. (2016a) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.
#' \emph{Statistics and Computing}. Advance online publication.
#'
#' Vehtari, A. Gelman, A. Gabry, J. (2016b) Pareto smoothed importance sampling. \emph{ArXiv} preprint.
#'
#' Gelman, A. Hwang, J. Vehtari, A. (2014) Understanding predictive information criteria for Bayesian models.
#' \emph{Statistics and Computing} 24, 997-1016.
#'
#' Watanable, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information
#' criterion in singular learning theory. \emph{Journal of Machine Learning Research} 11, 3571-3594.
#'
#' @export
#' @examples
#' # For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
#' # \code{\link{pattern}} or \code{\link{hurdle}}
#' #
#' #
pic <- function(x, criterion = "dic", module = "total") {
if(!isTRUE(requireNamespace("loo", quietly = TRUE))) {
stop("You need to install the R package 'loo'. Please run in your R terminal:\n install.packages('loo')")
}
if(!inherits(x, what = "missingHE")) {
stop("Only objects of class 'missingHE' can be used")
}
if(x$model_output$`model summary`$BUGSoutput$n.chains == 1){
stop("information criteria cannot be computed if n.chain=1")
}
if(is.character(criterion) == FALSE | is.character(module) == FALSE) {
stop("criterion and module must be character names")
}
criterion <- tolower(criterion)
module <- tolower(module)
if(!criterion %in% c("dic", "waic", "looic")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
if(x$data_format == "wide") {
if(!module %in% c("total", "e", "c", "both")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
m_e1 <- x$data_set$missing_effects$Control
loglik_e1 <- x$model_output$loglik$effects$control[, m_e1 == 0]
m_e2 <- x$data_set$missing_effects$Intervention
loglik_e2 <- x$model_output$loglik$effects$intervention[, m_e2 == 0]
m_c1 <- x$data_set$missing_costs$Control
loglik_c1 <- x$model_output$loglik$costs$control[, m_c1 == 0]
m_c2 <- x$data_set$missing_costs$Intervention
loglik_c2 <- x$model_output$loglik$costs$intervention[, m_c2 == 0]
if(x$model_output$type == "SELECTION" | x$model_output$type == "SELECTION_e" |
x$model_output$type == "SELECTION_c" | x$model_output$type == "SELECTION_ec" ) {
loglik_me1 <- x$model_output$loglik$`missing indicators effects`$control
loglik_me2 <- x$model_output$loglik$`missing indicators effects`$intervention
loglik_mc1 <- x$model_output$loglik$`missing indicators costs`$control
loglik_mc2 <- x$model_output$loglik$`missing indicators costs`$intervention
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_me <- rowSums(cbind(loglik_me1, loglik_me2))
loglik_mc <- rowSums(cbind(loglik_mc1, loglik_mc2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_me, loglik_mc))
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_me <- cbind(loglik_me1, loglik_me2)
loglik_mc <- cbind(loglik_mc1, loglik_mc2)
loglik_e_c <- cbind(loglik_e, loglik_c)
loglik_total <- cbind(loglik_e_c, loglik_me, loglik_mc)
}
}
if(x$model_output$type == "PATTERN" | x$model_output$type == "PATTERN_e" |
x$model_output$type == "PATTERN_c" | x$model_output$type == "PATTERN_ec" ) {
loglik_d1 <- x$model_output$loglik$`pattern indicators`$control
loglik_d2 <- x$model_output$loglik$`pattern indicators`$intervention
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_d <- rowSums(cbind(loglik_d1, loglik_d2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_d))
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_d <- cbind(loglik_d1, loglik_d2)
loglik_e_c <- cbind(loglik_e, loglik_c)
loglik_total <- cbind(loglik_e_c, loglik_d)
}
}
if(x$model_output$type == "HURDLE" | x$model_output$type == "HURDLE_e" |
x$model_output$type == "HURDLE_c" | x$model_output$type == "HURDLE_ec") {
if(x$model_output$type == "HURDLE_ec" | x$model_output$type == "HURDLE_e") {
m_de1 <- ifelse(is.na(x$data_set$structural_effects$Control) == TRUE, 1, 0)
loglik_e1 <- x$model_output$loglik$effects$control[, m_e1 == 0]
loglik_de1 <- x$model_output$loglik$`structural indicators effects`$control[, m_e1 == 0]
m_de2 <- ifelse(is.na(x$data_set$structural_effects$Intervention) == TRUE, 1, 0)
loglik_e2 <- x$model_output$loglik$effects$intervention[, m_e2 == 0]
loglik_de2 <- x$model_output$loglik$`structural indicators effects`$intervention[, m_e2 == 0]
}
if(x$model_output$type == "HURDLE_ec" | x$model_output$type == "HURDLE_c") {
m_dc1 <- ifelse(is.na(x$data_set$structural_costs$Control) == TRUE, 1, 0)
loglik_c1 <- x$model_output$loglik$costs$control[, m_c1 == 0]
loglik_dc1 <- x$model_output$loglik$`structural indicators costs`$control[, m_c1 == 0]
m_dc2 <- ifelse(is.na(x$data_set$structural_costs$Intervention) == TRUE, 1, 0)
loglik_c2 <- x$model_output$loglik$costs$intervention[, m_c2 == 0]
loglik_dc2 <- x$model_output$loglik$`structural indicators costs`$intervention[, m_c2 == 0]
}
if(criterion == "dic") {
loglik_e <- rowSums(cbind(loglik_e1, loglik_e2))
loglik_c <- rowSums(cbind(loglik_c1, loglik_c2))
loglik_e_c <- rowSums(cbind(loglik_e, loglik_c))
if(x$model_output$type == "HURDLE_ec") {
loglik_de <- rowSums(cbind(loglik_de1, loglik_de2))
loglik_dc <- rowSums(cbind(loglik_dc1, loglik_dc2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_de, loglik_dc))
} else if(x$model_output$type == "HURDLE_e") {
loglik_de <- rowSums(cbind(loglik_de1, loglik_de2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_de))
} else if(x$model_output$type == "HURDLE_c") {
loglik_dc <- rowSums(cbind(loglik_dc1, loglik_dc2))
loglik_total <- rowSums(cbind(loglik_e_c, loglik_dc))
}
} else if(criterion == "looic" | criterion == "waic"){
loglik_e <- cbind(loglik_e1, loglik_e2)
loglik_c <- cbind(loglik_c1, loglik_c2)
loglik_e_c <- cbind(loglik_e, loglik_c)
if(x$model_output$type == "HURDLE_ec") {
loglik_de <- cbind(loglik_de1, loglik_de2)
loglik_dc <- cbind(loglik_dc1, loglik_dc2)
loglik_total <- cbind(loglik_e_c, loglik_de, loglik_dc)
} else if(x$model_output$type == "HURDLE_e") {
loglik_de <- cbind(loglik_de1, loglik_de2)
loglik_total <- cbind(loglik_e_c, loglik_de)
} else if(x$model_output$type == "HURDLE_c") {
loglik_dc <- cbind(loglik_dc1, loglik_dc2)
loglik_total <- cbind(loglik_e_c, loglik_dc)
}
}
}
if(module == "total") {
loglik <- loglik_total
} else if(module == "e") {
loglik <- loglik_e
} else if(module == "c") {
loglik <- loglik_c
} else if(module == "both") {
loglik <- loglik_e_c
}
if(criterion == "dic") {
d_bar <- mean(-2 * loglik)
pD <- var(-2 * loglik) / 2
dic <- d_bar + pD
d_hat <- 2 * d_bar - dic
ic <- list("d_bar" = d_bar, "pD" = pD, "dic" = dic, "d_hat" = d_hat)
}
if(criterion == "waic") {
waic_l <- suppressWarnings(loo::waic(loglik))
elpd <- waic_l$estimates[1, 1]
elpd_se <- waic_l$estimates[1, 2]
p <- waic_l$estimates[2, 1]
p_se <- waic_l$estimates[2, 2]
waic <- waic_l$estimates[3, 1]
waic_se <- waic_l$estimates[3, 2]
pointwise <- waic_l$pointwise
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se,
"waic" = waic, "waic_se" = waic_se, "pointwise" = pointwise)
}
if(criterion == "looic") {
loo_l <- suppressWarnings(loo::loo(loglik))
elpd <- loo_l$estimates[1, 1]
elpd_se <- loo_l$estimates[1, 2]
p <- loo_l$estimates[2, 1]
p_se <- loo_l$estimates[2, 2]
looic <- loo_l$estimates[3, 1]
looic_se <- loo_l$estimates[3, 2]
pointwise <- loo_l$pointwise
pareto_k <- loo_l$pareto_k
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se,
"looic" = looic, "looic_se" = looic_se, "pointwise" = pointwise, "pareto_k" = pareto_k)
}
}
if(x$data_format == "long") {
if(!module %in% c("total", "u", "c", "both")) {
stop("you must select a character name among those available. For details type 'help(pic)'")
}
m_u1 <- is.na(x$data_set$effects$Control)
m_u2 <- is.na(x$data_set$effects$Intervention)
m_c1 <- is.na(x$data_set$costs$Control)
m_c2 <- is.na(x$data_set$costs$Intervention)
loglik_u1 <- replicate(dim(m_u1)[2], list())
loglik_u2 <- replicate(dim(m_u2)[2], list())
loglik_c1 <- replicate(dim(m_c1)[2], list())
loglik_c2 <- replicate(dim(m_c2)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u1[[time]] <- x$model_output$loglik$effects$control[, m_u1[, time] == FALSE, time]
loglik_u2[[time]] <- x$model_output$loglik$effects$intervention[, m_u2[, time] == FALSE, time]
loglik_c1[[time]] <- x$model_output$loglik$costs$control[, m_c1[, time] == FALSE, time]
loglik_c2[[time]] <- x$model_output$loglik$costs$intervention[, m_c2[, time] == FALSE, time]
}
if(x$model_output$type == "SELECTION" | x$model_output$type == "SELECTION_u" |
x$model_output$type == "SELECTION_c" | x$model_output$type == "SELECTION_uc" ) {
loglik_mu1 <- replicate(dim(m_u1)[2], list())
loglik_mu2 <- replicate(dim(m_u2)[2], list())
loglik_mc1 <- replicate(dim(m_c1)[2], list())
loglik_mc2 <- replicate(dim(m_c2)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_mu1[[time]] <- x$model_output$loglik$`missing indicators effects`$control[, , time]
loglik_mu2[[time]] <- x$model_output$loglik$`missing indicators effects`$intervention[, , time]
loglik_mc1[[time]] <- x$model_output$loglik$`missing indicators costs`$control[, , time]
loglik_mc2[[time]] <- x$model_output$loglik$`missing indicators costs`$intervention[, , time]
}
if(criterion == "dic") {
loglik_u <- replicate(dim(m_u1)[2], list())
loglik_c <- replicate(dim(m_u1)[2], list())
loglik_mu <- replicate(dim(m_u1)[2], list())
loglik_mc <- replicate(dim(m_u1)[2], list())
loglik_u_c <- replicate(dim(m_u1)[2], list())
loglik_total <- replicate(dim(m_u1)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]))
loglik_c[[time]] <- c(rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]))
loglik_mu[[time]] <- c(rowSums(loglik_mu1[[time]]), rowSums(loglik_mu2[[time]]))
loglik_mc[[time]] <- c(rowSums(loglik_mc1[[time]]), rowSums(loglik_mc2[[time]]))
loglik_u_c[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]), rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]))
loglik_total[[time]] <- c(rowSums(loglik_u1[[time]]), rowSums(loglik_u2[[time]]), rowSums(loglik_c1[[time]]), rowSums(loglik_c2[[time]]),
rowSums(loglik_mu1[[time]]), rowSums(loglik_mu2[[time]]), loglik_mc1[[time]], rowSums(loglik_mc2[[time]]))
}
} else if(criterion == "looic" | criterion == "waic"){
loglik_u <- replicate(dim(m_u1)[2], list())
loglik_c <- replicate(dim(m_u1)[2], list())
loglik_mu <- replicate(dim(m_u1)[2], list())
loglik_mc <- replicate(dim(m_u1)[2], list())
loglik_u_c <- replicate(dim(m_u1)[2], list())
loglik_total <- replicate(dim(m_u1)[2], list())
for(time in 1:dim(m_u1)[2]) {
loglik_u[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]])
loglik_c[[time]] <- cbind(loglik_c1[[time]], loglik_c2[[time]])
loglik_mu[[time]] <- cbind(loglik_mu1[[time]], loglik_mu2[[time]])
loglik_mc[[time]] <- cbind(loglik_mc1[[time]], loglik_mc2[[time]])
loglik_u_c[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]], loglik_c1[[time]], loglik_c2[[time]])
loglik_total[[time]] <- cbind(loglik_u1[[time]], loglik_u2[[time]], loglik_c1[[time]], loglik_c2[[time]],
loglik_mu1[[time]], loglik_mu2[[time]], loglik_mc1[[time]], loglik_mc2[[time]])
}
}
}
if(module == "total") {
loglik <- loglik_total
} else if(module == "u") {
loglik <- loglik_u
} else if(module == "c") {
loglik <- loglik_c
} else if(module == "both") {
loglik <- loglik_u_c
}
if(criterion == "dic") {
d_bar <- pD <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
d_bar[time] <- mean(-2 * loglik[[time]])
pD[time] <- var(-2 * loglik[[time]]) / 2
}
dic <- d_bar + pD
d_hat <- 2 * d_bar - dic
ic <- list("d_bar" = d_bar, "pD" = pD, "dic" = dic, "d_hat" = d_hat, "sum_dic" = sum(dic), "sum_pD" = sum(pD))
}
if(criterion == "waic") {
waic_l <- pointwise <- replicate(dim(m_u1)[2], list())
elpd <- elpd_se <- rep(NA, dim(m_u1)[2])
p <- p_se <- waic <- waic_se <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
waic_l[[time]] <- suppressWarnings(loo::waic(loglik[[time]]))
elpd[time] <- waic_l[[time]]$estimates[1, 1]
elpd_se[time] <- waic_l[[time]]$estimates[1, 2]
p[time] <- waic_l[[time]]$estimates[2, 1]
p_se[time] <- waic_l[[time]]$estimates[2, 2]
waic[time] <- waic_l[[time]]$estimates[3, 1]
waic_se[time] <- waic_l[[time]]$estimates[3, 2]
pointwise[[time]] <- waic_l[[time]]$pointwise
}
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se, "waic" = waic, "waic_se" = waic_se, "pointwise" = pointwise,
"sum_waic" = sum(waic), "sum_pwaic" = sum(p))
}
if(criterion == "looic") {
loo_l <- pointwise <- pareto_k <- replicate(dim(m_u1)[2], list())
elpd <- elpd_se <- rep(NA, dim(m_u1)[2])
p <- p_se <- looic <- looic_se <- rep(NA, dim(m_u1)[2])
for(time in 1:dim(m_u1)[2]) {
loo_l[[time]] <- suppressWarnings(loo::loo(loglik[[time]]))
elpd[time] <- loo_l[[time]]$estimates[1, 1]
elpd_se[time] <- loo_l[[time]]$estimates[1, 2]
p[time] <- loo_l[[time]]$estimates[2, 1]
p_se[time] <- loo_l[[time]]$estimates[2, 2]
looic[time] <- loo_l[[time]]$estimates[3, 1]
looic_se[time] <- loo_l[[time]]$estimates[3, 2]
pointwise[[time]] <- loo_l[[time]]$pointwise
pareto_k[[time]] <- loo_l[[time]]$pointwise
}
ic <- list("elpd" = elpd, "elpd_se" = elpd_se, "p" = p, "p_se" = p_se, "looic" = looic, "looic_se" = looic_se, "pointwise" = pointwise, "pareto_k" = pareto_k,
"sum_looic" = sum(looic), "sum_plooic" = sum(p))
}
}
return(ic)
}
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