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#' @title GGM: Estimation
#'
#' @description Estimate the conditional (in)dependence with either an analytic solution or efficiently
#' sampling from the posterior distribution. These methods were introduced in \insertCite{Williams2019;textual}{BGGM}.
#' The graph is selected with \code{\link{select.estimate}} and then plotted with \code{\link{plot.select}}.
#'
#' @name estimate
#'
#' @param Y Matrix (or data frame) of dimensions \emph{n} (observations) by \emph{p} (variables).
#'
#' @param formula An object of class \code{\link[stats]{formula}}. This allows for including
#' control variables in the model (i.e., \code{~ gender}). See the note for further details.
#'
#' @param type Character string. Which type of data for \code{Y} ? The options include \code{continuous},
#' \code{binary}, \code{ordinal}, or \code{mixed}. Note that mixed can be used for data with only
#' ordinal variables. See the note for further details.
#'
#' @param mixed_type Numeric vector. An indicator of length \emph{p} for which variables should be treated as ranks.
#' (1 for rank and 0 to assume normality). The default is currently to treat all integer variables as ranks
#' when \code{type = "mixed"} and \code{NULL} otherwise. See note for further details.
#'
#' @param analytic Logical. Should the analytic solution be computed (default is \code{FALSE})?
#'
#' @param prior_sd Scale of the prior distribution, approximately the standard deviation of a beta distribution
#' (defaults to sqrt(1/12)).
#'
#' @param iter Number of iterations (posterior samples; defaults to 5000).
#'
#' @param impute Logical. Should the missing values (\code{NA})
#' be imputed during model fitting (defaults to \code{TRUE}) ?
#'
#' @param progress Logical. Should a progress bar be included (defaults to \code{TRUE}) ?
#'
#' @param seed An integer for the random seed.
#'
#' @param ... Currently ignored.
#'
#' @references
#' \insertAllCited{}
#'
#'
#' @return The returned object of class \code{estimate} contains a lot of information that
#' is used for printing and plotting the results. For users of \strong{BGGM}, the following
#' are the useful objects:
#'
#' \itemize{
#'
#' \item \code{pcor_mat} Partial correltion matrix (posterior mean).
#'
#' \item \code{post_samp} An object containing the posterior samples.
#'
#' }
#'
#'
#' @details
#'
#' The default is to draw samples from the posterior distribution (\code{analytic = FALSE}). The samples are
#' required for computing edge differences (see \code{\link{ggm_compare_estimate}}), Bayesian R2 introduced in
#' \insertCite{gelman_r2_2019;textual}{BGGM} (see \code{\link{predictability}}), etc. If the goal is
#' to *only* determine the non-zero effects, this can be accomplished by setting \code{analytic = TRUE}.
#' This is particularly useful when a fast solution is needed (see the examples in \code{\link{ggm_compare_ppc}})
#'
#' \strong{Controlling for Variables}:
#'
#' When controlling for variables, it is assumed that \code{Y} includes \emph{only}
#' the nodes in the GGM and the control variables. Internally, \code{only} the predictors
#' that are included in \code{formula} are removed from \code{Y}. This is not behavior of, say,
#' \code{\link{lm}}, but was adopted to ensure users do not have to write out each variable that
#' should be included in the GGM. An example is provided below.
#'
#' \strong{Mixed Type}:
#'
#' The term "mixed" is somewhat of a misnomer, because the method can be used for data including \emph{only}
#' continuous or \emph{only} discrete variables. This is based on the ranked likelihood which requires sampling
#' the ranks for each variable (i.e., the data is not merely transformed to ranks). This is computationally
#' expensive when there are many levels. For example, with continuous data, there are as many ranks
#' as data points!
#'
#' The option \code{mixed_type} allows the user to determine which variable should be treated as ranks
#' and the "emprical" distribution is used otherwise \insertCite{hoff2007extending}{BGGM}. This is
#' accomplished by specifying an indicator vector of length \emph{p}. A one indicates to use the ranks,
#' whereas a zero indicates to "ignore" that variable. By default all integer variables are treated as ranks.
#'
#' \strong{Dealing with Errors}:
#'
#' An error is most likely to arise when \code{type = "ordinal"}. The are two common errors (although still rare):
#'
#' \itemize{
#'
#' \item The first is due to sampling the thresholds, especially when the data is heavily skewed.
#' This can result in an ill-defined matrix. If this occurs, we recommend to first try
#' decreasing \code{prior_sd} (i.e., a more informative prior). If that does not work, then
#' change the data type to \code{type = mixed} which then estimates a copula GGM
#' (this method can be used for data containing \strong{only} ordinal variable). This should
#' work without a problem.
#'
#' \item The second is due to how the ordinal data are categorized. For example, if the error states
#' that the index is out of bounds, this indicates that the first category is a zero. This is not allowed, as
#' the first category must be one. This is addressed by adding one (e.g., \code{Y + 1}) to the data matrix.
#'
#' }
#'
#' \strong{Imputing Missing Values}:
#'
#' Missing values are imputed with the approach described in \insertCite{hoff2009first;textual}{BGGM}.
#' The basic idea is to impute the missing values with the respective posterior pedictive distribution,
#' given the observed data, as the model is being estimated. Note that the default is \code{TRUE},
#' but this ignored when there are no missing values. If set to \code{FALSE}, and there are missing
#' values, list-wise deletion is performed with \code{na.omit}.
#'
#'
#' @note
#'
#' \strong{Posterior Uncertainty}:
#'
#' A key feature of \bold{BGGM} is that there is a posterior distribution for each partial correlation.
#' This readily allows for visiualizing uncertainty in the estimates. This feature works
#' with all data types and is accomplished by plotting the summary of the \code{estimate} object
#' (i.e., \code{plot(summary(fit))}). Several examples are provided below.
#'
#'
#'
#' \strong{Interpretation of Conditional (In)dependence Models for Latent Data}:
#'
#' See \code{\link{BGGM-package}} for details about interpreting GGMs based on latent data
#' (i.e, all data types besides \code{"continuous"})
#'
#' @examples
#' \donttest{
#' # note: iter = 250 for demonstrative purposes
#'
#' #########################################
#' ### example 1: continuous and ordinal ###
#' #########################################
#' # data
#' Y <- ptsd
#'
#' # continuous
#'
#' # fit model
#' fit <- estimate(Y, type = "continuous",
#' iter = 250)
#'
#' # summarize the partial correlations
#' summ <- summary(fit)
#'
#' # plot the summary
#' plt_summ <- plot(summary(fit))
#'
#' # select the graph
#' E <- select(fit)
#'
#' # plot the selected graph
#' plt_E <- plot(select(fit))
#'
#'
#' # ordinal
#'
#' # fit model (note + 1, due to zeros)
#' fit <- estimate(Y + 1, type = "ordinal",
#' iter = 250)
#'
#' # summarize the partial correlations
#' summ <- summary(fit)
#'
#' # plot the summary
#' plt <- plot(summary(fit))
#'
#' # select the graph
#' E <- select(fit)
#'
#' # plot the selected graph
#' plt_E <- plot(select(fit))
#'
#' ##################################
#' ## example 2: analytic solution ##
#' ##################################
#' # (only continuous)
#'
#' # data
#' Y <- ptsd
#'
#' # fit model
#' fit <- estimate(Y, analytic = TRUE)
#'
#' # summarize the partial correlations
#' summ <- summary(fit)
#'
#' # plot summary
#' plt_summ <- plot(summary(fit))
#'
#' # select graph
#' E <- select(fit)
#'
#' # plot the selected graph
#' plt_E <- plot(select(fit))
#'
#'}
#'
#' @export
estimate <- function(Y,
formula = NULL,
type = "continuous",
mixed_type = NULL,
analytic = FALSE,
prior_sd = sqrt(1/12),
iter = 5000,
impute = FALSE,
progress = TRUE,
seed = NULL,
...){
# temporary warning until missing data is fully implemented
if(!type %in% c("continuous", "mixed")){
if(impute){
warning(paste0("imputation during model fitting is\n",
"currently only implemented for 'continuous'
and 'mixed' data."))
}
}
## Random seed unless user provided
if(!is.null(seed) ) {
set.seed(seed)
}
# delta rho ~ beta(delta/2, delta/2)
delta <- delta_solve(prior_sd)
# nodes
p <- ncol(Y)
# sample posterior
if(!analytic){
if(isTRUE(progress)){
message(paste0("BGGM: Posterior Sampling ", ...))
}
# continuous
if(type == "continuous"){
# no control
if(is.null(formula)){
if(!impute){
# na omit
Y <- as.matrix(na.omit(Y))
Y_miss <- Y
} else {
Y_miss <- ifelse(is.na(Y), 1, 0)
if(sum(Y_miss) == 0){
impute <- FALSE
}
# impute means
for(i in 1:p){
Y[which(is.na(Y[,i])),i] <- mean(na.omit(Y[,i]))
}
}
# scale Y
Y <- scale(Y, scale = F)
# design matrix
X <- NULL
# number of variables
n <- nrow(Y)
# starting values
start <- solve(cov(Y))
# posterior sample
post_samp <- .Call(
'_BGGM_Theta_continuous',
PACKAGE = 'BGGM',
Y = Y,
iter = iter + 50,
delta = delta,
epsilon = 0.1,
prior_only = 0,
explore = 1,
start = start,
progress = progress,
impute = impute,
Y_miss = Y_miss
)
# control for variables
} else {
control_info <- remove_predictors_helper(list(as.data.frame(Y)),
formula = formula)
# data
Y <- as.matrix(scale(control_info$Y_groups[[1]], scale = F))
# nodes
p <- ncol(Y)
# observations
n <- nrow(Y)
# model matrix
X <- as.matrix(control_info$model_matrices[[1]])
start <- solve(cov(Y))
# posterior sample
post_samp <- .Call(
"_BGGM_mv_continuous",
Y = Y,
X = X,
delta = delta,
epsilon = 0.1,
iter = iter + 50,
start = start,
progress = progress
)
# end control
}
# binary
} else if (type == "binary") {
# intercept only
if (is.null(formula)) {
# data
Y <- as.matrix(na.omit(Y))
# obervations
n <- nrow(Y)
# nodes
p <- ncol(Y)
X <- matrix(1, n, 1)
formula <- ~ 1
start <- solve(cov(Y))
} else {
control_info <- remove_predictors_helper(list(as.data.frame(Y)),
formula = formula)
# data
Y <- as.matrix(control_info$Y_groups[[1]])
# observations
n <- nrow(Y)
# nodes
p <- ncol(Y)
# model matrix
X <- as.matrix(control_info$model_matrices[[1]])
start <- solve(cov(Y))
}
# posterior sample
post_samp <- .Call(
"_BGGM_mv_binary",
Y = Y,
X = X,
delta = delta,
epsilon = 0.1,
iter = iter + 50,
beta_prior = 0.0001,
cutpoints = c(-Inf, 0, Inf),
start = start,
progress = progress
)
# ordinal
} else if(type == "ordinal"){
# intercept only
if(is.null(formula)){
# data
Y <- as.matrix(na.omit(Y))
# obervations
n <- nrow(Y)
# nodes
p <- ncol(Y)
# intercept only
X <- matrix(1, n, 1)
formula <- ~ 1
start <- solve(cov(Y))
} else {
control_info <- remove_predictors_helper(list(as.data.frame(Y)),
formula = formula)
# data
Y <- as.matrix(control_info$Y_groups[[1]])
# observations
n <- nrow(Y)
# nodes
p <- ncol(Y)
# model matrix
X <- as.matrix(control_info$model_matrices[[1]])
start <- solve(cov(Y))
}
# categories
K <- max(apply(Y, 2, function(x) { length(unique(x)) } ))
# call c ++
post_samp <- .Call(
"_BGGM_mv_ordinal_albert",
Y = Y,
X = X,
iter = iter + 50,
delta = delta,
epsilon = 0.1,
K = K,
start = start,
progress = progress
)
} else if(type == "mixed"){
X <- NULL
# no control variables allowed
if(!is.null(formula)){
warning("formula ignored for mixed data at this time")
control_info <- remove_predictors_helper(list(as.data.frame(Y)),
formula = formula)
# data
Y <- as.matrix(control_info$Y_groups[[1]])
formula <- NULL
}
# default for ranks
if(is.null(mixed_type)) {
idx = rep(1, ncol(Y))
# user defined
} else {
idx = mixed_type
}
# observations
n <- nrow(Y)
# nodes
p <- ncol(Y)
# rank following hoff (2008)
rank_vars <- rank_helper(Y)
if(impute){
Y_missing <- ifelse(is.na(Y), 1, 0)
rank_vars$z0_start[is.na(rank_vars$z0_start)] <- rnorm(sum(Y_missing))
post_samp <- .Call(
"_BGGM_missing_copula",
Y = Y,
Y_missing = Y_missing,
z0_start = rank_vars$z0_start,
Sigma_start = cov(rank_vars$z0_start),
levels = rank_vars$levels,
iter_missing = iter + 50,
progress_impute = TRUE,
K = rank_vars$K,
idx = idx,
epsilon = 0.1,
delta = delta
)
} else {
## check if Y contains NA
contains_na <- any(is.na(Y))
if( contains_na ) {
Y <- na.omit(Y)
## Reassign values when NA are present
temp_helper <- rank_helper(Y)
rank_vars$z0_start <- temp_helper$z0_start
rank_vars$levels <- temp_helper$levels
warning("Observed variables contain missings:\n No imputation, cases are deleted listwise." )
}
post_samp <- .Call(
"_BGGM_copula",
z0_start = rank_vars$z0_start,
levels = rank_vars$levels,
K = rank_vars$K,
Sigma_start = cov(Y),
iter = iter + 50,
delta = delta,
epsilon = 0.1,
idx = idx,
progress = progress
)
}
} else {
stop("'type' not supported: must be continuous, binary, ordinal, or mixed.")
}
if(isTRUE(progress)){
message("BGGM: Finished")
}
pcor_mat <- post_samp$pcor_mat
results <- list(
pcor_mat = pcor_mat,
analytic = analytic,
formula = formula,
post_samp = post_samp,
type = type,
iter = iter,
Y = Y,
X = X,
call = match.call(),
prior_sd = prior_sd,
p = p,
n = n,
ppd_mean = post_samp$ppd_mean
)
# analytic
} else {
if(type != "continuous"){
warning("analytic solution only available for 'type = continuous'")
type <- "continuous"
}
if(!is.null(formula)){
stop("formula note permitted with the analytic solution")
}
Y <- na.omit(Y)
# observations
n <- nrow(Y)
p <- ncol(Y)
formula <- NULL
analytic_fit <- analytic_solve(Y)
results <- list(pcor_mat = analytic_fit$pcor_mat,
analytic_fit = analytic_fit,
analytic = analytic,
formula = formula,
type = type,
iter = iter,
Y = Y,
call = match.call(),
prior_sd = prior_sd,
p = p,
n = n)
} # end analytic
# removed per CRAN (8/12/21)
#.Random.seed <<- old
returned_object <- results
class(returned_object) <- c("BGGM",
"estimate",
"default")
return(returned_object)
}
#' @title Summary method for \code{estimate.default} objects
#'
#' @name summary.estimate
#'
#' @description Summarize the posterior distribution of each partial correlation
#' with the posterior mean and standard deviation.
#'
#'
#' @param object An object of class \code{estimate}
#'
#' @param col_names Logical. Should the summary include the column names (default is \code{TRUE})?
#' Setting to \code{FALSE} includes the column numbers (e.g., \code{1--2}).
#'
#' @param cred Numeric. The credible interval width for summarizing the posterior
#' distributions (defaults to 0.95; must be between 0 and 1).
#'
#' @param ... Currently ignored.
#'
#' @seealso \code{\link{estimate}}
#'
#' @return A dataframe containing the summarized posterior distributions.
#'
#' @examples
#' \donttest{
#' # data
#' Y <- ptsd[,1:5]
#'
#' fit <- estimate(Y, iter = 250,
#' progress = FALSE)
#'
#' summary(fit)
#'
#'}
#'
#' @export
summary.estimate <- function(object,
col_names = TRUE,
cred = 0.95, ...) {
# nodes
p <- object$p
# identity matrix
I_p <- diag(p)
# lower bound
lb <- (1 - cred) / 2
# upper bound
ub <- 1 - lb
# column names
cn <- colnames(object$Y)
if(is.null(cn) | isFALSE(col_names)){
mat_names <- sapply(1:p , function(x) paste(1:p, x, sep = "--"))[upper.tri(I_p)]
} else {
mat_names <- sapply(cn , function(x) paste(cn, x, sep = "--"))[upper.tri(I_p)]
}
if(isFALSE(object$analytic)){
post_mean <- round(object$pcor_mat[upper.tri(I_p)], 3)
post_sd <- round(apply(object$post_samp$pcors[,, 51:(object$iter + 50) ], 1:2, sd), 3)[upper.tri(I_p)]
post_lb <- round(apply( object$post_samp$pcors[,, 51:(object$iter + 50) ], 1:2, quantile, lb), 3)[upper.tri(I_p)]
post_ub <- round(apply( object$post_samp$pcors[,, 51:(object$iter + 50) ], 1:2, quantile, ub), 3)[upper.tri(I_p)]
dat_results <-
data.frame(
relation = mat_names,
post_mean = post_mean,
post_sd = post_sd,
post_lb = post_lb,
post_ub = post_ub
)
colnames(dat_results) <- c(
"Relation",
"Post.mean",
"Post.sd",
"Cred.lb",
"Cred.ub")
} else {
dat_results <-
data.frame(
relation = mat_names,
post_mean = object$pcor_mat[upper.tri(I_p)]
)
colnames(dat_results) <- c(
"Relation",
"Post.mean")
}
returned_object <- list(dat_results = dat_results,
object = object)
class(returned_object) <- c("BGGM", "estimate",
"summary_estimate",
"summary.estimate")
returned_object
}
print_summary_estimate <- function(x, ...) {
cat("BGGM: Bayesian Gaussian Graphical Models \n")
cat("--- \n")
cat("Type:", x$object$type, "\n")
cat("Analytic:", x$object$analytic, "\n")
cat("Formula:", paste(as.character(x$object$formula), collapse = " "), "\n")
# number of iterations
cat("Posterior Samples:", x$object$iter, "\n")
# number of observations
cat("Observations (n):\n")
# number of variables
cat("Nodes (p):", x$object$p, "\n")
# number of edges
cat("Relations:", .5 * (x$object$p * (x$object$p - 1)), "\n")
cat("--- \n")
cat("Call: \n")
print(x$object$call)
cat("--- \n")
cat("Estimates:\n")
print(x$dat_results, row.names = F)
cat("--- \n")
}
print_estimate <- function(x, ...){
cat("BGGM: Bayesian Gaussian Graphical Models \n")
cat("--- \n")
cat("Type:", x$type, "\n")
cat("Analytic:", x$analytic, "\n")
cat("Formula:", paste(as.character(x$formula), collapse = " "), "\n")
# number of iterations
cat("Posterior Samples:", x$iter, "\n")
# number of observations
cat("Observations (n):\n")
# number of variables
cat("Nodes (p):", x$p, "\n")
# number of edges
cat("Relations:", .5 * (x$p * (x$p-1)), "\n")
cat("--- \n")
cat("Call: \n")
print(x$call)
cat("--- \n")
cat("Date:", date(), "\n")
}
#' @title Plot \code{summary.estimate} Objects
#'
#' @description Visualize the posterior distributions for each partial correlation.
#'
#' @name plot.summary.estimate
#'
#' @param x An object of class \code{summary.estimate}
#'
#' @param size Numeric. The size for the points (defaults to \code{2}).
#'
#' @param color Character string. The color for the error bars.
#' (defaults to \code{"black"}).
#'
#' @param width Numeric. The width of error bar ends (defaults to \code{0}).
#'
#' @param ... Currently ignored
#'
#' @seealso \code{\link{estimate}}
#'
#' @return A \code{ggplot} object.
#'
#' @examples
#' \donttest{
#' # data
#' Y <- ptsd[,1:5]
#'
#' fit <- estimate(Y, iter = 250,
#' progress = FALSE)
#'
#'
#' plot(summary(fit))
#'
#' }
#'
#' @export
plot.summary.estimate <- function(x,
color = "black",
size = 2,
width = 0, ...){
dat_temp <- x$dat_results[order(x$dat_results$Post.mean,
decreasing = F), ]
dat_temp$Relation <-
factor(dat_temp$Relation,
levels = dat_temp$Relation,
labels = dat_temp$Relation)
if(isFALSE(x$object$analytic)){
ggplot(dat_temp,
aes(x = Relation,
y = Post.mean)) +
geom_errorbar(aes(ymax = dat_temp[, 4],
ymin = dat_temp[, 5]),
width = width,
color = color) +
geom_point(size = size) +
xlab("Index") +
theme(axis.text.x = element_text(
angle = 90,
vjust = 0.5,
hjust = 1
))
} else {
ggplot(dat_temp,
aes(x = Relation,
y = Post.mean)) +
geom_point(size = size) +
xlab("Index") +
theme(axis.text.x = element_text(
angle = 90,
vjust = 0.5,
hjust = 1
))
}
}
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