bayesfactor_savagedickey.R
#' Savage-Dickey density ratio Bayes Factor (BF)
#'
#' This method computes the ratio between the density of a single value (typically the null)
#' of two distributions. When the compared distributions are the posterior and the prior distributions,
#' this results is an approximation of a Bayes factor comparing the model against a model in which
#' the parameter of choice is restricted to the point null.
#' \cr \cr
#'
#' @param posterior A numerical vector, \code{stanreg} / \code{brmsfit} object, \code{emmGrid} or a data frame - representing a posterior distribution(s) from (see Details).
#' @param prior An object representing a prior distribution (see Details).
#' @param direction Test type (see details). One of \code{0}, \code{"two-sided"} (default, two tailed),
#' \code{-1}, \code{"left"} (left tailed) or \code{1}, \code{"right"} (right tailed).
#' @param hypothesis Value to be tested against (usually \code{0} in the context of null hypothesis testing).
#' @inheritParams hdi
#'
#' @return A data frame containing the Bayes factor representing evidence \emph{against} the (point) null effect model.
#'
#' @details This method is used to compute Bayes factors based on prior and posterior distributions.
#' When \code{posterior} is a model (\code{stanreg}, \code{brmsfit}), posterior and prior samples are
#' extracted for each parameter, and Savage-Dickey Bayes factors are computed for each parameter.
#'
#' \strong{NOTE:} For \code{brmsfit} models, the model must have been fitted with \emph{custom (non-default)} priors. See example below.
#'
#' \subsection{Setting the correct \code{prior}}{
#' It is important to provide the correct \code{prior} for meaningful results.
#' \itemize{
#'   \item When \code{posterior} is a numerical vector, \code{prior} should also be a numerical vector.
#'   \item When \code{posterior} is an \code{emmGrid} object based on a \code{stanreg} or \code{brmsfit} model, \code{prior} should be \emph{that model object} (see example).
#'   \item When \code{posterior} is a \code{stanreg} or \code{brmsfit} model, there is no need to specify \code{prior}, as prior samples are drawn internally.
#'   \item When \code{posterior} is a \code{data.frame}, \code{prior} should also be a \code{data.frame}, with matching column order.
#' }}
#' \subsection{One-sided Tests (setting an order restriction)}{
#' One sided tests (controlled by \code{direction}) are conducted by setting an order restriction on
#' the prior and posterior distributions (\cite{Morey & Wagenmakers, 2013}).
#' }
#' \subsection{Interpreting Bayes Factors}{
#' A Bayes factor greater than 1 can be interpereted as evidence against the null,
#' at which one convention is that a Bayes factor greater than 3 can be considered
#' as "substantial" evidence against the null (and vice versa, a Bayes factor
#' smaller than 1/3 indicates substantial evidence in favor of the null-hypothesis)
#' (\cite{Wetzels et al. 2011}).
#' }
#'
#' @examples
#' library(bayestestR)
#'
#' prior <- distribution_normal(1000, mean = 0, sd = 1)
#' posterior <- distribution_normal(1000, mean = .5, sd = .3)
#'
#' bayesfactor_savagedickey(posterior, prior)
#' \dontrun{
#' # rstanarm models
#' # ---------------
#' library(rstanarm)
#' stan_model <- stan_lmer(extra ~ group + (1 | ID), data = sleep)
#' bayesfactor_savagedickey(stan_model)
#'
#' # emmGrid objects
#' # ---------------
#' library(emmeans)
#' group_diff <- pairs(emmeans(stan_model, ~group))
#' bayesfactor_savagedickey(group_diff, prior = stan_model)
#'
#' # brms models
#' # -----------
#' library(brms)
#' my_custom_priors <-
#'   set_prior("student_t(3, 0, 1)", class = "b") +
#'   set_prior("student_t(3, 0, 1)", class = "sd", group = "ID")
#'
#' brms_model <- brm(extra ~ group + (1 | ID),
#'   data = sleep,
#'   prior = my_custom_priors
#' )
#' bayesfactor_savagedickey(brms_model)
#' }
#'
#' @references
#' \itemize{
#' \item Wagenmakers, E. J., Lodewyckx, T., Kuriyal, H., and Grasman, R. (2010). Bayesian hypothesis testing for psychologists: A tutorial on the Savage-Dickey method. Cognitive psychology, 60(3), 158-189.
#' \item Wetzels, R., Matzke, D., Lee, M. D., Rouder, J. N., Iverson, G. J., and Wagenmakers, E.-J. (2011). Statistical Evidence in Experimental Psychology: An Empirical Comparison Using 855 t Tests. Perspectives on Psychological Science, 6(3), 291–298. \doi{10.1177/1745691611406923}
#' \item Heck, D. W. (2019). A caveat on the Savage–Dickey density ratio: The case of computing Bayes factors for regression parameters. British Journal of Mathematical and Statistical Psychology, 72(2), 316-333.
#' \item Morey, R. D., & Wagenmakers, E. J. (2014). Simple relation between Bayesian order-restricted and point-null hypothesis tests. Statistics & Probability Letters, 92, 121-124.
#' }
#'
#' @author Mattan S. Ben-Shachar
#'
#' @export
bayesfactor_savagedickey <- function(posterior, prior = NULL, direction = "two-sided", hypothesis = 0, verbose = TRUE, ...) {
UseMethod("bayesfactor_savagedickey")
}

#' @rdname bayesfactor_savagedickey
#' @export
bayesfactor_savagedickey.numeric <- function(posterior, prior = NULL, direction = "two-sided", hypothesis = 0, verbose = TRUE, ...) {
# nm <- .safe_deparse(substitute(posterior))

# find direction
direction <- .get_direction(direction)

if (is.null(prior)) {
prior <- posterior
if (verbose) {
warning(
"Prior not specified! ",
"Please specify a prior (in the form 'prior = distribution_normal(1000, 0, 1)')",
" to get meaningful results."
)
}
}
prior <- data.frame(X = prior)
posterior <- data.frame(X = posterior)
colnames(posterior) <- colnames(prior) <- "X" # nm

# Get savage-dickey BFs
sdbf <- bayesfactor_savagedickey.data.frame(
posterior = posterior, prior = prior,
direction = direction, hypothesis = hypothesis
)
sdbf$Parameter <- NULL sdbf } #' @importFrom insight get_parameters #' @rdname bayesfactor_savagedickey #' @export bayesfactor_savagedickey.stanreg <- function(posterior, prior = NULL, direction = "two-sided", hypothesis = 0, verbose = TRUE, effects = c("fixed", "random", "all"), ...) { effects <- match.arg(effects) # Get Priors if (is.null(prior)) { prior <- .update_to_priors(posterior, verbose = verbose) } prior <- insight::get_parameters(prior, effects = effects) posterior <- insight::get_parameters(posterior, effects = effects) # Get savage-dickey BFs bayesfactor_savagedickey.data.frame( posterior = posterior, prior = prior, direction = direction, hypothesis = hypothesis ) } #' @rdname bayesfactor_savagedickey #' @export bayesfactor_savagedickey.brmsfit <- bayesfactor_savagedickey.stanreg #' @importFrom stats update #' @importFrom insight get_parameters #' @rdname bayesfactor_savagedickey #' @export bayesfactor_savagedickey.emmGrid <- function(posterior, prior = NULL, direction = "two-sided", hypothesis = 0, verbose = TRUE, ...) { if (!requireNamespace("emmeans")) { stop("Package \"emmeans\" needed for this function to work. Please install it.") } if (is.null(prior)) { prior <- posterior warning( "Prior not specified! ", "Please provide the original model to get meaningful results." ) } else { prior <- .update_to_priors(prior, verbose = verbose) prior <- insight::get_parameters(prior, effects = "fixed") prior <- stats::update(posterior, post.beta = as.matrix(prior)) } prior <- as.data.frame(as.matrix(emmeans::as.mcmc.emmGrid(prior, names = FALSE))) posterior <- as.data.frame(as.matrix(emmeans::as.mcmc.emmGrid(posterior, names = FALSE))) bayesfactor_savagedickey.data.frame( posterior = posterior, prior = prior, direction = direction, hypothesis = hypothesis ) } #' @rdname bayesfactor_savagedickey #' @export bayesfactor_savagedickey.data.frame <- function(posterior, prior = NULL, direction = "two-sided", hypothesis = 0, verbose = TRUE, ...) { # find direction direction <- .get_direction(direction) if (is.null(prior)) { prior <- posterior warning( "Prior not specified! ", "Please specify priors (with column order matching 'posterior')", " to get meaningful results." ) } sdbf <- numeric(ncol(posterior)) for (par in seq_along(posterior)) { sdbf[par] <- .bayesfactor_savagedickey( posterior[[par]], prior[[par]], direction = direction, hypothesis = hypothesis ) } bf_val <- data.frame( Parameter = colnames(posterior), BF = sdbf, stringsAsFactors = FALSE ) class(bf_val) <- unique(c( "bayesfactor_savagedickey", "see_bayesfactor_savagedickey", class(bf_val) )) attr(bf_val, "hypothesis") <- hypothesis attr(bf_val, "direction") <- direction attr(bf_val, "plot_data") <- .make_sdBF_plot_data(posterior, prior, direction, hypothesis) bf_val } #' @keywords internal #' @importFrom insight print_color .bayesfactor_savagedickey <- function(posterior, prior, direction = 0, hypothesis = 0) { if (isTRUE(all.equal(posterior, prior))) { return(1) } if (requireNamespace("logspline", quietly = TRUE)) { relative_density <- function(samples) { f_samples <- suppressWarnings(logspline::logspline(samples)) d_samples <- logspline::dlogspline(hypothesis, f_samples) if (direction < 0) { norm_samples <- logspline::plogspline(hypothesis, f_samples) } else if (direction > 0) { norm_samples <- 1 - logspline::plogspline(hypothesis, f_samples) } else { norm_samples <- 1 } d_samples / norm_samples } } else { insight::print_color("Consider installing the \"logspline\" package for a more robust estimate.\n", "red") relative_density <- function(samples) { d_samples <- density_at(samples, hypothesis) if (direction < 0) { norm_samples <- mean(samples < hypothesis) } else if (direction > 0) { norm_samples <- 1 - mean(samples < 0) } else { norm_samples <- 1 } d_samples / norm_samples } } relative_density(prior) / relative_density(posterior) } # UTILS ------------------------------------------------------------------- #' @keywords internal .get_direction <- function(direction) { if (length(direction) > 1) { warning("Using first 'direction' value.") direction <- direction[1] } String <- c("left", "right", "one-sided", "onesided", "two-sided", "twosided", "<", ">", "=", "-1", "0", "1", "+1") Value <- c(-1, 1, 1, 1, 0, 0, -1, 1, 0, -1, 0, 1, 1) ind <- String == direction if (length(ind) == 0) { stop("Unrecognized 'direction' argument.") } Value[ind] } #' @importFrom stats median mad approx #' @importFrom utils stack #' @keywords internal .make_sdBF_plot_data <- function(posterior, prior, direction, hypothesis) { if (requireNamespace("logspline", quietly = TRUE)) { density_method <- "logspline" } else { density_method <- "kernel" } estimate_samples_density <- function(samples) { nm <- .safe_deparse(substitute(samples)) samples <- utils::stack(samples) samples <- split(samples, samples$ind)

samples <- lapply(samples, function(data) {
# 1. estimate density
x <- data$values extend_scale <- 0.05 precision <- 2^8 x_range <- range(x) x_rangex <- stats::median(x) + 7 * stats::mad(x) * c(-1, 1) x_range <- c( max(c(x_range[1], x_rangex[1])), min(c(x_range[2], x_rangex[2])) ) extension_scale <- diff(x_range) * extend_scale x_range[1] <- x_range[1] - extension_scale x_range[2] <- x_range[2] + extension_scale if (requireNamespace("logspline", quietly = TRUE)) { x_axis <- seq(x_range[1], x_range[2], length.out = precision) y <- logspline::dlogspline(x_axis, logspline::logspline(x)) d_points <- data.frame(x = x_axis, y = y) } else { d_points <- as.data.frame(density( x, n = precision, bw = "SJ", from = x_range[1], to = x_range[2] )) } # 2. estimate points d_null <- stats::approx(d_points$x, d_points$y, xout = hypothesis) d_null$y[is.na(d_null$y)] <- 0 # 3. direction? if (direction > 0) { d_points <- d_points[d_points$x > hypothesis, , drop = FALSE]
d_points$y <- d_points$y / mean(data$values > hypothesis) d_null$y <- d_null$y / mean(data$values > hypothesis)
} else if (direction < 0) {
d_points <- d_points[d_points$x < hypothesis, , drop = FALSE] d_points$y <- d_points$y / mean(data$values < hypothesis)
d_null$y <- d_null$y / mean(data$values < hypothesis) } d_points$ind <- d_null$ind <- data$ind[1]
list(d_points, d_null)
})

# 4a. orgenize
point0 <- lapply(samples, function(.) as.data.frame(.[[2]]))
point0 <- do.call("rbind", point0)

samplesX <- lapply(samples, function(.) .[[1]])
samplesX <- do.call("rbind", samplesX)

samplesX$Distribution <- point0$Distribution <- nm
rownames(samplesX) <- rownames(point0) <- c()

list(samplesX, point0)
}

# 4b. orgenize
posterior <- estimate_samples_density(posterior)
prior <- estimate_samples_density(prior)

list(
plot_data = rbind(posterior[[1]], prior[[1]]),
d_points = rbind(posterior[[2]], prior[[2]])
)
}

#' @keywords internal
.update_to_priors <- function(model, verbose = TRUE) {
UseMethod(".update_to_priors")
}

#' @keywords internal
#' @importFrom stats update
#' @importFrom utils capture.output
.update_to_priors.stanreg <- function(model, verbose = TRUE) {
if (!requireNamespace("rstanarm")) {
stop("Package \"rstanarm\" needed for this function to work. Please install it.")
}

if (verbose) {
message("Computation of Bayes factors: sampling priors, please wait...")
}

utils::capture.output(
model_prior <- suppressWarnings(
stats::update(model, prior_PD = TRUE)
)
)

model_prior
}

#' @keywords internal
#' @importFrom stats update
#' @importFrom utils capture.output
#' @importFrom methods is
.update_to_priors.brmsfit <- function(model, verbose = TRUE) {
if (!requireNamespace("brms")) {
stop("Package \"brms\" needed for this function to work. Please install it.")
}

if (verbose) {
message("Computation of Bayes factors: sampling priors, please wait...")
}

utils::capture.output(
model_prior <- try(suppressMessages(suppressWarnings(
stats::update(model, sample_prior = "only")
)), silent = TRUE)
)

if (is(model_prior, "try-error")) {
if (grepl("proper priors", model_prior)) {
stop("Cannot compute BF for 'brmsfit' models fit with default priors.\n",
"See '?bayesfactor_savagedickey'",
call. = FALSE
)
} else {
stop(model_prior)
}
}

model_prior
}