#' Sequential Effect eXistence and sIgnificance Testing (SEXIT)
#'
#' The SEXIT is a new framework to describe Bayesian effects, guiding which
#' indices to use. Accordingly, the `sexit()` function returns the minimal (and
#' optimal) required information to describe models' parameters under a Bayesian
#' framework. It includes the following indices:
#' \itemize{
#' \item{Centrality: the median of the posterior distribution. In
#' probabilistic terms, there is `50%` of probability that the effect is higher
#' and lower. See [`point_estimate()`][point_estimate].}
#' \item{Uncertainty: the `95%` Highest Density Interval (HDI). In
#' probabilistic terms, there is `95%` of probability that the effect is
#' within this confidence interval. See [`ci()`][ci].}
#' \item{Existence: The probability of direction allows to quantify the
#' certainty by which an effect is positive or negative. It is a critical
#' index to show that an effect of some manipulation is not harmful (for
#' instance in clinical studies) or to assess the direction of a link. See
#' [`p_direction()`][p_direction].}
#' \item{Significance: Once existence is demonstrated with high certainty, we
#' can assess whether the effect is of sufficient size to be considered as
#' significant (i.e., not negligible). This is a useful index to determine
#' which effects are actually important and worthy of discussion in a given
#' process. See [`p_significance()`][p_significance].}
#' \item{Size: Finally, this index gives an idea about the strength of an
#' effect. However, beware, as studies have shown that a big effect size can
#' be also suggestive of low statistical power (see details section).}
#' }
#'
#' @inheritParams p_direction
#' @inheritParams hdi
#' @param significant,large The threshold values to use for significant and
#' large probabilities. If left to 'default', will be selected through
#' [`sexit_thresholds()`][sexit_thresholds]. See the details section below.
#'
#' @details
#'
#' \subsection{Rationale}{
#' The assessment of "significance" (in its broadest meaning) is a pervasive
#' issue in science, and its historical index, the p-value, has been strongly
#' criticized and deemed to have played an important role in the replicability
#' crisis. In reaction, more and more scientists have tuned to Bayesian methods,
#' offering an alternative set of tools to answer their questions. However, the
#' Bayesian framework offers a wide variety of possible indices related to
#' "significance", and the debate has been raging about which index is the best,
#' and which one to report.
#'
#' This situation can lead to the mindless reporting of all possible indices
#' (with the hopes that with that the reader will be satisfied), but often
#' without having the writer understanding and interpreting them. It is indeed
#' complicated to juggle between many indices with complicated definitions and
#' subtle differences.
#'
#' SEXIT aims at offering a practical framework for Bayesian effects reporting,
#' in which the focus is put on intuitiveness, explicitness and usefulness of
#' the indices' interpretation. To that end, we suggest a system of description
#' of parameters that would be intuitive, easy to learn and apply,
#' mathematically accurate and useful for taking decision.
#'
#' Once the thresholds for significance (i.e., the ROPE) and the one for a
#' "large" effect are explicitly defined, the SEXIT framework does not make any
#' interpretation, i.e., it does not label the effects, but just sequentially
#' gives 3 probabilities (of direction, of significance and of being large,
#' respectively) as-is on top of the characteristics of the posterior (using the
#' median and HDI for centrality and uncertainty description). Thus, it provides
#' a lot of information about the posterior distribution (through the mass of
#' different 'sections' of the posterior) in a clear and meaningful way.
#' }
#'
#' \subsection{Threshold selection}{
#' One of the most important thing about the SEXIT framework is that it relies
#' on two "arbitrary" thresholds (i.e., that have no absolute meaning). They
#' are the ones related to effect size (an inherently subjective notion),
#' namely the thresholds for significant and large effects. They are set, by
#' default, to `0.05` and `0.3` of the standard deviation of the outcome
#' variable (tiny and large effect sizes for correlations according to Funder
#' \& Ozer, 2019). However, these defaults were chosen by lack of a better
#' option, and might not be adapted to your case. Thus, they are to be handled
#' with care, and the chosen thresholds should always be explicitly reported
#' and justified.
#' \itemize{
#' \item For **linear models (lm)**, this can be generalised to \ifelse{html}{\out{0.05 * SD_{y}}}{\eqn{[0.05*SD_{y}]}} and \ifelse{html}{\out{0.3 * SD_{y}}}{\eqn{[0.3*SD_{y}]}} for significant and large effects, respectively.
#' \item For **logistic models**, the parameters expressed in log odds ratio can be converted to standardized difference through the formula \ifelse{html}{\out{π/√(3)}}{\eqn{\pi/\sqrt{3}}}, resulting a threshold of `0.09` and `0.54`.
#' \item For other models with **binary outcome**, it is strongly recommended to manually specify the rope argument. Currently, the same default is applied that for logistic models.
#' \item For models from **count data**, the residual variance is used. This is a rather experimental threshold and is probably often similar to `0.05` and `0.3`, but should be used with care!
#' \item For **t-tests**, the standard deviation of the response is used, similarly to linear models (see above).
#' \item For **correlations**,`0.05` and `0.3` are used.
#' \item For all other models, `0.05` and `0.3` are used, but it is strongly advised to specify it manually.
#' }
#' }
#' \subsection{Examples}{
#' The three values for existence, significance and size provide a useful description of the posterior distribution of the effects. Some possible scenarios include:
#' \itemize{
#' \item{The probability of existence is low, but the probability of being large is high: it suggests that the posterior is very wide (covering large territories on both side of 0). The statistical power might be too low, which should warrant any confident conclusion.}
#' \item{The probability of existence and significance is high, but the probability of being large is very small: it suggests that the effect is, with high confidence, not large (the posterior is mostly contained between the significance and the large thresholds).}
#' \item{The 3 indices are very low: this suggests that the effect is null with high confidence (the posterior is closely centred around 0).}}}
#'
#' @return A dataframe and text as attribute.
#'
#' @references \itemize{
#' \item{Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. Journal of Open Source Software, 4(40), 1541. \doi{10.21105/joss.01541}}
#' \item{Makowski D, Ben-Shachar MS, Chen SHA, Lüdecke D (2019) Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology 2019;10:2767. \doi{10.3389/fpsyg.2019.02767}}
#' }
#'
#' @examples
#' \dontrun{
#' library(bayestestR)
#'
#' s <- sexit(rnorm(1000, -1, 1))
#' s
#' print(s, summary = TRUE)
#'
#' s <- sexit(iris)
#' s
#' print(s, summary = TRUE)
#'
#' if (require("rstanarm")) {
#' model <- rstanarm::stan_glm(mpg ~ wt * cyl,
#' data = mtcars,
#' iter = 400, refresh = 0
#' )
#' s <- sexit(model)
#' s
#' print(s, summary = TRUE)
#' }
#' }
#' @export
sexit <- function(x, significant = "default", large = "default", ci = 0.95, ...) {
thresholds <- .sexit_preprocess(x, significant, large, ...)
significant <- thresholds$significant
large <- thresholds$large
thresholds_text <- thresholds$text
# Description
centrality <- point_estimate(x, "median")
centrality$Effects <- centrality$Component <- NULL
centrality_text <- paste0("Median = ", insight::format_value(centrality$Median))
direction <- ifelse(centrality$Median < 0, "negative", "positive")
uncertainty <- ci(x, ci = ci, method = "HDI", ...)[c("CI", "CI_low", "CI_high")]
uncertainty_text <- insight::format_ci(uncertainty$CI_low, uncertainty$CI_high, uncertainty$CI)
# Indices
existence_rez <- as.numeric(p_direction(x, ...))
existence_value <- insight::format_value(existence_rez, as_percent = TRUE)
existence_threshold <- ifelse(direction == "negative", "< 0", "> 0")
sig_rez <- as.numeric(p_significance(x, threshold = significant, ...))
sig_value <- insight::format_value(sig_rez, as_percent = TRUE)
sig_threshold <- ifelse(direction == "negative", -1 * significant, significant)
sig_threshold <- paste0(ifelse(direction == "negative", "< ", "> "), insight::format_value(sig_threshold))
large_rez <- as.numeric(p_significance(x, threshold = large, ...))
large_value <- insight::format_value(large_rez, as_percent = TRUE)
large_threshold <- ifelse(direction == "negative", -1 * large, large)
large_threshold <- paste0(ifelse(direction == "negative", "< ", "> "), insight::format_value(large_threshold))
if ("Parameter" %in% names(centrality)) {
parameters <- centrality$Parameter
} else {
parameters <- "The effect"
}
text_full <- paste0(
parameters,
" (",
centrality_text,
", ",
uncertainty_text,
") has a ",
existence_value,
" probability of being ",
direction,
" (",
existence_threshold,
"), ",
sig_value,
" of being significant (",
sig_threshold,
"), and ",
large_value,
" of being large (",
large_threshold,
")"
)
text_short <- paste0(
parameters,
" (",
centrality_text,
", ",
uncertainty_text,
") has ",
existence_value,
", ",
sig_value,
" and ",
large_value,
" probability of being ",
direction,
" (",
existence_threshold,
"), significant (",
sig_threshold,
") and large (",
large_threshold,
")"
)
out <- cbind(
centrality,
as.data.frame(uncertainty),
data.frame(Direction = existence_rez),
data.frame(Significance = sig_rez),
data.frame(Large = large_rez)
)
# Prepare output
attr(out, "sexit_info") <- "Following the Sequential Effect eXistence and sIgnificance Testing (SEXIT) framework, we report the median of the posterior distribution and its 95% CI (Highest Density Interval), along the probability of direction (pd), the probability of significance and the probability of being large."
attr(out, "sexit_ci_method") <- "HDI"
attr(out, "sexit_significance") <- significant
attr(out, "sexit_large") <- large
attr(out, "sexit_textlong") <- text_full
attr(out, "sexit_textshort") <- text_short
attr(out, "sexit_thresholds") <- thresholds_text
pretty_cols <- c(
"Median",
paste0(insight::format_value(ci * 100, protect_integers = TRUE), "% CI"),
"Direction",
paste0("Significance (> |", insight::format_value(significant), "|)"),
paste0("Large (> |", insight::format_value(large), "|)")
)
if ("Parameter" %in% names(out)) pretty_cols <- c("Parameter", pretty_cols)
attr(out, "pretty_cols") <- pretty_cols
attr(out, "data") <- x
class(out) <- unique(c("sexit", "see_sexit", class(out)))
out
}
#' @keywords internal
.sexit_preprocess <- function(x, significant = "default", large = "default", ...) {
thresholds <- sexit_thresholds(x)
if (significant == "default") significant <- thresholds[1]
if (large == "default") large <- thresholds[2]
suppressWarnings(resp <- tryCatch(insight::get_response(x), error = function(e) NULL))
suppressWarnings(info <- tryCatch(insight::model_info(x), error = function(e) NULL))
if (!is.null(resp) && !is.null(info) && info$is_linear) {
sd1 <- significant / stats::sd(resp, na.rm = TRUE)
sd2 <- large / stats::sd(resp, na.rm = TRUE)
text_sd <- paste0(
" (corresponding respectively to ",
insight::format_value(sd1),
" and ",
insight::format_value(sd2),
" of the outcome's SD)"
)
} else {
text_sd <- ""
}
thresholds <- paste0(
"The thresholds beyond which the effect is considered ",
"as significant (i.e., non-negligible) and large are |",
insight::format_value(significant),
"| and |",
insight::format_value(large),
"|",
text_sd,
"."
)
list(significant = significant, large = large, text = thresholds)
}
#' @export
print.sexit <- function(x, summary = FALSE, digits = 2, ...) {
orig_x <- x
# Long
if (isFALSE(summary)) {
insight::print_color(paste0("# ", attributes(x)$sexit_info, " ", attributes(x)$sexit_thresholds, "\n\n"), "blue")
text <- attributes(x)$sexit_textlong
if (length(text) > 1) text <- paste0(paste0("- ", text), collapse = "\n")
insight::print_color(text, "yellow")
cat("\n\n")
df <- data.frame(Median = x$Median, CI = insight::format_ci(x$CI_low, x$CI_high, NULL))
if ("Parameter" %in% names(x)) {
df <- cbind(data.frame(Parameter = x$Parameter), df, x[c("Direction", "Significance", "Large")])
} else {
df <- cbind(df, x[c("Direction", "Significance", "Large")])
}
names(df) <- attributes(x)$pretty_cols
print_data_frame(df, digits = digits, ...)
# Short
} else {
insight::print_color(paste0("# ", attributes(x)$sexit_thresholds, "\n\n"), "blue")
text <- attributes(x)$sexit_textshort
if (length(text) > 1) text <- paste0(paste0("- ", text), collapse = "\n")
cat(text)
}
invisible(orig_x)
}