#' Test for Practical Equivalence
#'
#' Perform a **Test for Practical Equivalence** for Bayesian and frequentist models.
#'
#' Documentation is accessible for:
#'
#' - [Bayesian models](https://easystats.github.io/bayestestR/reference/equivalence_test.html)
#' - [Frequentist models](https://easystats.github.io/parameters/reference/equivalence_test.lm.html)
#'
#' For Bayesian models, the **Test for Practical Equivalence** is based on the
#' *"HDI+ROPE decision rule"* (\cite{Kruschke, 2014, 2018}) to check whether
#' parameter values should be accepted or rejected against an explicitly
#' formulated "null hypothesis" (i.e., a ROPE). In other words, it checks the
#' percentage of the `89%` [HDI][hdi] that is the null region (the ROPE). If
#' this percentage is sufficiently low, the null hypothesis is rejected. If this
#' percentage is sufficiently high, the null hypothesis is accepted.
#'
#' @inheritParams rope
#'
#' @details Using the [ROPE][rope] and the [HDI][hdi], \cite{Kruschke (2018)}
#' suggests using the percentage of the `95%` (or `89%`, considered more stable)
#' HDI that falls within the ROPE as a decision rule. If the HDI
#' is completely outside the ROPE, the "null hypothesis" for this parameter is
#' "rejected". If the ROPE completely covers the HDI, i.e., all most credible
#' values of a parameter are inside the region of practical equivalence, the
#' null hypothesis is accepted. Else, it’s undecided whether to accept or
#' reject the null hypothesis. If the full ROPE is used (i.e., `100%` of the
#' HDI), then the null hypothesis is rejected or accepted if the percentage
#' of the posterior within the ROPE is smaller than to `2.5%` or greater than
#' `97.5%`. Desirable results are low proportions inside the ROPE (the closer
#' to zero the better).
#' \cr \cr
#' Some attention is required for finding suitable values for the ROPE limits
#' (argument `range`). See 'Details' in [`rope_range()`][rope_range]
#' for further information.
#' \cr \cr
#' **Multicollinearity: Non-independent covariates**
#' \cr \cr
#' When parameters show strong correlations, i.e. when covariates are not
#' independent, the joint parameter distributions may shift towards or
#' away from the ROPE. In such cases, the test for practical equivalence may
#' have inappropriate results. Collinearity invalidates ROPE and hypothesis
#' testing based on univariate marginals, as the probabilities are conditional
#' on independence. Most problematic are the results of the "undecided"
#' parameters, which may either move further towards "rejection" or away
#' from it (\cite{Kruschke 2014, 340f}).
#' \cr \cr
#' `equivalence_test()` performs a simple check for pairwise correlations
#' between parameters, but as there can be collinearity between more than two variables,
#' a first step to check the assumptions of this hypothesis testing is to look
#' at different pair plots. An even more sophisticated check is the projection
#' predictive variable selection (\cite{Piironen and Vehtari 2017}).
#'
#'
#' @references
#' - Kruschke, J. K. (2018). Rejecting or accepting parameter values in Bayesian estimation. Advances in Methods and Practices in Psychological Science, 1(2), 270-280. \doi{10.1177/2515245918771304}
#' - Kruschke, J. K. (2014). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press
#' - Piironen, J., & Vehtari, A. (2017). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3), 711–735. \doi{10.1007/s11222-016-9649-y}
#'
#' @return A data frame with following columns:
#'
#' - `Parameter` The model parameter(s), if `x` is a model-object. If `x` is a vector, this column is missing.
#' - `CI` The probability of the HDI.
#' - `ROPE_low`, `ROPE_high` The limits of the ROPE. These values are identical for all parameters.
#' - `ROPE_Percentage` The proportion of the HDI that lies inside the ROPE.
#' - `ROPE_Equivalence` The "test result", as character. Either "rejected", "accepted" or "undecided".
#' - `HDI_low` , `HDI_high` The lower and upper HDI limits for the parameters.
#'
#' @note There is a `print()`-method with a `digits`-argument to control
#' the amount of digits in the output, and there is a
#' [`plot()`-method](https://easystats.github.io/see/articles/bayestestR.html)
#' to visualize the results from the equivalence-test (for models only).
#'
#' @examplesIf require("rstanarm") && require("brms") && require("emmeans") && require("BayesFactor")
#' library(bayestestR)
#'
#' equivalence_test(x = rnorm(1000, 0, 0.01), range = c(-0.1, 0.1))
#' equivalence_test(x = rnorm(1000, 0, 1), range = c(-0.1, 0.1))
#' equivalence_test(x = rnorm(1000, 1, 0.01), range = c(-0.1, 0.1))
#' equivalence_test(x = rnorm(1000, 1, 1), ci = c(.50, .99))
#'
#' # print more digits
#' test <- equivalence_test(x = rnorm(1000, 1, 1), ci = c(.50, .99))
#' print(test, digits = 4)
#' \donttest{
#' model <- rstanarm::stan_glm(mpg ~ wt + cyl, data = mtcars)
#' equivalence_test(model)
#' # multiple ROPE ranges - asymmetric, symmetric, default
#' equivalence_test(model, range = list(c(10, 40), c(-5, -4), "default"))
#' # named ROPE ranges
#' equivalence_test(model, range = list(wt = c(-5, -4), `(Intercept)` = c(10, 40)))
#'
#' # plot result
#' test <- equivalence_test(model)
#' plot(test)
#'
#' equivalence_test(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#'
#' model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#' equivalence_test(model)
#'
#' bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
#' # equivalence_test(bf)
#' }
#' @export
equivalence_test <- function(x, ...) {
UseMethod("equivalence_test")
}
#' @rdname equivalence_test
#' @export
equivalence_test.default <- function(x, ...) {
NULL
}
#' @export
equivalence_test.numeric <- function(x, range = "default", ci = 0.95, verbose = TRUE, ...) {
rope_data <- rope(x, range = range, ci = ci, verbose = verbose)
out <- as.data.frame(rope_data)
if (all(ci < 1)) {
out$ROPE_Equivalence <- datawizard::recode_into(
out$ROPE_Percentage == 0 ~ "Rejected",
out$ROPE_Percentage == 1 ~ "Accepted",
default = "Undecided"
)
} else {
# Related to guidelines for full rope (https://easystats.github.io/bayestestR/articles/4_Guidelines.html)
out$ROPE_Equivalence <- datawizard::recode_into(
out$ROPE_Percentage < 0.025 ~ "Rejected",
out$ROPE_Percentage > 0.975 ~ "Accepted",
default = "Undecided"
)
}
out$HDI_low <- attr(rope_data, "HDI_area", exact = TRUE)$CI_low
out$HDI_high <- attr(rope_data, "HDI_area", exact = TRUE)$CI_high
# remove attribute
attr(out, "HDI_area") <- NULL
attr(out, "data") <- x
class(out) <- unique(c("equivalence_test", "see_equivalence_test", class(out)))
out
}
#' @rdname equivalence_test
#' @inheritParams p_direction
#' @export
equivalence_test.data.frame <- function(x, range = "default", ci = 0.95, rvar_col = NULL, verbose = TRUE, ...) {
obj_name <- insight::safe_deparse_symbol(substitute(x))
x_rvar <- .possibly_extract_rvar_col(x, rvar_col)
if (length(x_rvar) > 0L) {
cl <- match.call()
cl[[1]] <- bayestestR::equivalence_test
cl$x <- x_rvar
cl$rvar_col <- NULL
out <- eval.parent(cl)
attr(out, "object_name") <- sprintf('%s[["%s"]]', obj_name, rvar_col)
return(.append_datagrid(out, x))
}
# multiple ranges for the parameters - iterate over parameters and range
if (is.list(range)) {
# check if list of values contains only valid values
range <- .check_list_range(range, x)
# apply thresholds to each column
l <- insight::compact_list(mapply(
function(p, r) {
equivalence_test(
p,
range = r,
ci = ci,
verbose = verbose
)
},
x,
range,
SIMPLIFY = FALSE
))
} else {
l <- insight::compact_list(lapply(
x,
equivalence_test,
range = range,
ci = ci,
verbose = verbose
))
}
dat <- do.call(rbind, l)
out <- data.frame(
Parameter = rep(names(l), each = nrow(dat) / length(l)),
dat,
stringsAsFactors = FALSE
)
row.names(out) <- NULL
attr(out, "object_name") <- obj_name
class(out) <- unique(c("equivalence_test", "see_equivalence_test_df", class(out)))
out
}
#' @export
equivalence_test.draws <- function(x, range = "default", ci = 0.95, verbose = TRUE, ...) {
equivalence_test(.posterior_draws_to_df(x), range = range, ci = ci, verbose = verbose, ...)
}
#' @export
equivalence_test.rvar <- equivalence_test.draws
#' @export
equivalence_test.emmGrid <- function(x, range = "default", ci = 0.95, verbose = TRUE, ...) {
xdf <- insight::get_parameters(x)
out <- equivalence_test(xdf, range = range, ci = ci, verbose = verbose, ...)
out <- .append_datagrid(out, x)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.emm_list <- equivalence_test.emmGrid
#' @export
equivalence_test.slopes <- function(x, range = "default", ci = 0.95, verbose = TRUE, ...) {
xrvar <- .get_marginaleffects_draws(x)
out <- equivalence_test(xrvar, range = range, ci = ci, verbose = verbose, ...)
out <- .append_datagrid(out, x)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.comparisons <- equivalence_test.slopes
#' @export
equivalence_test.predictions <- equivalence_test.slopes
#' @export
equivalence_test.BFBayesFactor <- function(x, range = "default", ci = 0.95, verbose = TRUE, ...) {
out <- equivalence_test(insight::get_parameters(x), range = range, ci = ci, verbose = verbose, ...)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @keywords internal
.equivalence_test_models <- function(x,
range = "default",
ci = 0.95,
effects = "fixed",
component = "conditional",
parameters = NULL,
verbose = TRUE) {
if (all(range == "default")) {
range <- rope_range(x, verbose = verbose)
} else if (!is.list(range) && (!all(is.numeric(range)) || length(range) != 2L)) {
insight::format_error("`range` should be 'default' or a vector of 2 numeric values (e.g., c(-0.1, 0.1)).")
}
if (verbose && !inherits(x, "blavaan")) .check_multicollinearity(x)
params <- insight::get_parameters(
x,
component = component,
effects = effects,
parameters = parameters,
verbose = verbose
)
equivalence_test(params, range = range, ci = ci, verbose = verbose)
}
#' @rdname equivalence_test
#' @export
equivalence_test.stanreg <- function(x,
range = "default",
ci = 0.95,
effects = c("fixed", "random", "all"),
component = c(
"location",
"all",
"conditional",
"smooth_terms",
"sigma",
"distributional",
"auxiliary"
),
parameters = NULL,
verbose = TRUE,
...) {
effects <- match.arg(effects)
component <- match.arg(component)
out <- .equivalence_test_models(x, range, ci, effects, component, parameters, verbose)
out <- .prepare_output(
out,
insight::clean_parameters(x),
inherits(x, "stanmvreg")
)
class(out) <- unique(c("equivalence_test", "see_equivalence_test", class(out)))
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.stanfit <- equivalence_test.stanreg
#' @export
equivalence_test.blavaan <- equivalence_test.stanreg
#' @rdname equivalence_test
#' @export
equivalence_test.brmsfit <- function(x,
range = "default",
ci = 0.95,
effects = c("fixed", "random", "all"),
component = c("conditional", "zi", "zero_inflated", "all"),
parameters = NULL,
verbose = TRUE,
...) {
effects <- match.arg(effects)
component <- match.arg(component)
out <- .equivalence_test_models(x, range, ci, effects, component, parameters, verbose)
out <- .prepare_output(
out,
insight::clean_parameters(x),
inherits(x, "stanmvreg")
)
class(out) <- unique(c("equivalence_test", "see_equivalence_test", class(out)))
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.sim.merMod <- function(x,
range = "default",
ci = 0.95,
parameters = NULL,
verbose = TRUE,
...) {
out <- .equivalence_test_models(
x,
range,
ci,
effects = "fixed",
component = "conditional",
parameters,
verbose = verbose
)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.sim <- equivalence_test.sim.merMod
#' @export
equivalence_test.mcmc <- function(x,
range = "default",
ci = 0.95,
parameters = NULL,
verbose = TRUE,
...) {
out <- .equivalence_test_models(
as.data.frame(x),
range,
ci,
effects = "fixed",
component = "conditional",
parameters,
verbose = verbose
)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.bcplm <- function(x,
range = "default",
ci = 0.95,
parameters = NULL,
verbose = TRUE,
...) {
out <- .equivalence_test_models(
insight::get_parameters(x),
range,
ci,
effects = "fixed",
component = "conditional",
parameters,
verbose = verbose
)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
#' @export
equivalence_test.blrm <- equivalence_test.bcplm
#' @export
equivalence_test.mcmc.list <- equivalence_test.bcplm
#' @export
equivalence_test.bayesQR <- equivalence_test.bcplm
#' @export
equivalence_test.bamlss <- function(x,
range = "default",
ci = 0.95,
component = c("all", "conditional", "location"),
parameters = NULL,
verbose = TRUE,
...) {
component <- match.arg(component)
out <- .equivalence_test_models(
insight::get_parameters(x, component = component),
range,
ci,
effects = "fixed",
component = "conditional",
parameters,
verbose = verbose
)
attr(out, "object_name") <- insight::safe_deparse_symbol(substitute(x))
out
}
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