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#' @title p-value or consonance function
#' @name p_function
#'
#' @description Compute p-values and compatibility (confidence) intervals for
#' statistical models, at different levels. This function is also called
#' consonance function. It allows to see which estimates are compatible with
#' the model at various compatibility levels. Use `plot()` to generate plots
#' of the _p_ resp. _consonance_ function and compatibility intervals at
#' different levels.
#'
#' @param ci_levels Vector of scalars, indicating the different levels at which
#' compatibility intervals should be printed or plotted. In plots, these levels
#' are highlighted by vertical lines. It is possible to increase thickness for
#' one or more of these lines by providing a names vector, where the to be
#' highlighted values should be named `"emph"`, e.g
#' `ci_levels = c(0.25, 0.5, emph = 0.95)`.
#'
#' @inheritParams model_parameters
#' @inheritParams model_parameters.default
#' @inheritParams model_parameters.glmmTMB
#'
#' @note
#' Curently, `p_function()` computes intervals based on Wald t- or z-statistic.
#' For certain models (like mixed models), profiled intervals may be more
#' accurate, however, this is currently not supported.
#'
#' @details
#' ## Compatibility intervals and continuous _p_-values for different estimate values
#'
#' `p_function()` only returns the compatibility interval estimates, not the
#' related _p_-values. The reason for this is because the _p_-value for a
#' given estimate value is just `1 - CI_level`. The values indicating the lower
#' and upper limits of the intervals are the related estimates associated with
#' the _p_-value. E.g., if a parameter `x` has a 75% compatibility interval
#' of `(0.81, 1.05)`, then the _p_-value for the estimate value of `0.81`
#' would be `1 - 0.75`, which is `0.25`. This relationship is more intuitive and
#' better to understand when looking at the plots (using `plot()`).
#'
#' ## Conditional versus unconditional interpretation of _p_-values and intervals
#'
#' `p_function()`, and in particular its `plot()` method, aims at re-interpreting
#' _p_-values and confidence intervals (better named: _compatibility_ intervals)
#' in _unconditional_ terms. Instead of referring to the long-term property and
#' repeated trials when interpreting interval estimates (so-called "aleatory
#' probability", _Schweder 2018_), and assuming that all underlying assumptions
#' are correct and met, `p_function()` interprets _p_-values in a Fisherian way
#' as "_continuous_ measure of evidence against the very test hypothesis _and_
#' entire model (all assumptions) used to compute it"
#' (*P-Values Are Tough and S-Values Can Help*, lesslikely.com/statistics/s-values;
#' see also _Amrhein and Greenland 2022_).
#'
#' This interpretation as a continuous measure of evidence against the test
#' hypothesis and the entire model used to compute it can be seen in the
#' figure below (taken from *P-Values Are Tough and S-Values Can Help*,
#' lesslikely.com/statistics/s-values). The "conditional" interpretation of
#' _p_-values and interval estimates (A) implicitly assumes certain assumptions
#' to be true, thus the interpretation is "conditioned" on these assumptions
#' (i.e. assumptions are taken as given). The unconditional interpretation (B),
#' however, questions all these assumptions.
#'
#' \if{html}{\cr \figure{unconditional_interpretation.png}{options: alt="Conditional versus unconditional interpretations of P-values"} \cr}
#'
#' "Emphasizing unconditional interpretations helps avoid overconfident and
#' misleading inferences in light of uncertainties about the assumptions used
#' to arrive at the statistical results." (_Greenland et al. 2022_).
#'
#' **Note:** The term "conditional" as used by Rafi and Greenland probably has
#' a slightly different meaning than normally. "Conditional" in this notion
#' means that all model assumptions are taken as given - it should not be
#' confused with terms like "conditional probability". See also _Greenland et al. 2022_
#' for a detailed elaboration on this issue.
#'
#' In other words, the term compatibility interval emphasizes "the dependence
#' of the _p_-value on the assumptions as well as on the data, recognizing that
#' _p_<0.05 can arise from assumption violations even if the effect under
#' study is null" (_Gelman/Greenland 2019_).
#'
#' ## Probabilistic interpretation of compatibility intervals
#'
#' Schweder (2018) resp. Schweder and Hjort (2016) (and others) argue that
#' confidence curves (as produced by `p_function()`) have a valid probabilistic
#' interpretation. They distinguish between _aleatory probability_, which
#' describes the aleatory stochastic element of a distribution _ex ante_, i.e.
#' before the data are obtained. This is the classical interpretation of
#' confidence intervals following the Neyman-Pearson school of statistics.
#' However, there is also an _ex post_ probability, called _epistemic_ probability,
#' for confidence curves. The shift in terminology from _confidence_ intervals
#' to _compatibility_ intervals may help emphasizing this interpretation.
#'
#' In this sense, the probabilistic interpretation of _p_-values and
#' compatibility intervals is "conditional" - on the data _and_ model assumptions
#' (which is in line with the "unconditional" interpretation in the sense of
#' Rafi and Greenland).
#'
#' Ascribing a probabilistic interpretation to one realized confidence interval
#' is possible without repeated sampling of the specific experiment. Important
#' is the assumption that a _sampling distribution_ is a good description of the
#' variability of the parameter (_Vos and Holbert 2022_). At the core, the
#' interpretation of a confidence interval is "I assume that this sampling
#' distribution is a good description of the uncertainty of the parameter. If
#' that's a good assumption, then the values in this interval are the most
#' plausible or compatible with the data". The source of confidence in
#' probability statements is the assumption that the selected sampling
#' distribution is appropriate.
#'
#' "The realized confidence distribution is clearly an epistemic probability
#' distribution" (_Schweder 2018_). In Bayesian words, compatibility intervals
#' (or confidence distributons, or consonance curves) are "posteriors without
#' priors" (_Schweder, Hjort, 2003_). In this regard, interpretation of _p_-values
#' might be guided using [`bayestestR::p_to_pd()`].
#'
#' ## Compatibility intervals - is their interpretation conditional or not?
#'
#' The fact that the term "conditional" is used in different meanings, is
#' confusing and unfortunate. Thus, we would summarize the probabilistic
#' interpretation of compatibility intervals as follows: The intervals are built
#' from the data _and_ our modeling assumptions. The accuracy of the intervals
#' depends on our model assumptions. If a value is outside the interval, that
#' might be because (1) that parameter value isn't supported by the data, or
#' (2) the modeling assumptions are a poor fit for the situation. When we make
#' bad assumptions, the compatibility interval might be too wide or (more
#' commonly and seriously) too narrow, making us think we know more about the
#' parameter than is warranted.
#'
#' When we say "there is a 95% chance the true value is in the interval", that is
#' a statement of _epistemic probability_ (i.e. description of uncertainty related
#' to our knowledge or belief). When we talk about repeated samples or sampling
#' distributions, that is referring to _aleatoric_ (physical properties) probability.
#' Frequentist inference is built on defining estimators with known _aleatoric_
#' probability properties, from which we can draw _epistemic_ probabilistic
#' statements of uncertainty (_Schweder and Hjort 2016_).
#'
#' @return A data frame with p-values and compatibility intervals.
#'
#' @references
#' - Amrhein V, Greenland S. Discuss practical importance of results based on
#' interval estimates and p-value functions, not only on point estimates and
#' null p-values. Journal of Information Technology 2022;37:316–20.
#' \doi{10.1177/02683962221105904}
#'
#' - Fraser DAS. The P-value function and statistical inference. The American
#' Statistician. 2019;73(sup1):135-147. \doi{10.1080/00031305.2018.1556735}
#'
#' - Gelman A, Greenland S. Are confidence intervals better termed "uncertainty
#' intervals"? BMJ (2019)l5381. \doi{10.1136/bmj.l5381}
#'
#' - Greenland S, Rafi Z, Matthews R, Higgs M. To Aid Scientific Inference,
#' Emphasize Unconditional Compatibility Descriptions of Statistics. (2022)
#' https://arxiv.org/abs/1909.08583v7 (Accessed November 10, 2022)
#'
#' - Rafi Z, Greenland S. Semantic and cognitive tools to aid statistical
#' science: Replace confidence and significance by compatibility and surprise.
#' BMC Medical Research Methodology. 2020;20(1):244. \doi{10.1186/s12874-020-01105-9}
#'
#' - Schweder T. Confidence is epistemic probability for empirical science.
#' Journal of Statistical Planning and Inference (2018) 195:116–125.
#' \doi{10.1016/j.jspi.2017.09.016}
#'
#' - Schweder T, Hjort NL. Confidence and Likelihood. Scandinavian Journal of
#' Statistics. 2002;29(2):309-332. \doi{10.1111/1467-9469.00285}
#'
#' - Schweder T, Hjort NL. Frequentist analogues of priors and posteriors.
#' In Stigum, B. (ed.), Econometrics and the Philosophy of Economics: Theory
#' Data Confrontation in Economics, pp. 285-217. Princeton University Press,
#' Princeton, NJ, 2003
#'
#' - Schweder T, Hjort NL. Confidence, Likelihood, Probability: Statistical
#' inference with confidence distributions. Cambridge University Press, 2016.
#'
#' - Vos P, Holbert D. Frequentist statistical inference without repeated sampling.
#' Synthese 200, 89 (2022). \doi{10.1007/s11229-022-03560-x}
#'
#' @examplesIf requireNamespace("see")
#' model <- lm(Sepal.Length ~ Species, data = iris)
#' p_function(model)
#'
#' model <- lm(mpg ~ wt + as.factor(gear) + am, data = mtcars)
#' result <- p_function(model)
#'
#' # single panels
#' plot(result, n_columns = 2)
#'
#' # integrated plot, the default
#' plot(result)
#' @export
p_function <- function(model,
ci_levels = c(0.25, 0.5, 0.75, emph = 0.95),
exponentiate = FALSE,
effects = "fixed",
component = "all",
keep = NULL,
drop = NULL,
verbose = TRUE,
...) {
# degrees of freedom
dof <- insight::get_df(model, type = "wald")
# standard errors
se <- standard_error(
model,
effects = effects,
component = component
)$SE
if (is.null(dof) || length(dof) == 0 || .is_chi2_model(model, dof)) {
dof <- Inf
}
x <- do.call(rbind, lapply(seq(0, 1, 0.01), function(i) {
suppressMessages(.ci_dof(
model,
ci = i,
dof,
effects,
component,
method = "wald",
se = se,
vcov = NULL,
vcov_args = NULL,
verbose = TRUE
))
}))
# data for plotting
out <- x[!is.infinite(x$CI_low) & !is.infinite(x$CI_high), ]
out$CI <- round(out$CI, 2)
# most plausible value (point estimate)
point_estimate <- out$CI_low[which.min(out$CI)]
if (!is.null(keep) || !is.null(drop)) {
out <- .filter_parameters(out,
keep = keep,
drop = drop,
verbose = verbose
)
}
# transform non-Gaussian
if (isTRUE(exponentiate)) {
out$CI_low <- exp(out$CI_low)
out$CI_high <- exp(out$CI_high)
}
# data for p_function ribbon
data_ribbon <- datawizard::data_to_long(
out,
select = c("CI_low", "CI_high"),
values_to = "x"
)
# data for vertical CI level lines
out <- out[out$CI %in% ci_levels, ]
out$group <- 1
# emphasize focal hypothesis line
emphasize <- which(names(ci_levels) == "emph")
if (length(emphasize)) {
out$group[out$CI == ci_levels[emphasize]] <- 2
}
attr(out, "data") <- data_ribbon
attr(out, "point_estimate") <- point_estimate
attr(out, "pretty_names") <- suppressWarnings(format_parameters(model, ...))
class(out) <- c("parameters_p_function", "see_p_function", "data.frame")
out
}
#' @rdname p_function
#' @export
consonance_function <- p_function
#' @rdname p_function
#' @export
confidence_curve <- p_function
# methods ----------------------
#' @export
plot.parameters_p_function <- function(x, ...) {
insight::check_if_installed("see")
NextMethod()
}
#' @export
format.parameters_p_function <- function(x,
digits = 2,
format = NULL,
ci_width = NULL,
ci_brackets = TRUE,
pretty_names = TRUE,
...) {
# print
dat <- lapply(split(x, x$CI), function(i) {
ci <- as.character(i$CI)[1]
out <- datawizard::data_rename(
i,
pattern = c("CI_low", "CI_high"),
replacement = c(sprintf("CI_low_%s", ci), sprintf("CI_high_%s", ci))
)
out$CI <- NULL
out$group <- NULL
out
})
out <- do.call(datawizard::data_merge, list(dat, by = "Parameter"))
attr(out, "pretty_names") <- attributes(x)$pretty_names
insight::format_table(
out,
digits = digits,
ci_width = ci_width,
ci_brackets = ci_brackets,
format = format,
pretty_names = pretty_names
)
}
#' @export
print.parameters_p_function <- function(x,
digits = 2,
ci_width = "auto",
ci_brackets = TRUE,
pretty_names = TRUE,
...) {
cat(.print_p_function(
x,
digits,
ci_width,
ci_brackets,
pretty_names = pretty_names,
format = "text",
...
))
}
#' @export
print_md.parameters_p_function <- function(x,
digits = 2,
ci_width = "auto",
ci_brackets = c("(", ")"),
pretty_names = TRUE,
...) {
.print_p_function(x, digits, ci_width, ci_brackets, pretty_names, format = "markdown", ...)
}
#' @export
print_html.parameters_p_function <- function(x,
digits = 2,
ci_width = "auto",
ci_brackets = c("(", ")"),
pretty_names = TRUE,
...) {
.print_p_function(x, digits, ci_width, ci_brackets, pretty_names, format = "html", ...)
}
# helper ----------
.print_p_function <- function(x,
digits = 2,
ci_width = "auto",
ci_brackets = c("(", ")"),
pretty_names = TRUE,
format = "html",
...) {
formatted_table <- format(
x,
digits = digits,
format = format,
ci_width = ci_width,
ci_brackets = ci_brackets,
pretty_names = pretty_names,
...
)
insight::export_table(
formatted_table,
format = format,
caption = "Consonance Function",
...
)
}
# model <- lm(Sepal.Length ~ Species, data = iris)
# for later use: highlight p-value for secific parameter estimate values
# stat <- insight::get_statistic(model)
# se <- parameters::standard_error(model)
# estimate to test against - compute p-value for specific estimate
# null_estimate <- 1.5
# p <- 2 * stats::pt(abs(stat$Statistic[3]) - (null_estimate / se$SE[3]), df = 147, lower.tail = FALSE)
# bayestestR::p_to_pd(p)
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