diff --git a/R/p_direction.R b/R/p_direction.R index e989ecddb..7de3fe9f7 100644 --- a/R/p_direction.R +++ b/R/p_direction.R @@ -1,81 +1,91 @@ #' Probability of Direction (pd) #' -#' Compute the **Probability of Direction** (***pd***, also known -#' as the Maximum Probability of Effect - *MPE*). It varies between `50%` -#' and `100%` (*i.e.*, `0.5` and `1`) and can be interpreted as -#' the probability (expressed in percentage) that a parameter (described by its -#' posterior distribution) is strictly positive or negative (whichever is the -#' most probable). It is mathematically defined as the proportion of the -#' posterior distribution that is of the median's sign. Although differently -#' expressed, this index is fairly similar (*i.e.*, is strongly correlated) -#' to the frequentist **p-value**. -#' \cr\cr -#' Note that in some (rare) cases, especially when used with model averaged -#' posteriors (see [weighted_posteriors()] or -#' `brms::posterior_average`), `pd` can be smaller than `0.5`, -#' reflecting high credibility of `0`. +#' Compute the **Probability of Direction** (***pd***, also known as the Maximum +#' Probability of Effect - *MPE*). This can be interpreted as the probability +#' that a parameter (described by its posterior distribution) is strictly +#' positive or negative (whichever is the most probable). Although differently +#' expressed, this index is fairly similar (*i.e.*, is strongly correlated) to +#' the frequentist **p-value** (see details). #' -#' @param x Vector representing a posterior distribution. Can also be a Bayesian model (`stanreg`, `brmsfit` or `BayesFactor`). -#' @param method Can be `"direct"` or one of methods of [density estimation][estimate_density], such as `"kernel"`, `"logspline"` or `"KernSmooth"`. If `"direct"` (default), the computation is based on the raw ratio of samples superior and inferior to 0. Else, the result is based on the [Area under the Curve (AUC)][auc] of the estimated [density][estimate_density] function. -#' @param null The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios. +#' @param x A vector representing a posterior distribution, a data frame of +#' posterior draws (samples be parameter). Can also be a Bayesian model. +#' @param method Can be `"direct"` or one of methods of [`estimate_density()`], +#' such as `"kernel"`, `"logspline"` or `"KernSmooth"`. See details. +#' @param null The value considered as a "null" effect. Traditionally 0, but +#' could also be 1 in the case of ratios of change (OR, IRR, ...). #' @inheritParams hdi #' #' @details -#' \subsection{What is the *pd*?}{ +#' ## What is the *pd*? #' The Probability of Direction (pd) is an index of effect existence, ranging -#' from `50%` to `100%`, representing the certainty with which an effect goes in -#' a particular direction (*i.e.*, is positive or negative). Beyond its -#' simplicity of interpretation, understanding and computation, this index also -#' presents other interesting properties: -#' \itemize{ -#' \item It is independent from the model: It is solely based on the posterior -#' distributions and does not require any additional information from the data -#' or the model. -#' \item It is robust to the scale of both the response variable and the predictors. -#' \item It is strongly correlated with the frequentist p-value, and can thus +#' from 0 to 1, representing the certainty with which an effect goes in a +#' particular direction (*i.e.*, is positive or negative / has a sign). Beyond +#' its simplicity of interpretation, understanding and computation, this index +#' also presents other interesting properties: +#' - It is robust to the scale of both the response variable and the predictors. +#' - It is strongly correlated with the frequentist p-value, and can thus #' be used to draw parallels and give some reference to readers non-familiar -#' with Bayesian statistics. -#' } -#' } -#' \subsection{Relationship with the p-value}{ -#' In most cases, it seems that the *pd* has a direct correspondence with the frequentist one-sided *p*-value through the formula \ifelse{html}{\out{pone sided = 1 - p(d)/100}}{\eqn{p_{one sided}=1-\frac{p_{d}}{100}}} and to the two-sided p-value (the most commonly reported one) through the formula \ifelse{html}{\out{ptwo sided = 2 * (1 - p(d)/100)}}{\eqn{p_{two sided}=2*(1-\frac{p_{d}}{100})}}. Thus, a two-sided p-value of respectively `.1`, `.05`, `.01` and `.001` would correspond approximately to a *pd* of `95%`, `97.5%`, `99.5%` and `99.95%`. See also [pd_to_p()]. -#' } -#' \subsection{Methods of computation}{ -#' The most simple and direct way to compute the *pd* is to 1) look at the -#' median's sign, 2) select the portion of the posterior of the same sign and -#' 3) compute the percentage that this portion represents. This "simple" method -#' is the most straightforward, but its precision is directly tied to the -#' number of posterior draws. The second approach relies on [density -#' estimation][estimate_density]. It starts by estimating the density function -#' (for which many methods are available), and then computing the [area under -#' the curve][area_under_curve] (AUC) of the density curve on the other side of -#' 0. -#' } -#' \subsection{Strengths and Limitations}{ -#' **Strengths:** Straightforward computation and interpretation. Objective -#' property of the posterior distribution. 1:1 correspondence with the -#' frequentist p-value. -#' \cr \cr -#' **Limitations:** Limited information favoring the null hypothesis. -#' } +#' with Bayesian statistics (Makowski et al., 2019). See also [`pd_to_p()`]. #' -#' @return -#' Values between 0.5 and 1 corresponding to the probability of direction (pd). +#' ## Possible Range of Values +#' The largest value *pd* can take is 1 - the posterior is strictly directional. +#' However, the smallest value *pd* can take depends on the parameter space +#' represented by the posterior. #' \cr\cr -#' Note that in some (rare) cases, especially when used with model averaged -#' posteriors (see [weighted_posteriors()] or -#' `brms::posterior_average`), `pd` can be smaller than `0.5`, -#' reflecting high credibility of `0`. To detect such cases, the -#' `method = "direct"` must be used. +#' **For a continuous parameter space**, exact values of 0 (or any point null +#' value) are not possible, and so 100% of the posterior has _some_ sign, some +#' positive, some negative. Therefore, the smallest the *pd* can be is 0.5 - +#' with an equal posterior mass of positive and negative values. Values close to +#' 0.5 _cannot_ be used to support the null hypothesis (that the parameter does +#' _not_ have a direction) is a similar why to how large p-values cannot be used +#' to support the null hypothesis (see [`pd_tp_p()`]; Makowski et al., 2019). +#' \cr\cr +#' **For a discrete parameter space or a parameter space that is a mixture +#' between discrete and continuous spaces**, exact values of 0 (or any point +#' null value) _are_ possible! Therefore, the smallest the *pd* can be is 0 - +#' with 100% of the posterior mass on 0. Thus values close to 0 can be used to +#' support the null hypothesis (see van den Bergh et al., 2021). +#' \cr\cr +#' Examples of posteriors representing discrete parameter space: +#' - When a parameter can only take discrete values. +#' - When a mixture prior/posterior is used (such as the spike-and-slab prior; +#' see van den Bergh et al., 2021). +#' - When conducting Bayesian model averaging (e.g., [weighted_posteriors()] or +#' `brms::posterior_average`). +#' +#' ## Methods of computation +#' The *pd* is defined as: +#' \deqn{p_d = max({Pr(\hat{\theta} < \theta_{null}), Pr(\hat{\theta} > \theta_{null})})}{pd = max(mean(x < null), mean(x > null))} +#' \cr\cr +#' The most simple and direct way to compute the *pd* is to compute the +#' proportion of positive (or larger than `null`) posterior samples, the +#' proportion of negative (or smaller than `null`) posterior samples, and take +#' the larger of the two. This "simple" method is the most straightforward, but +#' its precision is directly tied to the number of posterior draws. +#' \cr\cr +#' The second approach relies on [`density estimation()`]: It starts by +#' estimating the continuous-smooth density function (for which many methods are +#' available), and then computing the [area under the curve][area_under_curve] +#' (AUC) of the density curve on either side of `null` and taking the maximum +#' between them. Note the this approach assumes a continuous density function, +#' and so **when the posterior represents a (partially) discrete parameter +#' space, only the direct method _must_ be used** (see above). +#' +#' @return +#' Values between 0.5 and 1 *or* between 0 and 1 (see above) corresponding to +#' the probability of direction (pd). #' #' @seealso [pd_to_p()] to convert between Probability of Direction (pd) and p-value. #' #' @note There is also a [`plot()`-method](https://easystats.github.io/see/articles/bayestestR.html) implemented in the \href{https://easystats.github.io/see/}{\pkg{see}-package}. #' #' @references -#' 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} +#' - Makowski, D., Ben-Shachar, M. S., Chen, S. A., & Lüdecke, D. (2019). +#' Indices of effect existence and significance in the Bayesian framework. +#' Frontiers in psychology, 10, 2767. \doi{10.3389/fpsyg.2019.02767} +#' - van den Bergh, D., Haaf, J. M., Ly, A., Rouder, J. N., & Wagenmakers, E. J. +#' (2021). A cautionary note on estimating effect size. Advances in Methods +#' and Practices in Psychological Science, 4(1). \doi{10.1177/2515245921992035} #' #' @examples #' library(bayestestR) diff --git a/man/p_direction.Rd b/man/p_direction.Rd index cf90a3531..ca54f27d4 100644 --- a/man/p_direction.Rd +++ b/man/p_direction.Rd @@ -48,13 +48,16 @@ pd(x, ...) \method{p_direction}{BFBayesFactor}(x, method = "direct", null = 0, ...) } \arguments{ -\item{x}{Vector representing a posterior distribution. Can also be a Bayesian model (\code{stanreg}, \code{brmsfit} or \code{BayesFactor}).} +\item{x}{A vector representing a posterior distribution, a data frame of +posterior draws (samples be parameter). Can also be a Bayesian model.} \item{...}{Currently not used.} -\item{method}{Can be \code{"direct"} or one of methods of \link[=estimate_density]{density estimation}, such as \code{"kernel"}, \code{"logspline"} or \code{"KernSmooth"}. If \code{"direct"} (default), the computation is based on the raw ratio of samples superior and inferior to 0. Else, the result is based on the \link[=auc]{Area under the Curve (AUC)} of the estimated \link[=estimate_density]{density} function.} +\item{method}{Can be \code{"direct"} or one of methods of \code{\link[=estimate_density]{estimate_density()}}, +such as \code{"kernel"}, \code{"logspline"} or \code{"KernSmooth"}. See details.} -\item{null}{The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios.} +\item{null}{The value considered as a "null" effect. Traditionally 0, but +could also be 1 in the case of ratios of change (OR, IRR, ...).} \item{effects}{Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated.} @@ -70,65 +73,81 @@ filtered by default, so only parameters that typically appear in the for the output.} } \value{ -Values between 0.5 and 1 corresponding to the probability of direction (pd). -\cr\cr -Note that in some (rare) cases, especially when used with model averaged -posteriors (see \code{\link[=weighted_posteriors]{weighted_posteriors()}} or -\code{brms::posterior_average}), \code{pd} can be smaller than \code{0.5}, -reflecting high credibility of \code{0}. To detect such cases, the -\code{method = "direct"} must be used. +Values between 0.5 and 1 \emph{or} between 0 and 1 (see above) corresponding to +the probability of direction (pd). } \description{ -Compute the \strong{Probability of Direction} (\emph{\strong{pd}}, also known -as the Maximum Probability of Effect - \emph{MPE}). It varies between \verb{50\%} -and \verb{100\%} (\emph{i.e.}, \code{0.5} and \code{1}) and can be interpreted as -the probability (expressed in percentage) that a parameter (described by its -posterior distribution) is strictly positive or negative (whichever is the -most probable). It is mathematically defined as the proportion of the -posterior distribution that is of the median's sign. Although differently -expressed, this index is fairly similar (\emph{i.e.}, is strongly correlated) -to the frequentist \strong{p-value}. -\cr\cr -Note that in some (rare) cases, especially when used with model averaged -posteriors (see \code{\link[=weighted_posteriors]{weighted_posteriors()}} or -\code{brms::posterior_average}), \code{pd} can be smaller than \code{0.5}, -reflecting high credibility of \code{0}. +Compute the \strong{Probability of Direction} (\emph{\strong{pd}}, also known as the Maximum +Probability of Effect - \emph{MPE}). This can be interpreted as the probability +that a parameter (described by its posterior distribution) is strictly +positive or negative (whichever is the most probable). Although differently +expressed, this index is fairly similar (\emph{i.e.}, is strongly correlated) to +the frequentist \strong{p-value} (see details). } \details{ \subsection{What is the \emph{pd}?}{ + The Probability of Direction (pd) is an index of effect existence, ranging -from \verb{50\%} to \verb{100\%}, representing the certainty with which an effect goes in -a particular direction (\emph{i.e.}, is positive or negative). Beyond its -simplicity of interpretation, understanding and computation, this index also -presents other interesting properties: +from 0 to 1, representing the certainty with which an effect goes in a +particular direction (\emph{i.e.}, is positive or negative / has a sign). Beyond +its simplicity of interpretation, understanding and computation, this index +also presents other interesting properties: \itemize{ -\item It is independent from the model: It is solely based on the posterior -distributions and does not require any additional information from the data -or the model. \item It is robust to the scale of both the response variable and the predictors. \item It is strongly correlated with the frequentist p-value, and can thus be used to draw parallels and give some reference to readers non-familiar -with Bayesian statistics. +with Bayesian statistics (Makowski et al., 2019). See also \code{\link[=pd_to_p]{pd_to_p()}}. +} } + +\subsection{Possible Range of Values}{ + +The largest value \emph{pd} can take is 1 - the posterior is strictly directional. +However, the smallest value \emph{pd} can take depends on the parameter space +represented by the posterior. +\cr\cr +\strong{For a continuous parameter space}, exact values of 0 (or any point null +value) are not possible, and so 100\% of the posterior has \emph{some} sign, some +positive, some negative. Therefore, the smallest the \emph{pd} can be is 0.5 - +with an equal posterior mass of positive and negative values. Values close to +0.5 \emph{cannot} be used to support the null hypothesis (that the parameter does +\emph{not} have a direction) is a similar why to how large p-values cannot be used +to support the null hypothesis (see \code{\link[=pd_tp_p]{pd_tp_p()}}; Makowski et al., 2019). +\cr\cr +\strong{For a discrete parameter space or a parameter space that is a mixture +between discrete and continuous spaces}, exact values of 0 (or any point +null value) \emph{are} possible! Therefore, the smallest the \emph{pd} can be is 0 - +with 100\% of the posterior mass on 0. Thus values close to 0 can be used to +support the null hypothesis (see van den Bergh et al., 2021). +\cr\cr +Examples of posteriors representing discrete parameter space: +\itemize{ +\item When a parameter can only take discrete values. +\item When a mixture prior/posterior is used (such as the spike-and-slab prior; +see van den Bergh et al., 2021). +\item When conducting Bayesian model averaging (e.g., \code{\link[=weighted_posteriors]{weighted_posteriors()}} or +\code{brms::posterior_average}). } -\subsection{Relationship with the p-value}{ -In most cases, it seems that the \emph{pd} has a direct correspondence with the frequentist one-sided \emph{p}-value through the formula \ifelse{html}{\out{pone sided = 1 - p(d)/100}}{\eqn{p_{one sided}=1-\frac{p_{d}}{100}}} and to the two-sided p-value (the most commonly reported one) through the formula \ifelse{html}{\out{ptwo sided = 2 * (1 - p(d)/100)}}{\eqn{p_{two sided}=2*(1-\frac{p_{d}}{100})}}. Thus, a two-sided p-value of respectively \code{.1}, \code{.05}, \code{.01} and \code{.001} would correspond approximately to a \emph{pd} of \verb{95\%}, \verb{97.5\%}, \verb{99.5\%} and \verb{99.95\%}. See also \code{\link[=pd_to_p]{pd_to_p()}}. } + \subsection{Methods of computation}{ -The most simple and direct way to compute the \emph{pd} is to 1) look at the -median's sign, 2) select the portion of the posterior of the same sign and -3) compute the percentage that this portion represents. This "simple" method -is the most straightforward, but its precision is directly tied to the -number of posterior draws. The second approach relies on \link[=estimate_density]{density estimation}. It starts by estimating the density function -(for which many methods are available), and then computing the \link[=area_under_curve]{area under the curve} (AUC) of the density curve on the other side of -0. -} -\subsection{Strengths and Limitations}{ -\strong{Strengths:} Straightforward computation and interpretation. Objective -property of the posterior distribution. 1:1 correspondence with the -frequentist p-value. -\cr \cr -\strong{Limitations:} Limited information favoring the null hypothesis. + +The \emph{pd} is defined as: +\deqn{p_d = max({Pr(\hat{\theta} < \theta_{null}), Pr(\hat{\theta} > \theta_{null})})}{pd = max(mean(x < null), mean(x > null))} +\cr\cr +The most simple and direct way to compute the \emph{pd} is to compute the +proportion of positive (or larger than \code{null}) posterior samples, the +proportion of negative (or smaller than \code{null}) posterior samples, and take +the larger of the two. This "simple" method is the most straightforward, but +its precision is directly tied to the number of posterior draws. +\cr\cr +The second approach relies on \code{\link[=density estimation]{density estimation()}}: It starts by +estimating the continuous-smooth density function (for which many methods are +available), and then computing the \link[=area_under_curve]{area under the curve} +(AUC) of the density curve on either side of \code{null} and taking the maximum +between them. Note the this approach assumes a continuous density function, +and so \strong{when the posterior represents a (partially) discrete parameter +space, only the direct method \emph{must} be used} (see above). } } \note{ @@ -184,9 +203,14 @@ if (require("BayesFactor")) { } } \references{ -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} +\itemize{ +\item Makowski, D., Ben-Shachar, M. S., Chen, S. A., & Lüdecke, D. (2019). +Indices of effect existence and significance in the Bayesian framework. +Frontiers in psychology, 10, 2767. \doi{10.3389/fpsyg.2019.02767} +\item van den Bergh, D., Haaf, J. M., Ly, A., Rouder, J. N., & Wagenmakers, E. J. +(2021). A cautionary note on estimating effect size. Advances in Methods +and Practices in Psychological Science, 4(1). \doi{10.1177/2515245921992035} +} } \seealso{ \code{\link[=pd_to_p]{pd_to_p()}} to convert between Probability of Direction (pd) and p-value. diff --git a/man/p_map.Rd b/man/p_map.Rd index 9e9e625fe..94d9c21f2 100644 --- a/man/p_map.Rd +++ b/man/p_map.Rd @@ -41,7 +41,8 @@ of models (see, for example, \code{methods("hdi")}) and not all of those are documented in the 'Usage' section, because methods for other classes mostly resemble the arguments of the \code{.numeric} or \code{.data.frame}methods.} -\item{null}{The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios.} +\item{null}{The value considered as a "null" effect. Traditionally 0, but +could also be 1 in the case of ratios of change (OR, IRR, ...).} \item{precision}{Number of points of density data. See the \code{n} parameter in \code{density}.} diff --git a/man/sexit.Rd b/man/sexit.Rd index 8b0f8029a..74d4cd5ce 100644 --- a/man/sexit.Rd +++ b/man/sexit.Rd @@ -7,7 +7,8 @@ sexit(x, significant = "default", large = "default", ci = 0.95, ...) } \arguments{ -\item{x}{Vector representing a posterior distribution. Can also be a Bayesian model (\code{stanreg}, \code{brmsfit} or \code{BayesFactor}).} +\item{x}{A vector representing a posterior distribution, a data frame of +posterior draws (samples be parameter). Can also be a Bayesian model.} \item{significant, large}{The threshold values to use for significant and large probabilities. If left to 'default', will be selected through