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Tip revision: d8462ad2168ad7ee61c0d7e679174e775f01a9be authored by Dominique Makowski on 18 January 2020, 07:10 UTC
version 0.5.0
Tip revision: d8462ad
% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/p_direction.R
\title{Probability of Direction (pd)}
p_direction(x, ...)

pd(x, ...)

\method{p_direction}{numeric}(x, method = "direct", ...)

\method{p_direction}{data.frame}(x, method = "direct", ...)

\method{p_direction}{MCMCglmm}(x, method = "direct", ...)

\method{p_direction}{emmGrid}(x, method = "direct", ...)

  effects = c("fixed", "random", "all"),
  parameters = NULL,
  method = "direct",

  effects = c("fixed", "random", "all"),
  component = c("conditional", "zi", "zero_inflated", "all"),
  parameters = NULL,
  method = "direct",

\method{p_direction}{BFBayesFactor}(x, method = "direct", ...)
\item{x}{Vector representing a posterior distribution. Can also be a Bayesian model (\code{stanreg}, \code{brmsfit} or \code{BayesFactor}).}

\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{effects}{Should results for fixed effects, random effects or both be returned?
Only applies to mixed models. May be abbreviated.}

\item{parameters}{Regular expression pattern that describes the parameters that
should be returned. Meta-parameters (like \code{lp__} or \code{prior_}) are
filtered by default, so only parameters that typically appear in the
\code{summary()} are returned. Use \code{parameters} to select specific parameters
for the output.}

\item{component}{Should results for all parameters, parameters for the conditional model
or the zero-inflated part of the model be returned? May be abbreviated. Only
applies to \pkg{brms}-models.}
Values between 0.5 and 1 corresponding to the probability of direction (pd).
Compute the \strong{Probability of Direction} (\strong{\emph{pd}}, also known as the Maximum Probability of Effect - \emph{MPE}). It varies between 50\% and 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. Altough differently expressed, this index is fairly similar (\emph{i.e.}, is strongly correlated) to the frequentist \strong{p-value}.
\subsection{What is the \emph{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 (\emph{i.e.}, is positive or negative). Beyond its simplicity of interpretation, understanding and computation, this index also presents other interesting properties:
  \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.
\subsection{Relationship with the p-value}{
In most cases, it seems that the \emph{pd} has a direct correspondance with the frequentist one-sided \emph{p}-value through the formula \ifelse{html}{\out{p<sub>one&nbsp;sided</sub>&nbsp;=&nbsp;1&nbsp;-&nbsp;<sup>p(<em>d</em>)</sup>/<sub>100</sub>}}{\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{p<sub>two&nbsp;sided</sub>&nbsp;=&nbsp;2&nbsp;*&nbsp;(1&nbsp;-&nbsp;<sup>p(<em>d</em>)</sup>/<sub>100</sub>)}}{\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 95\%, 97.5\%, 99.5\% and 99.95\%.
\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 straigtfoward, 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.

# Simulate a posterior distribution of mean 1 and SD 1
# ----------------------------------------------------
posterior <- rnorm(1000, mean = 1, sd = 1)
p_direction(posterior, method = "kernel")

# Simulate a dataframe of posterior distributions
# -----------------------------------------------
df <- data.frame(replicate(4, rnorm(100)))
p_direction(df, method = "kernel")

# rstanarm models
# -----------------------------------------------
model <- rstanarm::stan_glm(mpg ~ wt + cyl,
  data = mtcars,
  chains = 2, refresh = 0
p_direction(model, method = "kernel")
# emmeans
# -----------------------------------------------
p_direction(emtrends(model, ~1, "wt"))

# brms models
# -----------------------------------------------
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
p_direction(model, method = "kernel")

# BayesFactor objects
# -----------------------------------------------
bf <- ttestBF(x = rnorm(100, 1, 1))
p_direction(bf, method = "kernel")
Makowski D, Ben-Shachar MS, Chen SHA, L├╝decke D (2019) Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology 2019;10:2767. \doi{10.3389/fpsyg.2019.02767}
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