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.
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.
p_direction(x, ...)
pd(x, ...)
## S3 method for class 'numeric'
p_direction(x, method = "direct", null = 0, ...)
## S3 method for class 'data.frame'
p_direction(x, method = "direct", null = 0, ...)
## S3 method for class 'MCMCglmm'
p_direction(x, method = "direct", null = 0, ...)
## S3 method for class 'emmGrid'
p_direction(x, method = "direct", null = 0, ...)
## S3 method for class 'stanreg'
p_direction(
x,
effects = c("fixed", "random", "all"),
component = c("location", "all", "conditional", "smooth_terms", "sigma",
"distributional", "auxiliary"),
parameters = NULL,
method = "direct",
null = 0,
...
)
## S3 method for class 'brmsfit'
p_direction(
x,
effects = c("fixed", "random", "all"),
component = c("conditional", "zi", "zero_inflated", "all"),
parameters = NULL,
method = "direct",
null = 0,
...
)
## S3 method for class 'BFBayesFactor'
p_direction(x, method = "direct", null = 0, ...)x |
Vector representing a posterior distribution. Can also be a Bayesian model ( |
... |
Currently not used. |
method |
Can be |
null |
The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios. |
effects |
Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated. |
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 brms-models. |
parameters |
Regular expression pattern that describes the parameters that
should be returned. Meta-parameters (like |
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:
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.
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.
In most cases, it seems that the pd has a direct correspondence with the frequentist one-sided p-value through the formula pone sided = 1 - p(d)/100 and to the two-sided p-value (the most commonly reported one) through the formula ptwo sided = 2 * (1 - 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.
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. It starts by estimating the density function (for which many methods are available), and then computing the area under the curve (AUC) of the density curve on the other side of 0.
Strengths: Straightforward computation and interpretation. Objective property of the posterior distribution. 1:1 correspondence with the frequentist p-value.
Limitations: Limited information favoring the null hypothesis.
Values between 0.5 and 1 corresponding to the probability of direction (pd).
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.
There is also a plot()-method implemented in the see-package.
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
pd_to_p to convert between Probability of Direction (pd) and p-value.
library(bayestestR)
# Simulate a posterior distribution of mean 1 and SD 1
# ----------------------------------------------------
posterior <- rnorm(1000, mean = 1, sd = 1)
p_direction(posterior)
p_direction(posterior, method = "kernel")
# Simulate a dataframe of posterior distributions
# -----------------------------------------------
df <- data.frame(replicate(4, rnorm(100)))
p_direction(df)
p_direction(df, method = "kernel")
## Not run:
# rstanarm models
# -----------------------------------------------
if (require("rstanarm")) {
model <- rstanarm::stan_glm(mpg ~ wt + cyl,
data = mtcars,
chains = 2, refresh = 0
)
p_direction(model)
p_direction(model, method = "kernel")
}
# emmeans
# -----------------------------------------------
if (require("emmeans")) {
p_direction(emtrends(model, ~1, "wt"))
}
# brms models
# -----------------------------------------------
if (require("brms")) {
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
p_direction(model)
p_direction(model, method = "kernel")
}
# BayesFactor objects
# -----------------------------------------------
if (require("BayesFactor")) {
bf <- ttestBF(x = rnorm(100, 1, 1))
p_direction(bf)
p_direction(bf, method = "kernel")
}
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.