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.
Usage
p_direction(x, ...)
pd(x, ...)
# S3 method for numeric
p_direction(x, method = "direct", null = 0, ...)
# S3 method for data.frame
p_direction(x, method = "direct", null = 0, ...)
# S3 method for MCMCglmm
p_direction(x, method = "direct", null = 0, ...)
# S3 method for emmGrid
p_direction(x, method = "direct", null = 0, ...)
# S3 method for 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 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 BFBayesFactor
p_direction(x, method = "direct", null = 0, ...)Arguments
- x
Vector representing a posterior distribution. Can also be a Bayesian model (
stanreg,brmsfitorBayesFactor).- ...
Currently not used.
- method
Can be
"direct"or one of methods of density estimation, 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) of the estimated density function.- 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
lp__orprior_) are filtered by default, so only parameters that typically appear in thesummary()are returned. Useparametersto select specific parameters for the output.
Value
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.
Details
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:
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.
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 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().
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. 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.
Note
There is also a plot()-method implemented in the 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
See also
pd_to_p() to convert between Probability of Direction (pd) and p-value.
Examples
library(bayestestR)
# Simulate a posterior distribution of mean 1 and SD 1
# ----------------------------------------------------
posterior <- rnorm(1000, mean = 1, sd = 1)
p_direction(posterior)
#> Probability of Direction: 0.87
p_direction(posterior, method = "kernel")
#> Probability of Direction: 0.85
# Simulate a dataframe of posterior distributions
# -----------------------------------------------
df <- data.frame(replicate(4, rnorm(100)))
p_direction(df)
#> Probability of Direction
#>
#> Parameter | pd
#> ------------------
#> X1 | 52.00%
#> X2 | 56.00%
#> X3 | 61.00%
#> X4 | 59.00%
p_direction(df, method = "kernel")
#> Probability of Direction
#>
#> Parameter | pd
#> ------------------
#> X1 | 50.98%
#> X2 | 51.95%
#> X3 | 58.20%
#> X4 | 56.33%
# \dontrun{
# 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")
}
#> Probability of Direction
#>
#> Parameter | pd
#> ------------------
#> (Intercept) | 100%
#> wt | 100%
#> cyl | 100%
# emmeans
# -----------------------------------------------
if (require("emmeans")) {
p_direction(emtrends(model, ~1, "wt", data = mtcars))
}
#> Probability of Direction
#>
#> Parameter | pd
#> ----------------
#> overall | 100%
# brms models
# -----------------------------------------------
if (require("brms")) {
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
p_direction(model)
p_direction(model, method = "kernel")
}
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup)
#> Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup)
#> Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup)
#> Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup)
#> Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling)
#> Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling)
#> Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling)
#> Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling)
#> Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 1:
#> Chain 1: Elapsed Time: 0.036344 seconds (Warm-up)
#> Chain 1: 0.036091 seconds (Sampling)
#> Chain 1: 0.072435 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 7e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
#> Chain 2:
#> Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 2: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 2: Iteration: 400 / 2000 [ 20%] (Warmup)
#> Chain 2: Iteration: 600 / 2000 [ 30%] (Warmup)
#> Chain 2: Iteration: 800 / 2000 [ 40%] (Warmup)
#> Chain 2: Iteration: 1000 / 2000 [ 50%] (Warmup)
#> Chain 2: Iteration: 1001 / 2000 [ 50%] (Sampling)
#> Chain 2: Iteration: 1200 / 2000 [ 60%] (Sampling)
#> Chain 2: Iteration: 1400 / 2000 [ 70%] (Sampling)
#> Chain 2: Iteration: 1600 / 2000 [ 80%] (Sampling)
#> Chain 2: Iteration: 1800 / 2000 [ 90%] (Sampling)
#> Chain 2: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 2:
#> Chain 2: Elapsed Time: 0.034636 seconds (Warm-up)
#> Chain 2: 0.031971 seconds (Sampling)
#> Chain 2: 0.066607 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 8e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
#> Chain 3:
#> Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup)
#> Chain 3: Iteration: 600 / 2000 [ 30%] (Warmup)
#> Chain 3: Iteration: 800 / 2000 [ 40%] (Warmup)
#> Chain 3: Iteration: 1000 / 2000 [ 50%] (Warmup)
#> Chain 3: Iteration: 1001 / 2000 [ 50%] (Sampling)
#> Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling)
#> Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling)
#> Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling)
#> Chain 3: Iteration: 1800 / 2000 [ 90%] (Sampling)
#> Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 3:
#> Chain 3: Elapsed Time: 0.037821 seconds (Warm-up)
#> Chain 3: 0.034419 seconds (Sampling)
#> Chain 3: 0.07224 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 9e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
#> Chain 4:
#> Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup)
#> Chain 4: Iteration: 600 / 2000 [ 30%] (Warmup)
#> Chain 4: Iteration: 800 / 2000 [ 40%] (Warmup)
#> Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup)
#> Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling)
#> Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling)
#> Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling)
#> Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling)
#> Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling)
#> Chain 4: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 4:
#> Chain 4: Elapsed Time: 0.036868 seconds (Warm-up)
#> Chain 4: 0.033868 seconds (Sampling)
#> Chain 4: 0.070736 seconds (Total)
#> Chain 4:
#> Probability of Direction
#>
#> Parameter | pd
#> ------------------
#> (Intercept) | 100%
#> wt | 100%
#> cyl | 100%
# BayesFactor objects
# -----------------------------------------------
if (require("BayesFactor")) {
bf <- ttestBF(x = rnorm(100, 1, 1))
p_direction(bf)
p_direction(bf, method = "kernel")
}
#> Probability of Direction
#>
#> Parameter | pd
#> -----------------
#> Difference | 100%
# }