Compute the **Probability of Direction** (* pd*, also known
as the Maximum Probability of Effect -

`50%`

and `100%`

(`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 (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 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, ...)

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 |

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.

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 p_{one sided} = 1 - ^{p(d)}/_{100} and to the two-sided p-value (the most commonly reported one) through the formula p_{two 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.

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)#> Probability of Direction: 0.84p_direction(posterior, method = "kernel")#> Probability of Direction: 0.83# Simulate a dataframe of posterior distributions # ----------------------------------------------- df <- data.frame(replicate(4, rnorm(100))) p_direction(df)#> Probability of Direction #> #> Parameter | pd #> ------------------ #> X1 | 54.00% #> X2 | 51.00% #> X3 | 59.00% #> X4 | 50.00%p_direction(df, method = "kernel")#> Probability of Direction #> #> Parameter | pd #> ------------------ #> X1 | 52.92% #> X2 | 50.05% #> X3 | 56.44% #> X4 | 51.21%if (FALSE) { # 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") } }