Probability of Direction (pd) — p_direction • bayestestR

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).

Usage

p_direction(x, ...)

pd(x, ...)

# S3 method for class 'numeric'
p_direction(
  x,
  method = "direct",
  null = 0,
  as_p = FALSE,
  remove_na = TRUE,
  ...
)

# S3 method for class 'data.frame'
p_direction(
  x,
  method = "direct",
  null = 0,
  as_p = FALSE,
  remove_na = TRUE,
  rvar_col = NULL,
  ...
)

# S3 method for class 'brmsfit'
p_direction(
  x,
  effects = "fixed",
  component = "conditional",
  parameters = NULL,
  method = "direct",
  null = 0,
  as_p = FALSE,
  remove_na = TRUE,
  ...
)

# S3 method for class 'get_predicted'
p_direction(
  x,
  method = "direct",
  null = 0,
  as_p = FALSE,
  remove_na = TRUE,
  use_iterations = FALSE,
  verbose = TRUE,
  ...
)

Arguments

x

A vector representing a posterior distribution, a data frame of posterior draws (samples be parameter). Can also be a Bayesian model.

...

Currently not used.

method

Can be "direct" or one of methods of estimate_density(), such as "kernel", "logspline" or "KernSmooth". See details.

null

The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios of change (OR, IRR, ...).

as_p

If TRUE, the p-direction (pd) values are converted to a frequentist p-value using pd_to_p().

remove_na

Should missing values be removed before computation? Note that Inf (infinity) are not removed.

rvar_col

A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

effects

Should results for fixed effects ("fixed", the default), random effects ("random") or both ("all") be returned? Only applies to mixed models. May be abbreviated.

component

Which type of parameters to return, such as parameters for the conditional model, the zero-inflated part of the model, the dispersion term, etc. See details in section Model Components. May be abbreviated. Note that the conditional component also refers to the count or mean component - names may differ, depending on the modeling package. There are three convenient shortcuts (not applicable to all model classes):

component = "all" returns all possible parameters.
If component = "location", location parameters such as conditional, zero_inflated, smooth_terms, or instruments are returned (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
For component = "distributional" (or "auxiliary"), components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

parameters

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

use_iterations

Logical, if TRUE and x is a get_predicted object, (returned by insight::get_predicted()), the function is applied to the iterations instead of the predictions. This only applies to models that return iterations for predicted values (e.g., brmsfit models).

verbose

Toggle off warnings.

Value

Values between 0.5 and 1 or between 0 and 1 (see above) corresponding to the probability of direction (pd).

Note

There is also a plot()-method implemented in the see-package.

What is the pd?

The Probability of Direction (pd) is an index of effect existence, representing the certainty with which an effect goes in a particular direction (i.e., is positive or negative / has a sign), typically ranging from 0.5 to 1 (but see next section for cases where it can range between 0 and 1). Beyond its simplicity of interpretation, understanding and computation, this index also presents other interesting properties:

Like other posterior-based indices, pd is solely based on the posterior distributions and does not require any additional information from the data or the model (e.g., such as priors, as in the case of Bayes factors).
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 (Makowski et al., 2019).

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 (for two-sided p): p = 2 * (1 - p_d) 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 pd_to_p() for details.

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.

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_to_p(); Makowski et al., 2019).

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).

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: $$p_d = max({Pr(\hat{\theta} < \theta_{null}), Pr(\hat{\theta} > \theta_{null})})$$

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.

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 (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).

Model components

Possible values for the component argument depend on the model class. Following are valid options:

"all": returns all model components, applies to all models, but will only have an effect for models with more than just the conditional model component.
"conditional": only returns the conditional component, i.e. "fixed effects" terms from the model. Will only have an effect for models with more than just the conditional model component.
"smooth_terms": returns smooth terms, only applies to GAMs (or similar models that may contain smooth terms).
"zero_inflated" (or "zi"): returns the zero-inflation component.
"location": returns location parameters such as conditional, zero_inflated, or smooth_terms (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
"distributional" (or "auxiliary"): components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

For models of class brmsfit (package brms), even more options are possible for the component argument, which are not all documented in detail here. See also ?insight::find_parameters.

References

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)

# Simulate a posterior distribution of mean 1 and SD 1
# ----------------------------------------------------
posterior <- rnorm(1000, mean = 1, sd = 1)
p_direction(posterior)
#> Probability of Direction
#> 
#> Parameter |     pd
#> ------------------
#> Posterior | 84.50%
p_direction(posterior, method = "kernel")
#> Probability of Direction
#> 
#> Parameter |     pd
#> ------------------
#> Posterior | 83.17%

# Simulate a dataframe of posterior distributions
# -----------------------------------------------
df <- data.frame(replicate(4, rnorm(100)))
p_direction(df)
#> Probability of Direction
#> 
#> Parameter |     pd
#> ------------------
#> X1        | 51.00%
#> X2        | 52.00%
#> X3        | 51.00%
#> X4        | 58.00%
p_direction(df, method = "kernel")
#> Probability of Direction
#> 
#> Parameter |     pd
#> ------------------
#> X1        | 51.24%
#> X2        | 51.93%
#> X3        | 50.15%
#> X4        | 59.86%

# \donttest{
# rstanarm models
# -----------------------------------------------
model <- rstanarm::stan_glm(mpg ~ wt + cyl,
  data = mtcars,
  chains = 2, refresh = 0
)
p_direction(model)
#> Probability of Direction 
#> 
#> Parameter   |   pd
#> ------------------
#> (Intercept) | 100%
#> wt          | 100%
#> cyl         | 100%
p_direction(model, method = "kernel")
#> Probability of Direction 
#> 
#> Parameter   |      pd
#> ---------------------
#> (Intercept) | 100.00%
#> wt          |  99.98%
#> cyl         |  99.97%

# emmeans
# -----------------------------------------------
p_direction(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#> Probability of Direction
#> 
#> X1      |   pd
#> --------------
#> overall | 100%

# brms models
# -----------------------------------------------
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 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: 0.022 seconds (Warm-up)
#> Chain 1:                0.022 seconds (Sampling)
#> Chain 1:                0.044 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 3e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.03 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: 0.02 seconds (Warm-up)
#> Chain 2:                0.017 seconds (Sampling)
#> Chain 2:                0.037 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 3e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.03 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: 0.02 seconds (Warm-up)
#> Chain 3:                0.018 seconds (Sampling)
#> Chain 3:                0.038 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 3e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.03 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: 0.02 seconds (Warm-up)
#> Chain 4:                0.017 seconds (Sampling)
#> Chain 4:                0.037 seconds (Total)
#> Chain 4: 
p_direction(model)
#> Probability of Direction 
#> 
#> Parameter   |     pd
#> --------------------
#> (Intercept) |   100%
#> wt          |   100%
#> cyl         | 99.98%
p_direction(model, method = "kernel")
#> Probability of Direction 
#> 
#> Parameter   |     pd
#> --------------------
#> (Intercept) |   100%
#> wt          | 99.99%
#> cyl         | 99.97%

# BayesFactor objects
# -----------------------------------------------
bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
p_direction(bf)
#> Probability of Direction
#> 
#> Parameter  |   pd
#> -----------------
#> Difference | 100%
p_direction(bf, method = "kernel")
#> Probability of Direction
#> 
#> Parameter  |   pd
#> -----------------
#> Difference | 100%
# }
# Using "rvar_col"
x <- data.frame(mu = c(0, 0.5, 1), sigma = c(1, 0.5, 0.25))
x$my_rvar <- posterior::rvar_rng(rnorm, 3, mean = x$mu, sd = x$sigma)
x
#>    mu sigma      my_rvar
#> 1 0.0  1.00 -0.01 ± 0.98
#> 2 0.5  0.50  0.49 ± 0.50
#> 3 1.0  0.25  1.00 ± 0.25
p_direction(x, rvar_col = "my_rvar")
#> Probability of Direction
#> 
#> mu   | sigma |     pd
#> ---------------------
#> 0.00 |  1.00 | 50.10%
#> 0.50 |  0.50 | 83.90%
#> 1.00 |  0.25 |   100%