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Compute a Bayesian equivalent of the p-value, related to the odds that a parameter (described by its posterior distribution) has against the null hypothesis (h0) using Mills' (2014, 2017) Objective Bayesian Hypothesis Testing framework. It corresponds to the density value at the null (e.g., 0) divided by the density at the Maximum A Posteriori (MAP).

Usage

p_map(x, ...)

p_pointnull(x, ...)

# S3 method for numeric
p_map(x, null = 0, precision = 2^10, method = "kernel", ...)

# S3 method for get_predicted
p_map(
  x,
  null = 0,
  precision = 2^10,
  method = "kernel",
  use_iterations = FALSE,
  verbose = TRUE,
  ...
)

# S3 method for stanreg
p_map(
  x,
  null = 0,
  precision = 2^10,
  method = "kernel",
  effects = c("fixed", "random", "all"),
  component = c("location", "all", "conditional", "smooth_terms", "sigma",
    "distributional", "auxiliary"),
  parameters = NULL,
  ...
)

# S3 method for brmsfit
p_map(
  x,
  null = 0,
  precision = 2^10,
  method = "kernel",
  effects = c("fixed", "random", "all"),
  component = c("conditional", "zi", "zero_inflated", "all"),
  parameters = NULL,
  ...
)

Arguments

x

Vector representing a posterior distribution, or a data frame of such vectors. Can also be a Bayesian model. bayestestR supports a wide range of models (see, for example, methods("hdi")) and not all of those are documented in the 'Usage' section, because methods for other classes mostly resemble the arguments of the .numeric or .data.framemethods.

...

Currently not used.

null

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

precision

Number of points of density data. See the n parameter in density.

method

Density estimation method. Can be "kernel" (default), "logspline" or "KernSmooth".

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.

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__ 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.

Details

Note that this method is sensitive to the density estimation method (see the section in the examples below).

Strengths and Limitations

Strengths: Straightforward computation. Objective property of the posterior distribution.

Limitations: Limited information favoring the null hypothesis. Relates on density approximation. Indirect relationship between mathematical definition and interpretation. Only suitable for weak / very diffused priors.

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

  • Mills, J. A. (2018). Objective Bayesian Precise Hypothesis Testing. University of Cincinnati.

See also

Examples

library(bayestestR)

p_map(rnorm(1000, 0, 1))
#> MAP-based p-value
#> 
#> Parameter | p (MAP)
#> -------------------
#> Posterior |   0.971
p_map(rnorm(1000, 10, 1))
#> MAP-based p-value
#> 
#> Parameter | p (MAP)
#> -------------------
#> Posterior |  < .001
# \donttest{
model <- suppressWarnings(
  rstanarm::stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
)
p_map(model)
#> MAP-based p-value 
#> 
#> Parameter   | p (MAP)
#> ---------------------
#> (Intercept) |  < .001
#> wt          |  < .001
#> gear        |  0.832 

p_map(suppressWarnings(
  emmeans::emtrends(model, ~1, "wt", data = mtcars)
))
#> MAP-based p-value
#> 
#> Parameter | p (MAP)
#> -------------------
#> overall   |  < .001

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 7e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.07 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 4e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.04 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: 0.022 seconds (Warm-up)
#> Chain 2:                0.017 seconds (Sampling)
#> Chain 2:                0.039 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 4e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.04 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:                0.017 seconds (Sampling)
#> Chain 3:                0.037 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)
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#> Chain 4: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 4: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 4: 
#> Chain 4:  Elapsed Time: 0.021 seconds (Warm-up)
#> Chain 4:                0.018 seconds (Sampling)
#> Chain 4:                0.039 seconds (Total)
#> Chain 4: 
p_map(model)
#> MAP-based p-value 
#> 
#> Parameter   | p (MAP)
#> ---------------------
#> (Intercept) |  < .001
#> wt          |  < .001
#> cyl         |  0.003 

bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
p_map(bf)
#> MAP-based p-value
#> 
#> Parameter  | p (MAP)
#> --------------------
#> Difference |  < .001

# ---------------------------------------
# Robustness to density estimation method
set.seed(333)
data <- data.frame()
for (iteration in 1:250) {
  x <- rnorm(1000, 1, 1)
  result <- data.frame(
    Kernel = as.numeric(p_map(x, method = "kernel")),
    KernSmooth = as.numeric(p_map(x, method = "KernSmooth")),
    logspline = as.numeric(p_map(x, method = "logspline"))
  )
  data <- rbind(data, result)
}
data$KernSmooth <- data$Kernel - data$KernSmooth
data$logspline <- data$Kernel - data$logspline

summary(data$KernSmooth)
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#> -0.039649 -0.007867 -0.003854 -0.005315 -0.001114  0.056255 
summary(data$logspline)
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#> -0.092188 -0.008992  0.022235  0.026989  0.066329  0.166891 
boxplot(data[c("KernSmooth", "logspline")])

# }