p_map: Bayesian p-value based on the density at the Maximum A...

In bayestestR: Understand and Describe Bayesian Models and Posterior Distributions

Bayesian p-value based on the density at the Maximum A Posteriori (MAP)

Description

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 class 'numeric'
p_map(x, null = 0, precision = 2^10, method = "kernel", ...)

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

## S3 method for class 'data.frame'
p_map(x, null = 0, precision = 2^10, method = "kernel", rvar_col = NULL, ...)

## S3 method for class '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 class '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.
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, 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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3389/fpsyg.2019.02767")}

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

See Also

Jeff Mill's talk

Examples

library(bayestestR)

p_map(rnorm(1000, 0, 1))
p_map(rnorm(1000, 10, 1))

model <- suppressWarnings(
  rstanarm::stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
)
p_map(model)

p_map(suppressWarnings(
  emmeans::emtrends(model, ~1, "wt", data = mtcars)
))

model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
p_map(model)

bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
p_map(bf)

# ---------------------------------------
# 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$KernSmooth)
summary(data$logspline)
boxplot(data[c("KernSmooth", "logspline")])

bayestestR documentation built on Oct. 17, 2024, 5:07 p.m.