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, null = 0, precision = 2^10, method = "kernel", ...)
p_pointnull(x, null = 0, precision = 2^10, method = "kernel", ...)
# 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.frame`

methods.- null
The value considered as a "null" effect. Traditionally 0, but could also be 1 in the case of ratios.

- precision
Number of points of density data. See the

`n`

parameter in`density`

.- method
Density estimation method. Can be

`"kernel"`

(default),`"logspline"`

or`"KernSmooth"`

.- ...
Currently not used.

- 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.

## Examples

```
library(bayestestR)
p_map(rnorm(1000, 0, 1))
#> MAP-based p-value: 0.99
p_map(rnorm(1000, 10, 1))
#> MAP-based p-value: 0.00
# \dontrun{
library(rstanarm)
model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
#> Warning: The largest R-hat is 1.09, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
p_map(model)
#> MAP-based p-value
#>
#> Parameter | p (MAP)
#> ---------------------
#> (Intercept) | < .001
#> wt | < .001
#> gear | 0.979
library(emmeans)
p_map(emtrends(model, ~1, "wt"))
#> MAP-based p-value
#>
#> Parameter | p (MAP)
#> -------------------
#> overall | < .001
library(brms)
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.3e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.13 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: 0.026158 seconds (Warm-up)
#> Chain 1: 0.02337 seconds (Sampling)
#> Chain 1: 0.049528 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 8e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.08 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: 0.026034 seconds (Warm-up)
#> Chain 2: 0.027555 seconds (Sampling)
#> Chain 2: 0.053589 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: 0.027076 seconds (Warm-up)
#> Chain 3: 0.025258 seconds (Sampling)
#> Chain 3: 0.052334 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 8e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.08 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: 0.026197 seconds (Warm-up)
#> Chain 4: 0.025577 seconds (Sampling)
#> Chain 4: 0.051774 seconds (Total)
#> Chain 4:
p_map(model)
#> MAP-based p-value
#>
#> Parameter | p (MAP)
#> ---------------------
#> (Intercept) | < .001
#> wt | < .001
#> cyl | 0.008
library(BayesFactor)
bf <- 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" = p_map(x, method = "kernel"),
"KernSmooth" = p_map(x, method = "KernSmooth"),
"logspline" = 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")])
# }
```