This function attempts to return, or compute, p-values of a model's parameters. See the documentation for your object's class:

## Usage

p_value(model, ...)

# S3 method for default
p_value(
model,
dof = NULL,
method = NULL,
component = "all",
vcov = NULL,
vcov_args = NULL,
verbose = TRUE,
...
)

# S3 method for emmGrid
p_value(model, ci = 0.95, adjust = "none", ...)

## Arguments

model

A statistical model.

...

dof

Number of degrees of freedom to be used when calculating confidence intervals. If NULL (default), the degrees of freedom are retrieved by calling degrees_of_freedom() with approximation method defined in method. If not NULL, use this argument to override the default degrees of freedom used to compute confidence intervals.

method

Method for computing degrees of freedom for confidence intervals (CI) and the related p-values. Allowed are following options (which vary depending on the model class): "residual", "normal", "likelihood", "satterthwaite", "kenward", "wald", "profile", "boot", "uniroot", "ml1", "betwithin", "hdi", "quantile", "ci", "eti", "si", "bci", or "bcai". See section Confidence intervals and approximation of degrees of freedom in model_parameters() for further details.

component

Model component for which parameters should be shown. See the documentation for your object's class in model_parameters() or p_value() for further details.

vcov

Variance-covariance matrix used to compute uncertainty estimates (e.g., for robust standard errors). This argument accepts a covariance matrix, a function which returns a covariance matrix, or a string which identifies the function to be used to compute the covariance matrix.

• A covariance matrix

• A function which returns a covariance matrix (e.g., stats::vcov())

• A string which indicates the kind of uncertainty estimates to return.

• Heteroskedasticity-consistent: "vcovHC", "HC", "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5". See ?sandwich::vcovHC.

• Cluster-robust: "vcovCR", "CR0", "CR1", "CR1p", "CR1S", "CR2", "CR3". See ?clubSandwich::vcovCR.

• Bootstrap: "vcovBS", "xy", "residual", "wild", "mammen", "webb". See ?sandwich::vcovBS.

• Other sandwich package functions: "vcovHAC", "vcovPC", "vcovCL", "vcovPL".

vcov_args

List of arguments to be passed to the function identified by the vcov argument. This function is typically supplied by the sandwich or clubSandwich packages. Please refer to their documentation (e.g., ?sandwich::vcovHAC) to see the list of available arguments.

verbose

Toggle warnings and messages.

ci

Confidence Interval (CI) level. Default to 0.95 (95%).

Character value naming the method used to adjust p-values or confidence intervals. See ?emmeans::summary.emmGrid for details.

## Value

A data frame with at least two columns: the parameter names and the p-values. Depending on the model, may also include columns for model components etc.

## Confidence intervals and approximation of degrees of freedom

There are different ways of approximating the degrees of freedom depending on different assumptions about the nature of the model and its sampling distribution. The ci_method argument modulates the method for computing degrees of freedom (df) that are used to calculate confidence intervals (CI) and the related p-values. Following options are allowed, depending on the model class:

Classical methods:

Classical inference is generally based on the Wald method. The Wald approach to inference computes a test statistic by dividing the parameter estimate by its standard error (Coefficient / SE), then comparing this statistic against a t- or normal distribution. This approach can be used to compute CIs and p-values.

"wald":

• Applies to non-Bayesian models. For linear models, CIs computed using the Wald method (SE and a t-distribution with residual df); p-values computed using the Wald method with a t-distribution with residual df. For other models, CIs computed using the Wald method (SE and a normal distribution); p-values computed using the Wald method with a normal distribution.

"normal"

• Applies to non-Bayesian models. Compute Wald CIs and p-values, but always use a normal distribution.

"residual"

• Applies to non-Bayesian models. Compute Wald CIs and p-values, but always use a t-distribution with residual df when possible. If the residual df for a model cannot be determined, a normal distribution is used instead.

Methods for mixed models:

Compared to fixed effects (or single-level) models, determining appropriate df for Wald-based inference in mixed models is more difficult. See the R GLMM FAQ for a discussion.

Several approximate methods for computing df are available, but you should also consider instead using profile likelihood ("profile") or bootstrap ("boot") CIs and p-values instead.

"satterthwaite"

• Applies to linear mixed models. CIs computed using the Wald method (SE and a t-distribution with Satterthwaite df); p-values computed using the Wald method with a t-distribution with Satterthwaite df.

"kenward"

• Applies to linear mixed models. CIs computed using the Wald method (Kenward-Roger SE and a t-distribution with Kenward-Roger df); p-values computed using the Wald method with Kenward-Roger SE and t-distribution with Kenward-Roger df.

"ml1"

• Applies to linear mixed models. CIs computed using the Wald method (SE and a t-distribution with m-l-1 approximated df); p-values computed using the Wald method with a t-distribution with m-l-1 approximated df. See ci_ml1().

"betwithin"

• Applies to linear mixed models and generalized linear mixed models. CIs computed using the Wald method (SE and a t-distribution with between-within df); p-values computed using the Wald method with a t-distribution with between-within df. See ci_betwithin().

Likelihood-based methods:

Likelihood-based inference is based on comparing the likelihood for the maximum-likelihood estimate to the the likelihood for models with one or more parameter values changed (e.g., set to zero or a range of alternative values). Likelihood ratios for the maximum-likelihood and alternative models are compared to a $$\chi$$-squared distribution to compute CIs and p-values.

"profile"

• Applies to non-Bayesian models of class glm, polr or glmmTMB. CIs computed by profiling the likelihood curve for a parameter, using linear interpolation to find where likelihood ratio equals a critical value; p-values computed using the Wald method with a normal-distribution (note: this might change in a future update!)

"uniroot"

• Applies to non-Bayesian models of class glmmTMB. CIs computed by profiling the likelihood curve for a parameter, using root finding to find where likelihood ratio equals a critical value; p-values computed using the Wald method with a normal-distribution (note: this might change in a future update!)

Methods for bootstrapped or Bayesian models:

Bootstrap-based inference is based on resampling and refitting the model to the resampled datasets. The distribution of parameter estimates across resampled datasets is used to approximate the parameter's sampling distribution. Depending on the type of model, several different methods for bootstrapping and constructing CIs and p-values from the bootstrap distribution are available.

For Bayesian models, inference is based on drawing samples from the model posterior distribution.

"quantile" (or "eti")

• Applies to all models (including Bayesian models). For non-Bayesian models, only applies if bootstrap = TRUE. CIs computed as equal tailed intervals using the quantiles of the bootstrap or posterior samples; p-values are based on the probability of direction. See bayestestR::eti().

"hdi"

• Applies to all models (including Bayesian models). For non-Bayesian models, only applies if bootstrap = TRUE. CIs computed as highest density intervals for the bootstrap or posterior samples; p-values are based on the probability of direction. See bayestestR::hdi().

"bci" (or "bcai")

• Applies to all models (including Bayesian models). For non-Bayesian models, only applies if bootstrap = TRUE. CIs computed as bias corrected and accelerated intervals for the bootstrap or posterior samples; p-values are based on the probability of direction. See bayestestR::bci().

"si"

• Applies to Bayesian models with proper priors. CIs computed as support intervals comparing the posterior samples against the prior samples; p-values are based on the probability of direction. See bayestestR::si().

"boot"

• Applies to non-Bayesian models of class merMod. CIs computed using parametric bootstrapping (simulating data from the fitted model); p-values computed using the Wald method with a normal-distribution) (note: this might change in a future update!).

For all iteration-based methods other than "boot" ("hdi", "quantile", "ci", "eti", "si", "bci", "bcai"), p-values are based on the probability of direction (bayestestR::p_direction()), which is converted into a p-value using bayestestR::pd_to_p().

## Examples

data(iris)
model <- lm(Petal.Length ~ Sepal.Length + Species, data = iris)
p_value(model)
#>           Parameter            p
#> 1       (Intercept) 1.005180e-11
#> 2      Sepal.Length 1.121002e-28
#> 3 Speciesversicolor 9.645641e-67
#> 4  Speciesvirginica 4.917626e-71