bayesfactor_restricted: Bayes Factors (BF) for Order Restricted Models

In bayestestR: Understand and Describe Bayesian Models and Posterior Distributions

Bayes Factors (BF) for Order Restricted Models

Description

This method computes Bayes factors for comparing a model with an order restrictions on its parameters with the fully unrestricted model. Note that this method should only be used for confirmatory analyses.

The ⁠bf_*⁠ function is an alias of the main function.

For more info, see the Bayes factors vignette.

Usage

bayesfactor_restricted(posterior, ...)

bf_restricted(posterior, ...)

## S3 method for class 'stanreg'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  effects = "fixed",
  component = "conditional",
  ...
)

## S3 method for class 'brmsfit'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  effects = "fixed",
  component = "conditional",
  ...
)

## S3 method for class 'blavaan'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  ...
)

## S3 method for class 'emmGrid'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  ...
)

## S3 method for class 'matrix'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  ...
)

## S3 method for class 'data.frame'
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  rvar_col = NULL,
  ...
)

Arguments

posterior	A stanreg / brmsfit object, emmGrid or a data frame - representing a posterior distribution(s) from (see Details).
...	Currently not used.
hypothesis	A character vector specifying the restrictions as logical conditions (see examples below).
prior	An object representing a prior distribution (see Details).
verbose	Toggle off warnings.
effects	Should variables for fixed effects ("fixed"), random effects ("random") or both ("all") be returned? Only applies to mixed models. May be abbreviated. For models of from packages brms or rstanarm there are additional options: "fixed" returns fixed effects. "random_variance" return random effects parameters (variance and correlation components, e.g. those parameters that start with sd_ or cor_). "grouplevel" returns random effects group level estimates, i.e. those parameters that start with r_. "random" returns both "random_variance" and "grouplevel". "all" returns fixed effects and random effects variances. "full" returns all parameters.
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.
rvar_col	A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

Details

This method is used to compute Bayes factors for order-restricted models vs un-restricted models by setting an order restriction on the prior and posterior distributions (Morey & Wagenmakers, 2013).

(Though it is possible to use bayesfactor_restricted() to test interval restrictions, it is more suitable for testing order restrictions; see examples).

Additional methods

The resulting output is supported by the following methods:

• as.matrix(): Extract a full matrix of (log-)Bayes factors between all models (using the transitivity of Bayes factors).

• as.logical(): Extract boolean vectors indicating which (prior/posterior) samples are included in the hypothesized restriction.

• as.numeric(): Extract the (possibly log-)Bayes factor values.

See examples and bayesfactor_methods.

Value

A data frame containing the (log) Bayes factor representing evidence against the un-restricted model (Use as.numeric() to extract the non-log Bayes factors; see examples). (A bool_results attribute contains the results for each sample, indicating if they are included or not in the hypothesized restriction.)

Obtaining prior samples

It is important to provide the correct prior for meaningful results, to match the posterior-type input:

• A numeric vector - prior should also be a numeric vector, representing the prior-distribution

• A data frame - prior should also be a data frame, representing the prior-estimates, in matching column order.

  • If rvar_col is specified, prior should be the name of an rvar column that represents the prior-estimates.

• Supported Bayesian model (stanreg, brmsfit, etc.)

  • prior should be a model an equivalent model with MCMC samples from the priors only. See unupdate().

  • If prior is set to NULL, unupdate() is called internally (not supported for brmsfit_multiple model).

• Output from a {marginaleffects} function - prior should also be an equivalent output from a {marginaleffects} function based on a prior-model (See unupdate()).

• Output from an {emmeans} function

  • prior should also be an equivalent output from an {emmeans} function based on a prior-model (See unupdate()).

  • prior can also be the original (posterior) model, in which case the function will try to "unupdate" the estimates (not supported if the estimates have undergone any transformations – "log", "response", etc. – or any regriding).

Transitivity of Bayes factors

For multiple inputs (models or hypotheses), the function will return multiple Bayes factors between each model and the same reference model (the denominator or un-restricted model). However, we can take advantage of the transitivity of Bayes factors - where if we have two Bayes factors for Model A and model B against the same reference model C, we can obtain a Bayes factor for comparing model A to model B by dividing them:

BF_{AB} = \frac{BF_{AC}}{BF_{BC}} = \frac{\frac{ML_{A}}{ML_{C}}}{\frac{ML_{B}}{ML_{C}}} = \frac{ML_{A}}{ML_{B}}

(Where ML is the marginal likelihood.)

A full matrix comparing all models can be obtained with as.matrix().

Interpreting Bayes Factors

A Bayes factor greater than 1 can be interpreted as evidence against the null, at which one convention is that a Bayes factor greater than 3 can be considered as "substantial" evidence against the null (and vice versa, a Bayes factor smaller than 1/3 indicates substantial evidence in favor of the null-model). See also effectsize::interpret_bf().

References

• Morey, R. D., & Wagenmakers, E. J. (2014). Simple relation between Bayesian order-restricted and point-null hypothesis tests. Statistics & Probability Letters, 92, 121-124.

• Morey, R. D., & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. Psychological methods, 16(4), 406.

• Morey, R. D. (Jan, 2015). Multiple Comparisons with BayesFactor, Part 2 – order restrictions. Retrieved from https://richarddmorey.org/category/order-restrictions/.

See Also

Other Bayes factors: bayesfactor_inclusion(), bayesfactor_models(), bayesfactor_parameters()

Examples

set.seed(444)
library(bayestestR)
prior <- data.frame(
  A = rnorm(500),
  B = rnorm(500),
  C = rnorm(500)
)

posterior <- data.frame(
  A = rnorm(500, .4, 0.7),
  B = rnorm(500, -.2, 0.4),
  C = rnorm(500, 0, 0.5)
)

hyps <- c(
  "A > B & B > C",
  "A > B & A > C",
  "C > A"
)


(b <- bayesfactor_restricted(posterior, hypothesis = hyps, prior = prior))

# See the matrix of BFs
as.matrix(b)

bool <- as.logical(b, which = "posterior")
head(bool)



see::plots(
  plot(estimate_density(posterior)),
  # distribution **conditional** on the restrictions
  plot(estimate_density(posterior[bool[, hyps[1]], ])) + ggplot2::ggtitle(hyps[1]),
  plot(estimate_density(posterior[bool[, hyps[2]], ])) + ggplot2::ggtitle(hyps[2]),
  plot(estimate_density(posterior[bool[, hyps[3]], ])) + ggplot2::ggtitle(hyps[3]),
  guides = "collect"
)



# rstanarm models
# ---------------
data("mtcars")

fit_stan <- rstanarm::stan_glm(mpg ~ wt + cyl + am,
  data = mtcars, refresh = 0
)
hyps <- c(
  "am > 0 & cyl < 0",
  "cyl < 0",
  "wt - cyl > 0"
)

bayesfactor_restricted(fit_stan, hypothesis = hyps)




# emmGrid objects
# ---------------
# replicating http://bayesfactor.blogspot.com/2015/01/multiple-comparisons-with-bayesfactor-2.html
data("disgust")
contrasts(disgust$condition) <- contr.equalprior_pairs # see vignette
fit_model <- rstanarm::stan_glm(score ~ condition, data = disgust, family = gaussian())

em_condition <- emmeans::emmeans(fit_model, ~condition, data = disgust)
hyps <- c("lemon < control & control < sulfur")

bayesfactor_restricted(em_condition, prior = fit_model, hypothesis = hyps)
# > # Bayes Factor (Order-Restriction)
# >
# >                          Hypothesis P(Prior) P(Posterior)   BF
# >  lemon < control & control < sulfur     0.17         0.75 4.49
# > ---
# > Bayes factors for the restricted model vs. the un-restricted model.

bayestestR documentation built on May 22, 2026, 1:06 a.m.