bayesfactor_restricted: 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, in particular on specifying correct priors for factors with more than 2 levels, see the Bayes factors vignette.

Usage

bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  ...
)
bf_restricted(posterior, hypothesis, prior = NULL, verbose = TRUE, ...)
# S3 method for stanreg
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  effects = c("fixed", "random", "all"),
  component = c("conditional", "zi", "zero_inflated", "all"),
  ...
)
# S3 method for brmsfit
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  effects = c("fixed", "random", "all"),
  component = c("conditional", "zi", "zero_inflated", "all"),
  ...
)
# S3 method for emmGrid
bayesfactor_restricted(
  posterior,
  hypothesis,
  prior = NULL,
  verbose = TRUE,
  ...
)

Arguments

posterior

A stanreg / brmsfit object, emmGrid or a data frame - representing a posterior distribution(s) from (see Details).

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.

...

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.

Value

A data frame containing the Bayes factor representing evidence against the un-restricted model.

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).

For the computation of Bayes factors, the model priors must be proper priors (at the very least they should be not flat, and it is preferable that they be informative); As the priors for the alternative get wider, the likelihood of the null value(s) increases, to the extreme that for completely flat priors the null is infinitely more favorable than the alternative (this is called the Jeffreys-Lindley-Bartlett paradox). Thus, you should only ever try (or want) to compute a Bayes factor when you have an informed prior.

(Note that by default, brms::brm() uses flat priors for fixed-effects.)

Setting the correct `prior`

It is important to provide the correct prior for meaningful results.

When posterior is a data.frame, prior should also be a data.frame, with matching column order.
When posterior is a stanreg or brmsfit model:
- prior can be set to NULL, in which case prior samples are drawn internally.
- prior can also be a model equvilant to posterior but with samples from the priors only.
When posterior is an emmGrid object:
- prior should be the stanreg or brmsfit model used to create the emmGrid objects.
- prior can also be an emmGrid object equvilant to posterior but created with a model of priors samples only.

Interpreting Bayes Factors

A Bayes factor greater than 1 can be interpereted 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-hypothesis) (Wetzels et al. 2011).

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 <U+2013> order restrictions. Retrived from https://richarddmorey.org/category/order-restrictions/.

Examples

Run this code

# NOT RUN {
library(bayestestR)
prior <- data.frame(
  X = rnorm(100),
  X1 = rnorm(100),
  X3 = rnorm(100)
)

posterior <- data.frame(
  X = rnorm(100, .4),
  X1 = rnorm(100, -.2),
  X3 = rnorm(100)
)

hyps <- c(
  "X > X1 & X1 > X3",
  "X > X1"
)

bayesfactor_restricted(posterior, hypothesis = hyps, prior = prior)
# }
# NOT RUN {
# rstanarm models
# ---------------
if (require("rstanarm") && require("emmeans")) {
  fit_stan <- stan_glm(mpg ~ wt + cyl + am,
    data = mtcars
  )
  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
  disgust_data <- read.table(url("http://www.learnbayes.org/disgust_example.txt"), header = TRUE)

  contrasts(disgust_data$condition) <- contr.bayes # see vignette
  fit_model <- stan_glm(score ~ condition, data = disgust_data, family = gaussian())

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

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

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