# bayesfactor_restricted: Bayes Factors (BF) for Order Restricted Models In easystats/bayestestR: Understand and Describe Bayesian Models and Posterior Distributions

 bayesfactor_restricted R Documentation

## 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 class 'stanreg'
bayesfactor_restricted(
posterior,
hypothesis,
prior = NULL,
verbose = TRUE,
effects = c("fixed", "random", "all"),
component = c("conditional", "zi", "zero_inflated", "all"),
...
)

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

## 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 'bayesfactor_restricted'
as.logical(x, which = c("posterior", "prior"), ...)
``````

### 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. `x` An object of class `bayesfactor_restricted` `which` Should the logical matrix be of the posterior or prior distribution(s)?

### 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).

### 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.)

### Setting the correct `prior`

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; See example below.)

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

• When `posterior` is a numerical vector, `prior` should also be a numerical vector.

• When `posterior` is a `data.frame`, `prior` should also be a `data.frame`, with matching column order.

• When `posterior` is a `stanreg`, `brmsfit` or other supported Bayesian model:

• `prior` can be set to `NULL`, in which case prior samples are drawn internally.

• `prior` can also be a model equivalent to `posterior` but with samples from the priors only. See `unupdate()`.

• Note: When `posterior` is a `brmsfit_multiple` model, `prior` must be provided.

• When `posterior` is an `emmGrid` / `emm_list` object:

• `prior` should also be an `emmGrid` / `emm_list` object equivalent to `posterior` but created with a model of priors samples only. See `unupdate()`.

• `prior` can also be the original (posterior) model. If so, the function will try to update the `emmGrid` / `emm_list` to use the `unupdate()`d prior-model. (This cannot be done for `brmsfit` models.)

• Note: When the `emmGrid` has undergone any transformations (`"log"`, `"response"`, etc.), or `regrid`ing, then `prior` must be an `emmGrid` object, as stated above.

### 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) (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 – order restrictions. Retrieved from https://richarddmorey.org/category/order-restrictions/.

### 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))

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

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.

``````

easystats/bayestestR documentation built on Aug. 1, 2024, 9:41 a.m.