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Bayes Factors (BF) for Order Restricted Models

Source: R/bayesfactor_restricted.R
bayesfactor_restricted.Rd

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)?

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.)

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).

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 regriding, 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))
#> Bayes Factor (Order-Restriction)
#> 
#> Hypothesis    P(Prior) P(Posterior)    BF
#> A > B & B > C     0.16         0.23  1.39
#> A > B & A > C     0.36         0.59  1.61
#> C > A             0.46         0.34 0.742
#> 
#> * Bayes factors for the restricted model vs. the un-restricted model.

bool <- as.logical(b, which = "posterior")
head(bool)
#>      A > B & B > C A > B & A > C C > A
#> [1,]          TRUE          TRUE FALSE
#> [2,]          TRUE          TRUE FALSE
#> [3,]          TRUE          TRUE FALSE
#> [4,]         FALSE          TRUE FALSE
#> [5,]         FALSE         FALSE  TRUE
#> [6,]         FALSE          TRUE FALSE


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"
)

# \donttest{
# 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)
#> Sampling priors, please wait...
#> Bayes Factor (Order-Restriction)
#> 
#> Hypothesis       P(Prior) P(Posterior)    BF
#> am > 0 & cyl < 0     0.25         0.56  2.25
#> cyl < 0              0.50         1.00  1.99
#> wt - cyl > 0         0.50         0.10 0.197
#> 
#> * Bayes factors for the restricted model vs. the un-restricted model.
# }
# \donttest{
# 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())
#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
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#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 1e-05 seconds
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#> Chain 4: 
#> Chain 4: Gradient evaluation took 1.1e-05 seconds
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em_condition <- emmeans::emmeans(fit_model, ~condition, data = disgust)
hyps <- c("lemon < control & control < sulfur")

bayesfactor_restricted(em_condition, prior = fit_model, hypothesis = hyps)
#> Sampling priors, please wait...
#> Bayes Factor (Order-Restriction)
#> 
#> Hypothesis                         P(Prior) P(Posterior)   BF
#> lemon < control & control < sulfur     0.17         0.75 4.28
#> 
#> * Bayes factors for the restricted model vs. the un-restricted model.
# > # 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.
# }