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, ...)
bf_restricted(posterior, ...)
# 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 'data.frame'
bayesfactor_restricted(
posterior,
hypothesis,
prior = NULL,
rvar_col = NULL,
...
)
# 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).- ...
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 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.
- rvar_col
A single character - the name of an
rvar
column in the data frame to be processed. See example inp_direction()
.- 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,
to match the posterior
-type input:
A numeric vector -
prior
should also be a numeric vector, representing the prior-estimate.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 anrvar
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. Seeunupdate()
.If
prior
is set toNULL
,unupdate()
is called internally (not supported forbrmsfit_multiple
model).
Output from a
{marginaleffects}
function -prior
should also be an equivalent output from a{marginaleffects}
function based on a prior-model (Seeunupdate()
).Output from an
{emmeans}
functionprior
should also be an equivalent output from an{emmeans}
function based on a prior-model (Seeunupdate()
).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 anyregrid
ing).
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 2.2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 1.2e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1.1e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.11 seconds.
#> Chain 3: Adjust your expectations accordingly!
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#> Chain 3:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1.2e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 4: Adjust your expectations accordingly!
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#> Chain 4:
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.
# }