This method computes Bayes factors against the null (either a point or an
interval), based on prior and posterior samples of a single parameter. This
Bayes factor indicates the degree by which the mass of the posterior
distribution has shifted further away from or closer to the null value(s)
(relative to the prior distribution), thus indicating if the null value has
become less or more likely given the observed data.
When the null is an interval, the Bayes factor is computed by comparing the
prior and posterior odds of the parameter falling within or outside the null
interval (Morey & Rouder, 2011; Liao et al., 2020); When the null is a point,
a Savage-Dickey density ratio is computed, which is also an approximation of
a Bayes factor comparing the marginal likelihoods of the model against a
model in which the tested parameter has been restricted to the point null
(Wagenmakers et al., 2010; Heck, 2019).
Note that the logspline package is used for estimating densities and
probabilities, and must be installed for the function to work.
bayesfactor_pointnull() and bayesfactor_rope() are wrappers
around bayesfactor_parameters with different defaults for the null to
be tested against (a point and a range, respectively). Aliases of the main
functions are prefixed with bf_*, like bf_parameters() or
bf_pointnull().
For more info, in particular on specifying correct priors for factors
with more than 2 levels, see
the Bayes factors vignette.
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)bayesfactor_pointnull(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)
bayesfactor_rope(
posterior,
prior = NULL,
direction = "two-sided",
null = rope_range(posterior, verbose = FALSE),
...,
verbose = TRUE
)
bf_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)
bf_pointnull(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)
bf_rope(
posterior,
prior = NULL,
direction = "two-sided",
null = rope_range(posterior, verbose = FALSE),
...,
verbose = TRUE
)
# S3 method for numeric
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)
# S3 method for stanreg
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
effects = c("fixed", "random", "all"),
component = c("conditional", "location", "smooth_terms", "sigma", "zi",
"zero_inflated", "all"),
parameters = NULL,
...,
verbose = TRUE
)
# S3 method for brmsfit
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
effects = c("fixed", "random", "all"),
component = c("conditional", "location", "smooth_terms", "sigma", "zi",
"zero_inflated", "all"),
parameters = NULL,
...,
verbose = TRUE
)
# S3 method for blavaan
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
...,
verbose = TRUE
)
# S3 method for data.frame
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
rvar_col = NULL,
...,
verbose = TRUE
)
A data frame containing the (log) Bayes factor representing evidence
against the null (Use as.numeric() to extract the non-log Bayes
factors; see examples).
A numerical vector, stanreg / brmsfit object,
emmGrid or a data frame - representing a posterior distribution(s)
from (see 'Details').
An object representing a prior distribution (see 'Details').
Test type (see 'Details'). One of 0,
"two-sided" (default, two tailed), -1, "left" (left
tailed) or 1, "right" (right tailed).
Value of the null, either a scalar (for point-null) or a range (for a interval-null).
Arguments passed to and from other methods. (Can be used to pass
arguments to internal logspline::logspline().)
Toggle off warnings.
Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated.
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.
Regular expression pattern that describes the parameters
that should be returned. Meta-parameters (like lp__ or prior_) are
filtered by default, so only parameters that typically appear in the
summary() are returned. Use parameters to select specific parameters
for the output.
A single character - the name of an rvar column in the data
frame to be processed. See example in p_direction().
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 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).
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).
Mattan S. Ben-Shachar
This method is used to compute Bayes factors based on prior and posterior distributions.
One sided tests (controlled by direction) are conducted by restricting
the prior and posterior of the non-null values (the "alternative") to one
side of the null only (Morey & Wagenmakers, 2014). For example, if we
have a prior hypothesis that the parameter should be positive, the
alternative will be restricted to the region to the right of the null (point
or interval). For example, for a Bayes factor comparing the "null" of 0-0.1
to the alternative >0.1, we would set
bayesfactor_parameters(null = c(0, 0.1), direction = ">").
It is also possible to compute a Bayes factor for dividing
hypotheses - that is, for a null and alternative that are complementary,
opposing one-sided hypotheses (Morey & Wagenmakers, 2014). For
example, for a Bayes factor comparing the "null" of <0 to the alternative
>0, we would set bayesfactor_parameters(null = c(-Inf, 0)).
Wagenmakers, E. J., Lodewyckx, T., Kuriyal, H., and Grasman, R. (2010). Bayesian hypothesis testing for psychologists: A tutorial on the Savage-Dickey method. Cognitive psychology, 60(3), 158-189.
Heck, D. W. (2019). A caveat on the Savage–Dickey density ratio: The case of computing Bayes factors for regression parameters. British Journal of Mathematical and Statistical Psychology, 72(2), 316-333.
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.
Liao, J. G., Midya, V., & Berg, A. (2020). Connecting and contrasting the Bayes factor and a modified ROPE procedure for testing interval null hypotheses. The American Statistician, 1-19.
Wetzels, R., Matzke, D., Lee, M. D., Rouder, J. N., Iverson, G. J., and Wagenmakers, E.-J. (2011). Statistical Evidence in Experimental Psychology: An Empirical Comparison Using 855 t Tests. Perspectives on Psychological Science, 6(3), 291–298. tools:::Rd_expr_doi("10.1177/1745691611406923")
if (FALSE) { # require("logspline")
library(bayestestR)
prior <- distribution_normal(1000, mean = 0, sd = 1)
posterior <- distribution_normal(1000, mean = .5, sd = .3)
(BF_pars <- bayesfactor_parameters(posterior, prior, verbose = FALSE))
as.numeric(BF_pars)
}
if (FALSE) { # require("rstanarm") && require("emmeans") && require("logspline")
# \donttest{
# rstanarm models
# ---------------
contrasts(sleep$group) <- contr.equalprior_pairs # see vingette
stan_model <- suppressWarnings(stan_lmer(
extra ~ group + (1 | ID),
data = sleep,
refresh = 0
))
bayesfactor_parameters(stan_model, verbose = FALSE)
bayesfactor_parameters(stan_model, null = rope_range(stan_model))
# emmGrid objects
# ---------------
group_diff <- pairs(emmeans(stan_model, ~group, data = sleep))
bayesfactor_parameters(group_diff, prior = stan_model, verbose = FALSE)
# Or
# group_diff_prior <- pairs(emmeans(unupdate(stan_model), ~group))
# bayesfactor_parameters(group_diff, prior = group_diff_prior, verbose = FALSE)
# }
}
if (FALSE) { # require("brms") && require("logspline")
# brms models
# -----------
if (FALSE) {
contrasts(sleep$group) <- contr.equalprior_pairs # see vingette
my_custom_priors <-
set_prior("student_t(3, 0, 1)", class = "b") +
set_prior("student_t(3, 0, 1)", class = "sd", group = "ID")
brms_model <- suppressWarnings(brm(extra ~ group + (1 | ID),
data = sleep,
prior = my_custom_priors,
refresh = 0
))
bayesfactor_parameters(brms_model, verbose = FALSE)
}
}
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