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 = 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 = 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,
verbose = TRUE,
effects = c("fixed", "random", "all"),
component = c("conditional", "location", "smooth_terms", "sigma", "zi",
"zero_inflated", "all"),
parameters = NULL,
...
)
# S3 method for brmsfit
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
verbose = TRUE,
effects = c("fixed", "random", "all"),
component = c("conditional", "location", "smooth_terms", "sigma", "zi",
"zero_inflated", "all"),
parameters = NULL,
...
)
# S3 method for data.frame
bayesfactor_parameters(
posterior,
prior = NULL,
direction = "two-sided",
null = 0,
verbose = TRUE,
...
)
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).
Toggle off warnings.
Arguments passed to and from other methods. (Can be used to pass
arguments to internal logspline.)
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 data frame containing the Bayes factor representing evidence against the null.
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 or brmsfit 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 object:
prior should be the stanreg or brmsfit model used to create the emmGrid objects.
prior can also be an emmGrid object equivalent to posterior but created with a model of priors samples only.
Note: When the emmGrid has undergone any transformations ("log", "response", etc.), or regriding, then prior must be an emmGrid object, as stated above.
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).
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<U+2013>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<U+2013>298. 10.1177/1745691611406923
# NOT RUN {
library(bayestestR)
if (require("logspline")) {
prior <- distribution_normal(1000, mean = 0, sd = 1)
posterior <- distribution_normal(1000, mean = .5, sd = .3)
bayesfactor_parameters(posterior, prior)
}
# }
# NOT RUN {
# rstanarm models
# ---------------
if (require("rstanarm") && require("emmeans") && require("logspline")) {
contrasts(sleep$group) <- contr.bayes # see vingette
stan_model <- stan_lmer(extra ~ group + (1 | ID), data = sleep)
bayesfactor_parameters(stan_model)
bayesfactor_parameters(stan_model, null = rope_range(stan_model))
# emmGrid objects
# ---------------
group_diff <- pairs(emmeans(stan_model, ~group))
bayesfactor_parameters(group_diff, prior = stan_model)
}
# brms models
# -----------
if (require("brms")) {
contrasts(sleep$group) <- contr.bayes # 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 <- brm(extra ~ group + (1 | ID),
data = sleep,
prior = my_custom_priors
)
bayesfactor_parameters(brms_model)
}
# }
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