Bayes Factors (BF) for a Single Parameter

Source: R/bayesfactor_parameters.R

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 class 'numeric'
bayesfactor_parameters(
  posterior,
  prior = NULL,
  direction = "two-sided",
  null = 0,
  ...,
  verbose = TRUE
)

# S3 method for class 'stanreg'
bayesfactor_parameters(
  posterior,
  prior = NULL,
  direction = "two-sided",
  null = 0,
  effects = "fixed",
  component = "conditional",
  parameters = NULL,
  ...,
  verbose = TRUE
)

# S3 method for class 'data.frame'
bayesfactor_parameters(
  posterior,
  prior = NULL,
  direction = "two-sided",
  null = 0,
  rvar_col = NULL,
  ...,
  verbose = TRUE
)

Arguments

posterior

A numerical vector, stanreg / brmsfit object, emmGrid or a data frame - representing a posterior distribution(s) from (see 'Details').

prior

An object representing a prior distribution (see 'Details').

direction

Test type (see 'Details'). One of 0, "two-sided" (default, two tailed), -1, "left" (left tailed) or 1, "right" (right tailed).

null

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().)

verbose

Toggle off warnings.

effects

Should variables for fixed effects ("fixed"), random effects ("random") or both ("all") be returned? Only applies to mixed models. May be abbreviated.

For models of from packages brms or rstanarm there are additional options:

  • "fixed" returns fixed effects.

  • "random_variance" return random effects parameters (variance and correlation components, e.g. those parameters that start with sd_ or cor_).

  • "grouplevel" returns random effects group level estimates, i.e. those parameters that start with r_.

  • "random" returns both "random_variance" and "grouplevel".

  • "all" returns fixed effects and random effects variances.

  • "full" returns all parameters.

component

Which type of parameters to return, such as parameters for the conditional model, the zero-inflated part of the model, the dispersion term, etc. See details in section Model Components. May be abbreviated. Note that the conditional component also refers to the count or mean component - names may differ, depending on the modeling package. There are three convenient shortcuts (not applicable to all model classes):

  • component = "all" returns all possible parameters.

  • If component = "location", location parameters such as conditional, zero_inflated, smooth_terms, or instruments are returned (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).

  • For component = "distributional" (or "auxiliary"), components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

parameters

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.

rvar_col

A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

Value

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).

Details

This method is used to compute Bayes factors based on prior and posterior distributions.

One-sided & Dividing Tests (setting an order restriction)

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)).

Note

There is also a plot()-method implemented in the see-package.

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 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).

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).

Model components

Possible values for the component argument depend on the model class. Following are valid options:

  • "all": returns all model components, applies to all models, but will only have an effect for models with more than just the conditional model component.

  • "conditional": only returns the conditional component, i.e. "fixed effects" terms from the model. Will only have an effect for models with more than just the conditional model component.

  • "smooth_terms": returns smooth terms, only applies to GAMs (or similar models that may contain smooth terms).

  • "zero_inflated" (or "zi"): returns the zero-inflation component.

  • "location": returns location parameters such as conditional, zero_inflated, or smooth_terms (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).

  • "distributional" (or "auxiliary"): components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

For models of class brmsfit (package brms), even more options are possible for the component argument, which are not all documented in detail here. See also ?insight::find_parameters.

References

  • 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. doi:10.1177/1745691611406923

Author

Mattan S. Ben-Shachar

Examples

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))
#> Bayes Factor (Savage-Dickey density ratio)
#> 
#> BF  
#> ----
#> 1.21
#> 
#> * Evidence Against The Null: 0

as.numeric(BF_pars)
#> [1] 1.212843
# \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)
#> Bayes Factor (Savage-Dickey density ratio)
#> 
#> Parameter   |   BF
#> ------------------
#> (Intercept) | 3.23
#> group1      | 2.77
#> 
#> * Evidence Against The Null: 0
bayesfactor_parameters(stan_model, null = rope_range(stan_model))
#> Sampling priors, please wait...
#> Bayes Factor (Null-Interval)
#> 
#> Parameter   |   BF
#> ------------------
#> (Intercept) | 3.35
#> group1      | 2.52
#> 
#> * Evidence Against The Null: [-0.202, 0.202]

# emmGrid objects
# ---------------
group_diff <- pairs(emmeans(stan_model, ~group, data = sleep))
bayesfactor_parameters(group_diff, prior = stan_model, verbose = FALSE)
#> Bayes Factor (Savage-Dickey density ratio)
#> 
#> contrast        |   BF
#> ----------------------
#> group1 - group2 | 3.16
#> 
#> * Evidence Against The Null: 0

# Or
# group_diff_prior <- pairs(emmeans(unupdate(stan_model), ~group))
# bayesfactor_parameters(group_diff, prior = group_diff_prior, verbose = FALSE)
# }
# brms models
# -----------
if (FALSE) { # \dontrun{
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)
} # }