This function computes or extracts Bayes factors from fitted
models.
The bf_* function is an alias of the main function.
For more info, see the Bayes factors vignette.
Usage
bayesfactor_models(..., denominator = 1, verbose = TRUE)
bf_models(..., denominator = 1, verbose = TRUE)
# Default S3 method
bayesfactor_models(..., denominator = 1, verbose = TRUE)Arguments
- ...
Fitted models (see details), all fit on the same data, or a single
BFBayesFactorobject (see 'Details'). Ignored inas.matrix(),update(). If the following named arguments are present, they are passed toinsight::get_loglikelihood()(see details):estimator(defaults to"ML")check_response(defaults toFALSE)
- denominator
Either an integer indicating which of the models to use as the denominator, or a model to be used as a denominator. Ignored for
BFBayesFactor.- verbose
Toggle off warnings.
Value
A data frame containing the models' formulas (reconstructed fixed and
random effects) and their log(BF)s (Use as.numeric() to extract the
non-log Bayes factors; see examples), that prints nicely.
For as.matrix() a square matrix of (log) Bayes factors, with rows as
denominators and columns as numerators.
Details
If the passed models are supported by insight the DV of all models will be tested for equality (else this is assumed to be true), and the models' terms will be extracted (allowing for follow-up analysis with bayesfactor_inclusion).
For
brmsfitorstanregmodels, Bayes factors are computed using the bridgesampling package.brmsfitmodels must have been fitted withsave_pars = save_pars(all = TRUE).stanregmodels must have been fitted with a defineddiagnostic_file.
For
BFBayesFactor,bayesfactor_models()is a wraparoundBayesFactor::extractBF().For all other model types, Bayes factors are computed using the BIC approximation. Note that BICs are extracted from using insight::get_loglikelihood, see documentation there for options for dealing with transformed responses and REML estimation.
Additional methods
The resulting output is supported by the following methods:
as.matrix(): Extract a full matrix of (log-)Bayes factors between all models (using the transitivity of Bayes factors).update(): subset and/or re-reference the Bayes factors to a different model.as.numeric(): Extract the (possibly log-)Bayes factor values.
See examples and bayesfactor_methods.
Note
There is also a plot()-method
implemented in the see-package.
Prior and posterior considerations
In order to correctly and precisely estimate Bayes factors, a rule of thumb
are the 4 P's: Proper Priors and Plentiful
Posteriors.
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) (Note that by default, brms::brm() uses flat priors for
fixed-effects); Wide priors result in smaller marginal likelihoods, and thus
models with wider priors are trivially less likely than models with narrower
priors - where, at the extreme, that a model with completely flat priors is
infinitely less favorable than a point null model (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.
Additionally, for models using MCMC estimation the number of posterior
samples needed for testing is substantially larger than for estimation (the
default of 4000 samples may not be enough in many cases). A conservative rule
of thumb is to obtain 10 times more samples than would be required for
estimation (Gronau, Singmann, & Wagenmakers, 2017). If less than 40,000
samples are detected, a warning is issued.
Transitivity of Bayes factors
For multiple inputs (models or hypotheses), the function will return multiple
Bayes factors between each model and the same reference model (the
denominator or un-restricted model). However, we can take advantage of the
transitivity of Bayes factors - where if we have two Bayes factors for Model
A and model B against the same reference model C, we can obtain a Bayes
factor for comparing model A to model B by dividing them:
$$BF_{AB} = \frac{BF_{AC}}{BF_{BC}} = \frac{\frac{ML_{A}}{ML_{C}}}{\frac{ML_{B}}{ML_{C}}} = \frac{ML_{A}}{ML_{B}}$$
(Where ML is the marginal likelihood.)
A full matrix comparing all models can be obtained with as.matrix().
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). See also effectsize::interpret_bf().
References
Gronau, Q. F., Singmann, H., & Wagenmakers, E. J. (2017). Bridgesampling: An R package for estimating normalizing constants. arXiv preprint arXiv:1710.08162.
Kass, R. E., and Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773-795.
Robert, C. P. (2016). The expected demise of the Bayes factor. Journal of Mathematical Psychology, 72, 33–37.
Wagenmakers, E. J. (2007). A practical solution to the pervasive problems of p values. Psychonomic bulletin & review, 14(5), 779-804.
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
See also
bayesfactor_inclusion() for testing predictors across Bayesian models.
Other Bayes factors:
bayesfactor_inclusion(),
bayesfactor_parameters(),
bayesfactor_restricted()
Examples
# With lm objects:
# ----------------
lm1 <- lm(mpg ~ 1, data = mtcars)
lm2 <- lm(mpg ~ hp, data = mtcars)
lm3 <- lm(mpg ~ hp + drat, data = mtcars)
lm4 <- lm(mpg ~ hp * drat, data = mtcars)
(BFM <- bayesfactor_models(lm1, lm2, lm3, lm4, denominator = 1))
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [2] hp 4.54e+05
#> [3] hp + drat 7.70e+07
#> [4] hp * drat 1.59e+07
#>
#> * Against Denominator: [1] (Intercept only)
#> * Bayes Factor Type: BIC approximation
# bayesfactor_models(lm2, lm3, lm4, denominator = lm1) # same result
# bayesfactor_models(lm1, lm2, lm3, lm4, denominator = lm1) # same result
update(BFM, reference = "bottom")
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [2] hp 4.54e+05
#> [3] hp + drat 7.70e+07
#> [4] hp * drat 1.59e+07
#>
#> * Against Denominator: [1] (Intercept only)
#> * Bayes Factor Type: BIC approximation
as.matrix(BFM)
#> # Bayes Factors for Model Comparison
#>
#> Denominator\Numerator | [1] | [2] | [3] | [4]
#> -----------------------------------------------------------------
#> [1] (Intercept only) | 1 | 4.54e+05 | 7.70e+07 | 1.59e+07
#> [2] hp | 2.20e-06 | 1 | 169.72 | 35.09
#> [3] hp + drat | 1.30e-08 | 0.006 | 1 | 0.207
#> [4] hp * drat | 6.28e-08 | 0.028 | 4.84 | 1
as.numeric(BFM)
#> [1] 1.0 453874.3 77029881.3 15925712.4
lm2b <- lm(sqrt(mpg) ~ hp, data = mtcars)
# Set check_response = TRUE for transformed responses
bayesfactor_models(lm2b, denominator = lm2, check_response = TRUE)
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [1] hp 6.94
#>
#> * Against Denominator: [2] hp
#> * Bayes Factor Type: BIC approximation
# \donttest{
# With lmerMod objects:
# ---------------------
lmer1 <- lme4::lmer(Sepal.Length ~ Petal.Length + (1 | Species), data = iris)
lmer2 <- lme4::lmer(Sepal.Length ~ Petal.Length + (Petal.Length | Species), data = iris)
#> boundary (singular) fit: see help('isSingular')
lmer3 <- lme4::lmer(
Sepal.Length ~ Petal.Length + (Petal.Length | Species) + (1 | Petal.Width),
data = iris
)
#> boundary (singular) fit: see help('isSingular')
bayesfactor_models(lmer1, lmer2, lmer3,
denominator = 1,
estimator = "REML"
)
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [2] Petal.Length + (Petal.Length | Species) 0.058
#> [3] Petal.Length + (Petal.Length | Species) + (1 | Petal.Width) 0.005
#>
#> * Against Denominator: [1] Petal.Length + (1 | Species)
#> * Bayes Factor Type: BIC approximation
# rstanarm models
# ---------------------
# (note that a unique diagnostic_file MUST be specified in order to work)
stan_m0 <- suppressWarnings(rstanarm::stan_glm(Sepal.Length ~ 1,
data = iris,
family = gaussian(),
diagnostic_file = file.path(tempdir(), "df0.csv")
))
#>
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stan_m1 <- suppressWarnings(rstanarm::stan_glm(Sepal.Length ~ Species,
data = iris,
family = gaussian(),
diagnostic_file = file.path(tempdir(), "df1.csv")
))
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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stan_m2 <- suppressWarnings(rstanarm::stan_glm(Sepal.Length ~ Species + Petal.Length,
data = iris,
family = gaussian(),
diagnostic_file = file.path(tempdir(), "df2.csv")
))
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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bayesfactor_models(stan_m1, stan_m2, denominator = stan_m0, verbose = FALSE)
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [1] Species 6.27e+27
#> [2] Species + Petal.Length 2.25e+53
#>
#> * Against Denominator: [3] (Intercept only)
#> * Bayes Factor Type: marginal likelihoods (bridgesampling)
# brms models
# --------------------
# (note the save_pars MUST be set to save_pars(all = TRUE) in order to work)
brm1 <- brms::brm(Sepal.Length ~ 1, data = iris, save_pars = save_pars(all = TRUE))
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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brm2 <- brms::brm(Sepal.Length ~ Species, data = iris, save_pars = save_pars(all = TRUE))
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 8e-06 seconds
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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brm3 <- brms::brm(
Sepal.Length ~ Species + Petal.Length,
data = iris,
save_pars = save_pars(all = TRUE)
)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.1e-05 seconds
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bayesfactor_models(brm1, brm2, brm3, denominator = 1, verbose = FALSE)
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [2] Species 5.86e+29
#> [3] Species + Petal.Length 7.50e+55
#>
#> * Against Denominator: [1] (Intercept only)
#> * Bayes Factor Type: marginal likelihoods (bridgesampling)
# BayesFactor
# ---------------------------
data(puzzles)
BF <- BayesFactor::anovaBF(RT ~ shape * color + ID,
data = puzzles,
whichRandom = "ID", progress = FALSE
)
BF
#> Bayes factor analysis
#> --------------
#> [1] shape + ID : 2.841658 ±0.92%
#> [2] color + ID : 2.830879 ±0.86%
#> [3] shape + color + ID : 11.75567 ±1.98%
#> [4] shape + color + shape:color + ID : 4.371906 ±1.99%
#>
#> Against denominator:
#> RT ~ ID
#> ---
#> Bayes factor type: BFlinearModel, JZS
#>
bayesfactor_models(BF) # basically the same
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [2] shape + ID 2.84
#> [3] color + ID 2.83
#> [4] shape + color + ID 11.76
#> [5] shape + color + shape:color + ID 4.37
#>
#> * Against Denominator: [1] ID
#> * Bayes Factor Type: JZS (BayesFactor)
# }