This function computes or extracts Bayes factors from fitted
models. The bf_* function is an alias of the main function.
Usage
bayesfactor_models(..., denominator = 1, verbose = TRUE)
bf_models(..., denominator = 1, verbose = TRUE)
# Default S3 method
bayesfactor_models(..., denominator = 1, verbose = TRUE)
# S3 method for class 'bayesfactor_models'
update(object, subset = NULL, reference = NULL, ...)
# S3 method for class 'bayesfactor_models'
as.matrix(x, ...)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.
- object, x
A
bayesfactor_models()object.- subset
Vector of model indices to keep or remove.
- reference
Index of model to reference to, or
"top"to reference to the best model, or"bottom"to reference to the worst model.
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 mostly 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.
In order to correctly and precisely estimate Bayes factors, a rule of thumb
are the 4 P's: Proper Priors and Plentiful
Posteriors. How many? 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, bayesfactor_models() gives a warning.
See also the Bayes factors vignette.
Note
There is also a plot()-method
implemented in the see-package.
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:
$$BF_{AB} = \frac{BF_{AC}}{BF_{BC}} = \frac{\frac{ML_{A}}{ML_{C}}}{\frac{ML_{B}}{ML_{C}}} = \frac{ML_{A}}{ML_{B}}$$
A full matrix comparing all models can be obtained with as.matrix() (see
examples).
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
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
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
#> [lm2] hp 4.54e+05
#> [lm3] hp + drat 7.70e+07
#> [lm4] hp * drat 1.59e+07
#>
#> * Against Denominator: [lm1] (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
#> [lm2] hp 4.54e+05
#> [lm3] hp + drat 7.70e+07
#> [lm4] hp * drat 1.59e+07
#>
#> * Against Denominator: [lm1] (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
#> [lm2b] hp 6.94
#>
#> * Against Denominator: [lm2] 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
#> [lmer2] Petal.Length + (Petal.Length | Species) 0.058
#> [lmer3] Petal.Length + (Petal.Length | Species) + (1 | Petal.Width) 0.005
#>
#> * Against Denominator: [lmer1] 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")
))
#>
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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
#>
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brm2 <- brms::brm(Sepal.Length ~ Species, data = iris, save_pars = save_pars(all = TRUE))
#> Compiling Stan program...
#> Start sampling
#>
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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).
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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)
# }