BF: Bayes factors for Bayesian exploratory and confirmatory...

In BFpack: Flexible Bayes Factor Testing of Scientific Expectations

Bayes factors for Bayesian exploratory and confirmatory hypothesis testing

Description

The BF function can be used for hypothesis testing and model selection using the Bayes factor. By default exploratory hypothesis tests are performed of whether each model parameter equals zero, is negative, or is positive. Confirmatory hypothesis tests can be executed by specifying hypotheses with equality and/or order constraints on the parameters of interest. Depending on the class of the fitted model different Bayes factors are used as described in the output.

Usage

## Default S3 method:
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  Sigma,
  n,
  ...
)

## S3 method for class 'lm'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type = NULL,
  iter = 10000,
  ...
)

## S3 method for class 'rma.uni'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type,
  iter = 20000,
  ...
)

## S3 method for class 't_test'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type = NULL,
  iter = 1e+06,
  ...
)

Arguments

x	An R object containing the outcome of a statistical analysis. An R object containing the outcome of a statistical analysis. Currently, the following objects can be processed: t_test(), bartlett_test(), lm(), aov(), manova(), cor_test(), lmer() (only for testing random intercep variances), glm(), coxph(), survreg(), polr(), zeroinfl(), rma(), ergm(), bergm(), or named vector objects. In the case x is a named vector, the arguments Sigma and n are also needed. See vignettes for elaborations.
hypothesis	A character string containing the constrained (informative) hypotheses to evaluate in a confirmatory test. The default is NULL, which will result in standard exploratory testing under the model x.
prior.hyp.explo	The prior probabilities of the hypotheses in the exploratory tests. Except for objects of class aov (for (M)ANOVA, etc.), this argument should be a vector with three elements reflecting the prior probability of a zero effect, a negative effect, and a positive effect, respectively. For objects of class aov, the argument should be a list where the first element should be a vector of length 3 specifying the prior probabilities of each parameter being zero, negative, or positive, the second element should be a vector of length 2 specifying the prior probabilities of a model where is no main effect for a factor and the full model, and the third element should be a vector of length 2 specifying the prior probabilities of a model where is no interaction effect (if present) for two factors and the full model. The default (NULL) specifies equal prior probabilities for each hypothesis per exploratory test.
prior.hyp.conf	The prior probabilities of the constrained hypotheses in the confirmatory test.
prior.hyp	Deprecated. Please use the argument prior.hyp.conf.
complement	a logical specifying whether the complement should be added to the tested hypothesis under hypothesis.
log	a logical specifying whether the Bayes factors should be computed on a log scale. Default is FALSE.
cov.prob	coverage probability of the Bayesian credibility interval in the estimates element of the output object. The argument is only used for objects for which a Bayesion estimation algorithm is used, such as a cor_test or rma.uni object. The default coverage probability is 0.95.
Sigma	An approximate posterior covariance matrix (e.g,. error covariance matrix) of the parameters of interest. This argument is only required when x is a named vector.
n	The (effective) sample size that was used to acquire the estimates in the named vector x and the error covariance matrix Sigma. This argument is only required when x is a named vector.
...	Parameters passed to and from other functions.
BF.type	For certain object classes of x, different types of Bayes factor tests are supported. This can be specified using this argument. For models of class 'lm' and 't_test', setting this argument to 'FBF' the fractional Bayes factor (O'Hagan, 1995) is used and setting this argument to 'AFBF' the adjusted fractional Bayes factor (Mulder, 2014) is used. The default for these model classes is the fractional Bayes factor. For models of class 'rma.uni' (meta-analyses), 'BF.type' controls the prior that is used for the test. Certain defaults are provided for a standardized effect (BF.type="stand.effect"), a normal prior with mean 0 and sd 0.5 is used, for log odds (BF.type="log.odds"), a Student prior with mean 0, scale 2.36, and 13.1 degrees of freedom is used (corresponding to uniform priors on the success probabilities), for a correlation (BF.type="correlation"), a logistic prior is used for the Fisher transformed correlation having scale 0.5 (corresponding to a uniform prior for the correlation in (-1,1)), for a unit-information prior (BF.type="unit.info"), the total sample size sum(ni) is used to scale a normal unit information prior, and for a manually specified prior BF.type needs to be an object of class prior from the metaBMA package.
iter	Number of iterations that are used to compute the Monte Carlo estimates. Only used for certain hypothesis tests of class mlm (multivariate regression) where constraints are formulated across different dependent variables as well as different predictors.

Details

The function requires a fitted modeling object. Current analyses that are supported: t_test, bartlett_test, aov, manova, lm, mlm, glm, hetcor, lmer, coxph, survreg, ergm, bergm, zeroinfl, rma and polr.

For testing parameters from the results of t_test(), lm(), aov(), manova(), and bartlett_test(), hypothesis testing is done using adjusted fractional Bayes factors are computed (using minimal fractions). For testing measures of association (e.g., correlations) via cor_test(), Bayes factors are computed using joint uniform priors under the correlation matrices. For testing intraclass correlations (random intercept variances) via lmer(), Bayes factors are computed using uniform priors for the intraclass correlations. For all other tests, approximate adjusted fractional Bayes factors (with minimal fractions) are computed using Gaussian approximations, similar as a classical Wald test.

Value

The output is an object of class BF. The object has elements:

• BFtu_exploratory: The Bayes factors of the constrained hypotheses against the unconstrained hypothesis in the exploratory test.

• BFtu_main (only for aov objects with predictors of class factor): The Bayes factors of a constrained model where all levels of a factor are assumed to have the same effect on the outcome variable versus an unconstrained (full) model with no constraints.

• BFtu_interaction (only for aov objects with interaction effects with predictors of class factor): The Bayes factors of a constrained model where the effect of the dummy variables corresponding to an interaction effects are assumed to be zero versus an unconstrained (full) model with no constraints.

• PHP_exploratory: The posterior probabilities of the constrained hypotheses in the exploratory test.

• PHP_main (only for aov objects with predictors of class factor): The posterior probabilities a constrained model where all levels of a factor are assumed to have the same effect on the outcome variable versus an unconstrained (full) model with no constraints.

• PHP_interaction (only for aov objects with interaction effects with predictors of class factor): The posterior probabilities of a constrained model where the effect of the dummy variables corresponding to an interaction effects are assumed to be zero versus an unconstrained (full) model with no constraints.

• BFtu_confirmatory: The Bayes factors of the constrained hypotheses against the unconstrained hypothesis in the confirmatory test using the hypothesis argument.

• PHP_confirmatory: The posterior probabilities of the constrained hypotheses in the confirmatory test using the hypothesis argument.

• BFmatrix_confirmatory: The evidence matrix which contains the Bayes factors between all possible pairs of hypotheses in the confirmatory test.

• BFtable_confirmatory: The Specification table (output when printing the summary of a BF for a confirmatory test) which contains the different elements of the extended Savage Dickey density ratio where

  • The first column 'complex=' quantifies the relative complexity of the equality constraints of a hypothesis (the prior density at the equality constraints in the extended Savage Dickey density ratio).

  • The second column 'complex>' quantifies the relative complexity of the order constraints of a hypothesis (the prior probability of the order constraints in the extended Savage Dickey density ratio).

  • The third column 'fit=' quantifies the relative fit of the equality constraints of a hypothesis (the posterior density at the equality constraints in the extended Savage Dickey density ratio).

  • The fourth column 'fit>' quantifies the relative fit of the order constraints of a hypothesis (the posterior probability of the order constraints in the extended Savage Dickey density ratio)

  • The fifth column 'BF=' contains the Bayes factor of the equality constraints against the unconstrained hypothesis.

  • The sixth column 'BF>' contains the Bayes factor of the order constraints against the unconstrained hypothesis.

  • The seventh column 'BF' contains the Bayes factor of the constrained hypothesis against the unconstrained hypothesis.

  • The eighth column 'PHP' contains the posterior probabilities of the hypotheses.

• prior.hyp.explo: The prior probabilities of the constrained hypotheses in the exploratory tests.

• prior.hyp.conf: The prior probabilities of the constrained hypotheses in the confirmatory test.

• hypotheses: The tested constrained hypotheses in a confirmatory test.

• estimates: Descriptives of unconstrained estimates based on flat priors (also for rma.uni objects for Bayesian meta-analyses).

• model: The input model x.

• bayesfactor: The type of Bayes factor that is used for this model.

• parameter: The type of parameter that is tested.

• log: logical whether the Bayes factors were reported on a log scale.

• fraction_number_groupIDs (only for objects of class lm): The number of 'group identifiers' that were identified based on the number of unique combinations of levels of predictor variables of class factor in the data. These group identifiers are used to automatically specify the minimal fractions that are used to compute (adjusted) fractional Bayes factors.

• fraction_groupID_observations (only for objects of class lm): A vector that specifies to which 'group identifier' an observation belongs. The group identifiers are constructed based on the unique combination of the levels based on the predictor variables of class factor of the observations.

• call: The call of the BF function.

Methods (by class)

• BF(default): S3 method for a named vector 'x'

• BF(lm): S3 method for an object of class 'lm'

• BF(rma.uni): BF S3 method for an object of class 'rma.uni'

• BF(t_test): BF S3 method for an object of class 't_test'

References

Mulder, Williams, Gu, Tomarken, Böing-Messing, Olsson-Collentine, Meyerink, Menke, Fox, Rosseel, Wagenmakers, Hoijtink, and van Lissa (2021). BFpack: Flexible Bayes Factor Testing of Scientific Theories in R. Journal of Statistical Software, 100. <https://doi.org/10.18637/jss.v100.i18>

Mulder and Xin (2022). Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models. <https://doi.org/10.1080/00273171.2021.1904809>

Mulder and Gelissen (2021). Bayes factor testing of equality and order constraints on measures of association in social research. Journal of Applied Statistics, 50. <https://doi.org/10.1080/02664763.2021.1992360>

Mulder and Fox (2019). Bayes Factor Testing of Multiple Intraclass Correlations. Bayesian Analysis, 14. <http://doi.org/10.1214/18-BA1115>

Hoijtink, Mulder, van Lissa, and Gu (2018). A tutorial on testing hypotheses using the Bayes factor. Psychological Methods, 24(5), 539–556. <http://doi.org/10.1037/met0000201>

Boeing-Messing, van Assen, Hofman, Hoijtink, and Mulder (2017). Bayesian evaluation of constrained hypotheses on variances of multiple independent groups. Psycholological Methods, 22(2), 262-287. <https://doi.org/10.1037/met0000116>

van Aert and Mulder (2021). Bayesian hypothesis testing and estimation under the marginalized random-effects meta-analysis model. Psychonomic Bulletin and Review, 29, 55–69. <https://doi.org/10.3758/s13423-021-01918-9>

Examples

# EXAMPLE 1. One-sample t test
ttest1 <- t_test(therapeutic, mu = 5)
print(ttest1)
# confirmatory Bayesian one sample t test
BF1 <- BF(ttest1, hypothesis = "mu = 5")
summary(BF1)
# exploratory Bayesian one sample t test
BF(ttest1)

# EXAMPLE 2. ANOVA
aov1 <- aov(price ~ anchor * motivation,data = tvprices)
BF1 <- BF(aov1, hypothesis = "anchorrounded = motivationlow;
                              anchorrounded < motivationlow")
summary(BF1)

# EXAMPLE 3. linear regression
lm1 <- lm(mpg ~ cyl + hp + wt, data = mtcars)
BF(lm1, hypothesis = "wt < cyl < hp = 0")

# EXAMPLE 4. Logistic regression
fit <- glm(sent ~ ztrust + zfWHR + zAfro + glasses + attract + maturity +
   tattoos, family = binomial(), data = wilson)
BF1 <- BF(fit, hypothesis = "ztrust > zfWHR > 0;
                             ztrust > 0 & zfWHR = 0")
summary(BF1)

# EXAMPLE 5. Correlation analysis
set.seed(123)
cor1 <- cor_test(memory[1:20,c(1,2,6)])
BF1 <- BF(cor1)
summary(BF1)
BF2 <- BF(cor1, hypothesis = "Rat_with_Im > Rat_with_Del > 0;
                             Rat_with_Im = Rat_with_Del = 0")
summary(BF2)
# correlations can also be computed between continuous/ordinal variables
memory_test <- memory[1:20,c(1,2,6)]
memory_test[,3] <- as.ordered(memory_test[,3])
cor2 <- cor_test(memory_test)
BF(cor2)

# EXAMPLE 6. Bayes factor testing on a named vector
# A Poisson regression model is used to illustrate the computation
# of Bayes factors with a named vector as input
poisson1 <- glm(formula = breaks ~ wool + tension,
  data = datasets::warpbreaks, family = poisson)
# extract estimates, error covariance matrix, and sample size:
estimates <- poisson1$coefficients
covmatrix <- vcov(poisson1)
samplesize <- nobs(poisson1)
# compute Bayes factors on equal/order constrained hypotheses on coefficients
BF1 <- BF(estimates, Sigma = covmatrix, n = samplesize, hypothesis =
"woolB > tensionM > tensionH; woolB = tensionM = tensionH")
summary(BF1)

BFpack documentation built on April 12, 2025, 1:40 a.m.