BFpack (version 1.2.2)

BF.default: 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.

Usage

# S3 method for default
BF(x, hypothesis = NULL, prior.hyp = NULL, complement = TRUE, Sigma, n, ...)
# S3 method for lm
BF(x, hypothesis = NULL, prior.hyp = NULL, complement = TRUE, BF.type = 2, ...)
# S3 method for t_test
BF(x, hypothesis = NULL, prior.hyp = NULL, complement = TRUE, BF.type = 2, ...)

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.

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

• 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 `BF=' contains the posterior probabilities of the constrained hypotheses.

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

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

• estimates: The unconstrained estimates.

• model: The input model x.

• call: The call of the BF function.

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(), 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

A vector specifying the prior probabilities of the hypotheses. The default is NULL which will specify equal prior probabilities.

complement

a logical specifying whether the complement should be added to the tested hypothesis under hypothesis.

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

An integer that specified the type of Bayes factor (or prior) that is used for the test. Currently, this argument is only used for models of class 'lm' and 't_test', where BF.type=2 implies an adjusted fractional Bayes factor with a 'fractional prior mean' at the null value (Mulder, 2014), and BF.type=1 implies a regular fractional Bayes factor (based on O'Hagan (1995)) with a 'fractional prior mean' at the MLE.

Methods (by class)

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

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

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

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, 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.

References

Mulder, J., D.R. Williams, Gu, X., A. Tomarken, F. Böing-Messing, J.A.O.C. Olsson-Collentine, Marlyne Meyerink, J. Menke, J.-P. Fox, Y. Rosseel, E.J. Wagenmakers, H. Hoijtink., and van Lissa, C. (2021). BFpack: Flexible Bayes Factor Testing of Scientific Theories in R. Journal of Statistical Software. <DOI:10.18637/jss.v100.i18>

Examples

# \dontshow{
# 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. 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 4. Correlation analysis
set.seed(123)
cor1 <- cor_test(memory[1:20,1:3])
BF1 <- BF(cor1)
summary(BF1)
BF2 <- BF(cor1, hypothesis = "Wmn_with_Im > Wmn_with_Del > 0;
                              Wmn_with_Im = Wmn_with_Del = 0")
summary(BF2)

# EXAMPLE 5. 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)
# }
# \donttest{
# EXAMPLE 1. One-sample t test
ttest1 <- bain::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)
# check the names of the model parameters
names(aov1$coefficients)
BF1 <- BF(aov1, hypothesis = "anchorrounded = motivationlow;
                              anchorrounded < motivationlow;
                              anchorrounded > motivationlow")
summary(BF1)

# EXAMPLE 3. Logistic regression
fit <- glm(sent ~ ztrust + zfWHR + zAfro + glasses + attract + maturity +
   tattoos, family = binomial(), data = wilson)
BF1 <- BF(fit, hypothesis = "ztrust > (zfWHR, zAfro) > 0;
                             ztrust > 0 & zfWHR=zAfro= 0")
summary(BF1)

# EXAMPLE 4. Correlation analysis
set.seed(123)
cor1 <- cor_test(memory[1:20,1:3])
BF1 <- BF(cor1)
summary(BF1)
BF2 <- BF(cor1, hypothesis = "Wmn_with_Im > Wmn_with_Del > 0;
                              Wmn_with_Im = Wmn_with_Del = 0")
summary(BF2)

# EXAMPLE 5. Bayes factor testing on a named vector
# We illustrate the computation of Bayes factors using a named vector
# as input on a Poisson regression model
poisson1 <- glm(formula = breaks ~ wool + tension,
  data = datasets::warpbreaks, family = poisson)
# extract estimates, error covariance matrix, and sample size,
# from fitted object
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)
# }