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The R package BFpack contains a set of functions for exploratory hypothesis testing (e.g., equal vs negative vs postive) and confirmatory hypothesis testing (with equality and/or order constraints) using Bayes factors and posterior probabilities under commonly used statistical models, including (but not limited to) Bayesian t testing, (M)AN(C)OVA, multivariate/univariate linear regression, correlation analysis, multilevel analysis, or generalized linear models (e.g., logistic regression). The main function BF needs a fitted model (e.g., an object of class lm for a linear regression model) and (optionally) the argument hypothesis, a string which specifies a set of equality/order constraints on the parameters. By applying the function get_estimateson a fitted model, the names of the parameters are returned on which constrained hypotheses can be formulated. Bayes factors and posterior probabilities are computed for the hypotheses of interest.


Install the latest release version of BFpack from CRAN:


The current developmental version can be installed with

if (!requireNamespace("remotes")) { 

Example analyses

Below several example analyses are provided using BFpack.

Bayesian t testing

First a classical one sample t test is executed on the test value (&mu = 5) on the therapeutic data (part of BFpack). Here a right one-tailed classical test is executed:

ttest1 <- bain::t_test(therapeutic, alternative = "greater", mu = 5)

The t_test function is part of the bain package. The function is equivalent to the standard t.test function with the addition that the returned object contains additional output than the standard t.test function.

To perform a Bayesian t test plug the fitted object into the BF function.

BF1 <- BF(ttest1)

This executes an exploratoory (‘exhaustive’) test around the null value: H1: mu = 5 versus H2: mu < 5 versus H3: mu > 5 assuming equal prior probabilities for H1, H2, and H3 of 1/3. The output presents the posterior probabilities for the three hypotheses.

The same test would be executed when the same hypotheses are explicitly specified using the hypothesis argument.

hypothesis <- "mu = 5; mu < 5; mu > 5"
BF(ttest1, hypothesis = hypothesis)

When testing hypotheses via the hypothesis argument, the output also presents an Evidence matrix containing the Bayes factors between the hypotheses.

The argument prior.hyp can be used to specify different prior probabilities for the hypotheses. For example, when the left one-tailed hypothesis is not possible based on prior considerations (e.g., see preprint) while the precise (null) hypothesis and the right one-tailed hypothesis are equally likely, the argument prior.hyp should be a vector specifying the prior probabilities of the respective hypotheses

BF(ttest1, hypothesis = "mu = 5; mu < 5; mu > 5", prior.hyp = c(.5,0,.5))

Analysis of variance

First an analysis of variance (ANOVA) model is fitted using the aov fuction in R.

aov1 <- aov(price ~ anchor * motivation, data = tvprices)

Next a Bayesian test can be performed on the fitted object.


By default posterior probabilities are computed of whether main effects and interaction effects are present. Alternative constrained hypotheses can be tested on the model parameters get_estimates(aov1).

Logistic regression

An example hypothesis test is consdered under a logistic regression model. First a logistic regression model is fitted using the glm function

fit_glm <- glm(sent ~ ztrust + zfWHR + zAfro + glasses + attract + maturity +
               tattoos, family = binomial(), data = wilson)

The names of the regression coefficients on which constrained hypotheses can be formualted can be extracted using the get_estimates function.


Two different hypotheses are formulated with competing equality and/or order constraints on the parameters of interest. These hypotheses are motivated in Mulder et al. (2019)

BF_glm <- BF(fit_glm, hypothesis = "ztrust > (zfWHR, zAfro) > 0;
             ztrust > zfWHR = zAfro = 0")

By calling the summary function on the output object of class BF, the results of the exploratory tests are presented of whether each separate parameter is zero, negative, or positive, and the results of the confirmatory test of the hypotheses under the hypothesis argument are presented. When the hypotheses do not cover the complete parameter space, by default the complement hypothesis is added which covers the remaining parameter space that is not covered by the constraints under the hypotheses of interest. In the above example, the complement hypothesis covers the parameter space where neither "ztrust > (zfWHR, zAfro) > 0" holds, nor where "ztrust > zfWHR = zAfro = 0" holds.

Correlation analysis

By default BF performs exhaustice tests of whether the separate correlations are zero, negative, or positive. The name of the correlations is constructed using the names of the variables separated by _with_.

cor1 <- cor_test(memory[,1:3])
BF1 <- BF(cor1)

Constraints can also be tested between correlations, e.g., whether all correlations are equal and positive versus an unconstrained complement.

BF2 <- BF(cor1, hypothesis = "Del_with_Im = Wmn_with_Im = Wmn_with_Del > 0")

Univariate/Multivariate multiple regression

For a univariate regression model, by default an exhaustive test is executed of whether an effect is zero, negative, or postive.

lm1 <- lm(Superficial ~ Face + Vehicle, data = fmri)
BF1 <- BF(lm1)

Hypotheses can be tested with equality and/or order constraints on the effects of interest. If prefered the complement hypothesis can be omitted using the complement argument

BF2 <- BF(lm1, hypothesis = "Vehicle > 0 & Face < 0; Vehicle = Face = 0",
          complement = FALSE)

In a multivariate regression model hypotheses can be tested on the effects on the same dependent variable, and on effects across different dependent variables. The name of an effect is constructed as the name of the predictor variable and the dependent variable separated by _on_. Testing hypotheses with both constraints within a dependent variable and across dependent variables makes use of a Monte Carlo estimate which may take a few seconds.

lm2 <- lm(cbind(Superficial, Middle, Deep) ~ Face + Vehicle,
              data = fmri)
constraint2 <- "Face_on_Deep = Face_on_Superficial = Face_on_Middle < 0 <
     Vehicle_on_Deep = Vehicle_on_Superficial = Vehicle_on_Middle;
     Face_on_Deep < Face_on_Superficial = Face_on_Middle < 0 < Vehicle_on_Deep =
     Vehicle_on_Superficial = Vehicle_on_Middle"
BF3 <- BF(lm2, hypothesis = constraint2)

Running BF on a named vector

The input for the BF function can also be a named vector containing the estimates of the parameters of interest. In this case the error covariance matrix of the estimates is also needed via the Sigma argument, as well as the sample size that was used for obtaining the estimates via the n argument. Bayes factors are then computed using Gaussian approximations of the likelihood (and posterior), similar as in classical Wald test.

We illustrate this for a Poisson regression model

poisson1 <- glm(formula = breaks ~ wool + tension, data = datasets::warpbreaks,
             family = poisson)

The estimates, the error covariance matrix, and the sample size are extracted from the fitted model

estimates <- poisson1$coefficients
covmatrix <- vcov(poisson1)
samplesize <- nobs(poisson1)

Constrained hypotheses on the parameters names(estimates) can then be tested as follows

BF1 <- BF(estimates, Sigma = covmatrix, n = samplesize, hypothesis = 
  "woolB > tensionM > tensionH; woolB = tensionM = tensionH")

Note that the same hypothesis test would be executed when calling

BF2 <- BF(poisson1, hypothesis = "woolB > tensionM > tensionH;
          woolB = tensionM = tensionH")

because the same Bayes factor is used when running BF on an object of class glm (see Method: Bayes factor using Gaussian approximations when calling print(BF11) and print(BF2)).

Citing BFpack

You can cite the package and the paper using the following reference

Mulder, J., van Lissa, C., Gu, X., Olsson-Collentine, A., Boeing-Messing, F., Williams, D. R., Fox, J.-P., Menke, J., et al. (2020). BFpack: Flexible Bayes Factor Testing of Scientific Expectations. (Version 0.3.1) [R package].

Mulder, J., Williams, D. R., Gu, X., Olsson-Collentine, A., Tomarken, A., Böing-Messing, F., Hoijtink, H., . . . van Lissa, C. (2019). BFpack: Flexible Bayes factor testing of scientific theories in R. Retrieved from

Contributing and Contact Information

If you have ideas, please get involved. You can contribute by opening an issue on GitHub, or sending a pull request with proposed features.

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