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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. As input the main function BF() requires a fitted model which which the necessary elements are extracted to compute Bayes factors and posterior probabilities for the hypotheses.

Bayesian t testing

Univariate t testing

First a classical one sample t test needs 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.

Two default Bayes factors are implemented in BFpack to execute a t test: the fractional Bayes factor (O’Hagan, 1995) and the prior adjusted fractional Bayes factor (Mulder, 2014). Both do not require a prior to be manually specified as a default prior is implicitly constructed using a minimal fraction of the data. The remaining fraction is used for hypothesis testing. The fractional Bayes factor behaves similar as the JZS Bayes factor (Rouder et al., 2009, as implemented in the BayesFactor package) for standard null hypothesis testing and the prior adjusted Bayes factor was specifically designed for testing one-sided hypotheses. When setting the argument BF.type=1 or BF.type=2, the fractional Bayes factor and the prior adjusted fractional Bayes factor is used, respectively. The default choice is BF.type=2.

To perform a Bayesian t test, the BF function is run on the fitted object.

BF1 <- BF(ttest1)

This executes an exploratory 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))

Multivariate t testing

Bayesian multivariate t tests can be executed by first fitting a multivariate (regression) model using the lm function, and subsequently, the means of the dependent variables (or other coefficients) in the model can be tested using the BF() function. Users have to be aware that the (adjusted) means are modeled using intercepts which are named (Intercept) by default by lm while the hypothesis argument in BF() does not allow effect names that include brackets (i.e., ( or )). To circumvent this, one can create a vector of 1s, with name (say) ones, which can replace the intercept but results in an equivalent model.

For example, let us consider a multivariate normal model for the dependent variables Superficial, Middle, and Deep in the fmri data set:

fmri1 <- cbind(fmri,ones=1)
mlm1 <- lm(cbind(Superficial,Middle,Deep) ~ -1 + ones, data = fmri1)

Next, we can (for instance) test whether all means are equal to 0 (H1), whether all means are positive (H2), or neither (complement):

BFmlm1 <- BF(mlm1,
             complement = TRUE)

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).

Similar as for the Bayesian t test, two default Bayes factors are implemented in BFpack for (multivariate) analysis of (co)variance: the fractional Bayes factor (O’Hagan, 1995) and the prior adjusted fractional Bayes factor (Mulder, 2014, Mulder and Gu, 2022). The default choice is BF.type=2, which uses the prior adjusted fractional Bayes factor.

Univariate/Multivariate multiple regression

As an example we consider the fmri data (McGuigin et al, 2020) as discussed in Mulder et al. (2021). First, a classical linear regression is fitted with dependent variable Deep and predictor variables Face and Vehicle:

lm1 <- lm(Deep ~ Face + Vehicle, data = fmri)

When not using the hypothesis argument, Bayes factors and posterior probabilities are given of whether each predictor has a zero, negative, or positive effect (assuming equal prior probabilities) against the full model:


As motivated in Mulder et al. (2021), it was expected that Face had a negative effect on Deep and Vehicle had a positive effect on Deep. This (combined) one-sided hypothesis can be tested against its complement according to

BF(lm1, hypothesis = "Face < 0 < Vehicle")

The hypothesis of interest receives clear support from the data.

In a multivariate multiple 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)

Finally note that for (multivariate) multiple regression again note that two default Bayes factors are implemented in BFpack: the fractional Bayes factor (O’Hagan, 1995) and the prior adjusted fractional Bayes factor (Mulder, 2014; Mulder and Gu, 2022) which can be chosen using the argument BF.type=1 and BF.type=2, respectively. The default choice is the prior adjusted fractional Bayes factor. This criterion was specifically designed for testing one-sided and order constrained hypotheses.

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.

The Bayes factors and posterior posterior probabilities are based on the approximated adjusted default Bayes factor (Gu et al., 2018).

Correlation analysis

Bayes factors and posterior posterior probabilities among constrained hypotheses on measures of association are computed using uniform prior for the correlations (Mulder and Gelissen, 2023). 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_. To compute Bayes factors and posterior probabilities, first the unconstrained model needs to be fit using the cor_test() function. The resulting object can be added to the BF() function:

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. The function get_estimates() gives the names of the correlations on which constrained hypotheses can be formulated:

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

Depending on the class of the variables (numeric, ordered, factor with 2 levels), biserial, polyserial, polychoric, tetrachoric, or product-moment correlations are tested. As an illustration of these other types of measures of association, we change the measurement levels of a subset of the mtcars data:

mtcars_test <- mtcars[,c(1,2,9,10)]
mtcars_test[,2] <- as.ordered(mtcars_test[,2])
mtcars_test[,3] <- as.factor(mtcars_test[,3])
mtcars_test[,4] <- as.integer(mtcars_test[,4])

To compute the Bayes factors and posterior probabilities, again we first the full unconstrained model using the cor_test() function. The resulting object is placed in the BF() function to obtain the Bayes factors and posterior proabilities:

cor2 <- cor_test(mtcars_test)
BF2 <- BF(cor2, hypothesis = "0 < am_with_mpg = gear_with_mpg")

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)).

The Bayes factors and posterior posterior probabilities on named vectors are based on the adjusted default Bayes factor using Gaussian approximations (Gu et al., 2018).

Citing BFpack

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

Software paper > Mulder, J., Williams, D. R., Gu, X., Olsson-Collentine, A., Tomarken, > A., Böing-Messing, F., Hoijtink, H., . . . van Lissa (2021). > BFpack: Flexible Bayes factor testing of scientific theories in R. > Journal of Statistical Software, 100(18). > Retrieved from

Software package > 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 1.3.0) [R package]. >

Other references with technical details of the methodology, please see

Bayes factors for (multivariate) t tests, (M)AN(C)OVA, (multivariate) regression > Mulder, J. (2014). Prior adjusted default Bayes factors for testing (in)equality > constrained hypotheses. Computational Statistics and Data Analysis, 71, 448–463. >

Mulder, J. & Gu, X. (2022) Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models, Multivariate Behavioral Research, 57:5, 767-783.

Bayes factors for testing measures of association (e.g., correlations) > Mulder, J., & Gelissen, J. P. (2023). Bayes factor testing of equality > and order constraints on measures of association in social research. > Journal of Applied Statistics, 50(2), 315-351. >

Default Bayes factors using Gaussian approximations > Gu. X., Mulder, J., & Hoijtink, J. (2018). Approximated adjusted > fractional Bayes factors: A general method for testing informative > hypotheses. British Journal of Mathematical and Statistical > Psychology. >

Bayes factors under exponential random graphs > Mulder, J., Friel, N., & Leifeld, P. (2023). Bayesian Testing of Scientific > Expectations Under Exponential Random Graph Models. >

Bayes factors of intraclass correlations > Mulder, J., & Fox, J.-P. (2019). Bayes Factor Testing of Multiple > Intraclass Correlations. Bayesian Analysis. 14(2), 521-552. >

Bayes factors for testing group variances > Böing-Messing, F., van Assen, M. A. L. M., Hofman, A. D., Hoijtink, H., > & Mulder, J. (2017). Bayesian evaluation of constrained hypotheses on > variances of multiple independent groups. Psychological Methods, 22(2), > 262–287. >

Böing-Messing, F. & Mulder, J. (2018). Automatic Bayes factors for testing equality-and inequality-constrained hypotheses on variances. Psychometrika, 83, 586–617.

Bayes factors for meta-analyses > Van Aert R.C.M. & Mulder, J. (2022). Bayesian hypothesis testing and > estimation under the marginalized random-effects meta-analysis model. > Psychonomic Bulletin & Review, 29, 55–69. >

Contributing and Contact Information

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