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:
install.packages("BFpack")The current developmental version can be installed with
if (!requireNamespace("remotes")) {
install.packages("remotes")
}
remotes::install_github("jomulder/BFpack")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.
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.
library(BFpack)
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))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,
hypothesis="ones_on_Superficial=ones_on_Middle=ones_on_Deep=0;
(ones_on_Superficial,ones_on_Middle,ones_on_Deep)>0",
complement = TRUE)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.
BF(aov1)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.
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:
BF(lm1)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"
set.seed(123)
BF3 <- BF(lm2, hypothesis = constraint2)
summary(BF3)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.
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.
get_estimates(fit_glm)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")
summary(BF_glm)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).
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:
set.seed(123)
cor1 <- cor_test(memory[,1:3])
BF1 <- BF(cor1)
print(BF1)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:
get_estimates(cor1)
BF2 <- BF(cor1, hypothesis = "Del_with_Im = Wmn_with_Im = Wmn_with_Del > 0")
print(BF2)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")
print(BF2)BF on a named
vectorThe 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).
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 https://arxiv.org/abs/1911.07728
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]. > https://CRAN.R-project.org/package=BFpack
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. > https://doi.org/10.1016/j.csda.2013.07.017
Mulder, J. & Gu, X. (2022) Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models, Multivariate Behavioral Research, 57:5, 767-783. https://doi.org/10.1080/00273171.2021.1904809
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. > https://doi.org/10.1080/02664763.2021.1992360
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. > https://doi.org/10.1111/bmsp.12110
Bayes factors under exponential random graphs > Mulder, J., Friel, N., & Leifeld, P. (2023). Bayesian Testing of Scientific > Expectations Under Exponential Random Graph Models. > https://doi.org/10.48550/arXiv.2304.14750
Bayes factors of intraclass correlations > Mulder, J., & Fox, J.-P. (2019). Bayes Factor Testing of Multiple > Intraclass Correlations. Bayesian Analysis. 14(2), 521-552. > https://doi.org/10.1214/18-BA1115
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. > https://doi.org/10.1037/met0000116
Böing-Messing, F. & Mulder, J. (2018). Automatic Bayes factors for testing equality-and inequality-constrained hypotheses on variances. Psychometrika, 83, 586–617. https://link.springer.com/article/10.1007/s11336-018-9615-z
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. > https://doi.org/10.3758/s13423-021-01918-9
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