`BFpack`

contains a collection of functions for Bayesian
hypothesis testing using Bayes factors and posterior probabilities in R.
The main function `BF`

needs a fitted model `x`

as
input argument. Depending on the class of the fitted model, a standard
hypothesis test is executed by default. For example, if `x`

is a fitted regression model of class `lm`

then posterior
probabilities are computed of whether each separate coefficient is zero,
negative, or positive (assuming equal prior probabilities). If one has
specific hypotheses with equality and/or order constraints on the
parameters under the fitted model `x`

then these can be
formulated using the `hypothesis`

argument (a character
string), possibly together prior probabilities for the hypotheses via
the `prior.hyp`

argument (default all hypotheses are equally
likely a priori), and the `complement`

argument which is a
logical stating whether the complement hypotheses should be included in
the case (`TRUE`

by default).

Alternatively, when the model of interest is not of a class that is
currently supported, `x`

can also be a named numeric vector
containing the estimates of the model parameters of interest, together
with the error covariance matrix in the argument `Sigma`

, and
the sample size used to obtain the estimates, to perform an approximate
Bayes factor test using large sample theory.

The key references for the package are

Mulder, J., Williams, D. R., Gu, X., Tomarken, A., Boeing-Messing,
F., Olsson-Collentine, A., Meijerink, M., Menke, J., van Aert, R., Fox,
J.-P., Hoijtink, H., Rosseel, Y., Wagenmakers, E.-J., and van Lissa, C.
(2021). BFpack: Flexible Bayes Factor Testing of Scientific Theories in
R. *Journal of Statistical Software*. https://www.jstatsoft.org/article/view/v100i18

Mulder, J., van Lissa, C., Gu, X., Olsson-Collentine, A., Boeing-Messing, F., Williams, D. R., Fox, J.-P., Menke, J., et al. (2021). BFpack: Flexible Bayes Factor Testing of Scientific Expectations. (Version 0.3.2) https://CRAN.R-project.org/package=BFpack

`BF(x, hypothesis, prior.hyp = NULL, complement = TRUE, ...)`

`x`

, a fitted model object that is obtained using a R-function. The object can be obtained via the following R functions:`t_test`

for t testing,`bartlett_test`

for testing independent group variances,`aov`

for AN(C)OVA testing,`manova`

for MAN(C)OVA testing,`lm`

for linear regresssion analysis,`cor_test`

for correlation analysis,`lmer`

currently for testing intraclass correlations in random intercept models,`glm`

for generalized linear models,`coxph`

for survival analysis,`survreg`

for survival analysis,`polr`

for ordinal regression,`zeroinfl`

for zero-inflated regression,`rma`

for meta-analysis,`ergm`

or`bergm`

for an exponential random graph,`x`

can also be a named vector with estimates of the key parameters.

`hypothesis`

, a character string specifying the hypotheses with equality and/or order constraints on the key parameters of interest.- By default
`hypothesis = NULL`

which executes exploratory hypothesis tests (examples below). - The parameter names are based on the names of the estimated key
parameters. An overview of the key parameters is given using the
function
`get_estimates`

, e.g.,`get_estimates(model1),`

where`model1`

is a fitted model object. - Separate constraints within a hypothesis are separated with an
ampersand
`&`

. Hypotheses are separated using a semi-colon`;`

. For example`hypothesis = "weight > height & height > 0; weight = height = 0"`

implies that the first hypothesis assumes that the parameter`weight`

is larger than the parameter`height`

and that the parameter`height`

is positive, and the second hypothesis assumes that the two parameters are equal to zero. Note that the first hypothesis could equivalently have been written as`weight > height > 0`

.

- By default
`prior.hyp`

, a numeric vector specifying the prior probabilities of the hypotheses of the`hypothesis`

argument. The default setting is`prior.hyp = NULL`

which sets equal prior probabilities.`complement`

, a logical value which specified if a complement hypothesis is included in the tested hypotheses specified under`hypothesis`

. The default setting is`TRUE`

. The complement hypothesis covers the remaining parameters space that is not covered by the constrained hypotheses. For example, if an equality hypothesis and an order hypothesis are formulated, say,`hypothesis = "weight = height = length; weight > height > length"`

, the complement hypothesis covers the remaining subspace where neither`"weight = height = length"`

holds, nor`"weight > height > length"`

holds.

Alternatively if one is interested in testing hypotheses under a model class which that is currently not supported, an approximate Bayesian test can be executed with the following (additional) arguments

`x`

, a named numeric vector of the estimates (e.g., MLE) of the parameters of interest where the labels are equal to the names of the parameters which are used for the`hypothesis`

argument.`Sigma`

, the approximate posterior covariance matrix (e.g,. error covariance matrix) of the parameters of interest.`n`

, the sample size that was used to acquire the estimates and covariance matrix.

The output is of class `BF`

. By running the
`print`

function on the `BF`

object, a short
overview of the results are presented. By running the
`summary`

function on the `BF`

object, a
comprehensive overview of the results are presented.

First a classical one sample t test is executed for the test value \(\mu = 5\) on the therapeutic data

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 see which parameters can be tested on this object run

which shows that the only parameter that can be tested is the
population mean which has name `mu`

.

To perform an exploratory Bayesian t test of whether the population
mean is equal to, smaller than, or larger than the null value (which is
`5`

here, as specified when defining the `ttest1`

object), one needs to run `BF`

function on the object.

This executes an exploratory (‘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.

In the above test the complement hypothesis is excluded automatically
as the formualted hypothesis under the `hypothesis`

argument
cover the complete parameter space. Furthermore, when testing hypotheses
via the `hypothesis`

argument, the output also presents an
`Evidence matrix`

containing the Bayes factors between the
hypotheses formulated in the `hypothesis`

argument.

A standard two-sided test around the null value `mu`

is
executed by setting the hypothesis argument equal to the precise null
hypothesis so that the complement hypothesis (which is included by
default) corresponds to the hypothesis that assumes that the population
mean is anything but the null value

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 Mulder et
al. (2021, Section 4.1)) 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

For more information about the methodology, we refer the interested reader to Mulder et al. (2021) and Mulder and Gu (2021).

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 however that 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`

, to replace the intercept. For example, let
us consider a multivariate normal model for the dependent variables
`Superficial`

, `Middle`

, and `Deep`

in
the `fmri`

data set:

Next, we can (for instance) test whether all means equal 0
(`H1`

), whether all means are positive (`H2`

), or
none of these two hypotheses (`complement`

):

First an analysis of variance (ANOVA) model is fitted using the
`aov`

fuction in `R`

.

Next a Bayesian test can be performed on the fitted object. By default exploratory tests are executed of whether the individual main and interaction effects are zero or not (corresponding to the full model) (see Mulder et al. (2021, Section 4.2))

One can also test for specific equal/order hypotheses based on
scientific expectations such as whether `anchorrounded`

is
positive, `motivationlow`

is negative, and the interaction
effect `anchorrounded:motivationlow`

is negative (see Mulder et
al. (2021, Section 4.2)) versus null hypothesis versus the
complement hypothesis (which assumes that the constraints of neither two
hypotheses hold). This test can be executed as follows:

```
constraints2 <- "anchorrounded > 0 & motivationlow < 0 &
anchorrounded:motivationlow < 0; anchorrounded = 0 &
motivationlow = 0 & anchorrounded:motivationlow = 0"
set.seed(1234)
BF(aov1, hypothesis = constraints2)
```

For more information about the methodology, we refer the interested reader to Mulder et al. (2021) and Mulder and Gu (2021).

First a classical significance test is executed using the
`bartlett_test`

function, which is part of the
** BFpack** package. The function is equivalent to
the standard

`bartlett.test`

function with the addition that
the returned object contains additional output needed for the test using
the `BF`

function.On an object of this class, by default `BF`

executes an
exploratory test of homogeneity (equality) of variances against an
unconstrained (full) model

The group variances have names `ADHD`

,
`Controls`

, and `TS`

. This can be retrieved by
running

Let’s say we want to test whether a hypothesis (H1) which assumes
that group variances of groups `Controls`

and `TS`

are equal and smaller than the group variance of the `ADHD`

group, a hypothesis (H2) which assumes that the group variances of
`ADHD`

and `TS`

are equal and larger than the
`Controls`

group, a hypothesis (H3) which assumes all group
variances are equal, and a complement hypothesis (H4). To do this we run
the following:

```
hypothesis <- "Controls = TS < ADHD; Controls < TS = ADHD; Controls = TS = ADHD"
set.seed(358)
BF_var <- BF(bartlett1, hypothesis)
```

A comprehensive output of this analysis can be obtained by running:

which presents the results of an exploratory analysis and the results
of a confirmatory analysis (based on the hypotheses formulated under the
`hypothesis`

argument). The exploratory analysis tests a
hypothesis which assumes that the variances are equal across groups
(homogeneity of variances) versus an alternative unrestricted
hypothesis. The output shows that the posterior probabilities of these
two hypotheses are approximately 0.803 and 0.197 (assuming equal priori
probabilities). Note that the p value in the classical Bartlett test for
these data equals 0.1638 which implies that the hypothesis of
homogeneity of variances cannot be rejected using common significance
levels, such as 0.05 or 0.01. Note however that this p value cannot be
used as a measure for the evidence in the data in favor of homogeneity
of group variances. This can be done using the proposed Bayes factor
test which shows that the probability that the variances are equal is
approximately 0.803. Also note that the exploratory test could
equivalently tested via the `hypothesis`

argument by running
`BF(bartlett1, "Controls = TS = ADHD")`

.

The confirmatory test shows that H1 receives strongest support from the data, but H2 and H3 are viable competitors. It appears that even the complement H3 cannot be ruled out entirely given a posterior prob- ability of 0.058. To conclude, the results indicate that TS population are as heterogeneous in their attentional performances as the healthy control population in this specific task, but further research would be required to obtain more conclusive evidence.

For more information about the methodology, we refer the interested reader to Boing-Messing et al. (2017)

An example hypothesis test is considered 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)
```

By default exploratory exhaustive tests are executed of whether the separate regression coefficients are zero, negative, or positive:

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 regression coefficients of interest Mulder et al. (2021, Section 4.4)

```
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.

For more information about the methodology, we refer the interested reader to Gu et al. (2018) and Mulder et al. (2021)

By default `BF`

performs exhaustice tests of whether the
separate correlations are zero, negative, or positive.

The names of the correlations is constructed using the names of the
variables separated by `_with_`

:

Specific hypotheses based on prior/theoretical considerations can be
tested using the `hypothesis`

argument. As an example we show
here how to test whether all correlations are equal and positive versus
its complement.

We can also test equality and order constraints on correlations
across different groups. As the seventh column of the
`memory`

object is a group indicator, let us first create
different objects for the two different groups, and perform Bayesian
estimation on the correlation matrices of the two different groups

```
memoryHC <- subset(memory,Group=="HC")[,-(4:7)]
memorySZ <- subset(memory,Group=="SZ")[,-(4:7)]
set.seed(123)
cor1 <- cor_test(memoryHC,memorySZ)
```

In this case with multiple groups by default exploratory tests are
executed of whether the correlations are zero, negative, or positive for
each separate group (e.g., correlations in group `gr1`

are
denoted by `_in_gr1`

at the end of the name)

Next we test the one-sided hypothesis that the respective
correlations in the first group (`g1`

) are larger than the
correlations in the second group (`g2`

) via

```
set.seed(123)
BF6_cor <- BF(cor1, hypothesis =
"Del_with_Im_in_g1 > Del_with_Im_in_g2 &
Del_with_Wmn_in_g1 > Del_with_Wmn_in_g2 &
Im_with_Wmn_in_g1 > Im_with_Wmn_in_g2")
```

By running `print(BF6_cor)`

, the output shows that the
one-sided hypothesis received a posterior probability of 0.991 and the
alternative received a posterior probability of .009 (assuming equal
prior probabilities).

For more information about the methodology, we refer the interested reader to Mulder (2016) and Mulder and Gelissen (2019)

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

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)
print(BF2)
```

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"
set.seed(123)
BF3 <- BF(lm2, hypothesis = constraint2)
summary(BF3)
```

For more information about the methodology, we refer the interested reader to Mulder and Olsson-Collentine (2019) and Mulder and Gu (2021)

For illustrative purposes we generate a hypothetical simulated dataset

```
set.seed(123)
tau2 <- 0.05
vi <- runif(50, min = 0.01, max = 0.2)
yi <- rnorm(50, mean = 0, sd = sqrt(vi+tau2))
```

where `tau2`

denotes the true between-study heterogeneity,
`vi`

is a vector containing the squared standard errors of 50
studies, and `yi`

is a vector containing the estimated
effects sizes in the 50 studies. To test the overall effect size and the
between-study heterogeneity using `BFpack`

, an initial
meta-analysis needs to be executed using the `metafor`

package. Subsequently the output is plugged into the `BF`

function:

The `summary`

output gives the posterior probabilities for
a zero, negative, and positive between-study heterogeneity
`I^2`

and overall effect size `mu`

assuming equal
prior probabilities:

The results indicate evidence for positive between-study heterogeneity (suggesting that a random effects meta-analysis model is appropriate) and for a zero overall effect size.

The unconstrained estimates (posterior mean and median) and the lower and upper bound of the 95% Bayesian credible intervals can be obtained by calling:

For more information about the methodology, we refer the interested reader to Van Aert and Mulder (2021)

`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

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

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

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(BF1)`

and `print(BF2)`

).

For more information about the methodology, we refer the interested reader to Gu et al. (2018)