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_estimates`

on 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")
} ::install_github("jomulder/BFpack") remotes
```

Below several example analyses are provided using
**BFpack**.

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:

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

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.

```
library(BFpack)
<- BF(ttest1) BF1
```

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.

```
<- "mu = 5; mu < 5; mu > 5"
hypothesis 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))`

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

fuction in `R`

.

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

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)`

.

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

function

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

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(fit_glm, hypothesis = "ztrust > (zfWHR, zAfro) > 0;
BF_glm 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.

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_`

.

```
set.seed(123)
<- cor_test(memory[,1:3])
cor1 <- BF(cor1)
BF1 print(BF1)
```

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

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

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

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

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

```
<- BF(lm1, hypothesis = "Vehicle > 0 & Face < 0; Vehicle = Face = 0",
BF2 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.

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

`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

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

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

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

Constrained hypotheses on the parameters
`names(estimates)`

can then be tested as follows

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

Note that the same hypothesis test would be executed when calling

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

).

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]. https://CRAN.R-project.org/package=BFpack

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 https://arxiv.org/abs/1911.07728

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

By participating in this project, you agree to abide by the Contributor Code of Conduct v2.0.