# Inclusion Bayes Factors for testing predictors across Bayesian models

Source:`R/bayesfactor_inclusion.R`

`bayesfactor_inclusion.Rd`

The `bf_*`

function is an alias of the main function.

For more info, see the Bayes factors vignette.

## Usage

```
bayesfactor_inclusion(models, match_models = FALSE, prior_odds = NULL, ...)
bf_inclusion(models, match_models = FALSE, prior_odds = NULL, ...)
```

## Arguments

- models
An object of class

`bayesfactor_models()`

or`BFBayesFactor`

.- match_models
See details.

- prior_odds
Optional vector of prior odds for the models. See

`BayesFactor::priorOdds<-`

.- ...
Arguments passed to or from other methods.

## Value

a data frame containing the prior and posterior probabilities, and
log(BF) for each effect (Use `as.numeric()`

to extract the non-log Bayes
factors; see examples).

## Details

Inclusion Bayes factors answer the question: Are the observed data more probable under models with a particular effect, than they are under models without that particular effect? In other words, on average - are models with effect \(X\) more likely to have produced the observed data than models without effect \(X\)?

### Match Models

If `match_models=FALSE`

(default), Inclusion BFs are computed by comparing
all models with a term against all models without that term. If `TRUE`

,
comparison is restricted to models that (1) do not include any interactions
with the term of interest; (2) for interaction terms, averaging is done only
across models that containe the main effect terms from which the interaction
term is comprised.

## Note

Random effects in the `lmer`

style are converted to interaction terms:
i.e., `(X|G)`

will become the terms `1:G`

and `X:G`

.

## Interpreting Bayes Factors

A Bayes factor greater than 1 can be interpreted as evidence against the null, at which one convention is that a Bayes factor greater than 3 can be considered as "substantial" evidence against the null (and vice versa, a Bayes factor smaller than 1/3 indicates substantial evidence in favor of the null-model) (Wetzels et al. 2011).

## References

Hinne, M., Gronau, Q. F., van den Bergh, D., and Wagenmakers, E. (2019, March 25). A conceptual introduction to Bayesian Model Averaging. doi:10.31234/osf.io/wgb64

Clyde, M. A., Ghosh, J., & Littman, M. L. (2011). Bayesian adaptive sampling for variable selection and model averaging. Journal of Computational and Graphical Statistics, 20(1), 80-101.

Mathot, S. (2017). Bayes like a Baws: Interpreting Bayesian Repeated Measures in JASP. Blog post.

## See also

`weighted_posteriors()`

for Bayesian parameter averaging.

## Examples

```
library(bayestestR)
# Using bayesfactor_models:
# ------------------------------
mo0 <- lm(Sepal.Length ~ 1, data = iris)
mo1 <- lm(Sepal.Length ~ Species, data = iris)
mo2 <- lm(Sepal.Length ~ Species + Petal.Length, data = iris)
mo3 <- lm(Sepal.Length ~ Species * Petal.Length, data = iris)
BFmodels <- bayesfactor_models(mo1, mo2, mo3, denominator = mo0)
(bf_inc <- bayesfactor_inclusion(BFmodels))
#> Inclusion Bayes Factors (Model Averaged)
#>
#> P(prior) P(posterior) Inclusion BF
#> Species 0.75 1.00 2.02e+55
#> Petal.Length 0.50 1.00 3.58e+26
#> Petal.Length:Species 0.25 0.04 0.113
#>
#> * Compared among: all models
#> * Priors odds: uniform-equal
as.numeric(bf_inc)
#> [1] 2.021143e+55 3.575448e+26 1.131202e-01
# \donttest{
# BayesFactor
# -------------------------------
BF <- BayesFactor::generalTestBF(len ~ supp * dose, ToothGrowth, progress = FALSE)
bayesfactor_inclusion(BF)
#> Inclusion Bayes Factors (Model Averaged)
#>
#> P(prior) P(posterior) Inclusion BF
#> supp 0.60 0.98 34.83
#> dose 0.60 1.00 5.71e+12
#> dose:supp 0.20 0.55 4.92
#>
#> * Compared among: all models
#> * Priors odds: uniform-equal
# compare only matched models:
bayesfactor_inclusion(BF, match_models = TRUE)
#> Inclusion Bayes Factors (Model Averaged)
#>
#> P(prior) P(posterior) Inclusion BF
#> supp 0.40 0.43 22.88
#> dose 0.40 0.45 3.84e+12
#> dose:supp 0.20 0.55 1.28
#>
#> * Compared among: matched models only
#> * Priors odds: uniform-equal
# }
```