This function computes Bayes factors for all main-effects and interaction contrasts in an ANOVA design.
anovaBF(formula, data, whichRandom = NULL, whichModels = "withmain",
iterations = 10000, progress = getOption("BFprogress", interactive()),
rscaleFixed = "medium", rscaleRandom = "nuisance", rscaleEffects = NULL,
multicore = FALSE, method = "auto", noSample = FALSE,
callback = function(...) as.integer(0))a formula containing all factors to include in the analysis (see Examples)
a data frame containing data for all factors in the formula
a character vector specifying which factors are random
which set of models to compare; see Details
How many Monte Carlo simulations to generate, if relevant
if TRUE, show progress with a text progress bar
prior scale for standardized, reduced fixed effects. A number of preset values can be given as strings; see Details.
prior scale for standardized random effects
A named vector of prior settings for individual factors, overriding rscaleFixed and rscaleRandom. Values are scales, names are factor names.
if TRUE use multiple cores through the doMC
package. Unavailable on Windows.
approximation method, if needed. See nWayAOV for
details.
if TRUE, do not sample, instead returning NA.
callback function for third-party interfaces
An object of class BFBayesFactor, containing the computed
model comparisons
Models, priors, and methods of computation are provided in Rouder et al. (2012).
The ANOVA model for a vector of observations \(y\) is $$ y = \mu + X_1
\theta_1 + \ldots + X_p\theta_p +\epsilon,$$ where
\(\theta_1,\ldots,\theta_p\) are vectors of main-effect and interaction
effects, \(X_1,\ldots,X_p\) are corresponding design matrices, and
\(\epsilon\) is a vector of zero-centered noise terms with variance
\(\sigma^2\). Zellner and Siow (1980) inspired g-priors are placed on
effects, but with a separate g-prior parameter for each covariate:
$$\theta_1~N(0,g_1\sigma^2), \ldots, \theta_p~N(0,g_p \sigma^2).$$ A
Jeffries prior is placed on \(\mu\) and \(\sigma^2\). Independent
scaled inverse-chi-square priors with one degree of freedom are placed on
\(g_1,\ldots,g_p\). The square-root of the scale for g's corresponding to
fixed and random effects is given by rscaleFixed and
rscaleRandom, respectively.
When a factor is treated as random, there are as many main effect terms in the vector \(\theta\) as levels. When a factor is treated as fixed, the sums-to-zero linear constraint is enforced by centering the corresponding design matrix, and there is one fewer main effect terms as levels. The Cornfield-Tukey model of interactions is assumed. Details are provided in Rouder et al. (2012)
Bayes factors are computed by integrating the likelihood with respect to the
priors on parameters. The integration of all parameters except
\(g_1,\ldots,g_p\) may be expressed in closed-form; the integration of
\(g_1,\ldots,g_p\) is performed through Monte Carlo sampling, and
iterations is the number of iterations used to estimate the Bayes
factor.
anovaBF computes Bayes factors for either all submodels or select
submodels missing a single main effect or covariate, depending on the
argument whichModels. If no random factors are specified, the null
model assumed by anovaBF is the grand-mean only model. If random
factors are specified, the null model is the model with an additive model on
all random factors, plus a grand mean. Thus, anovaBF does not
currently test random factors. Testing random factors is possible with
lmBF.
The argument whichModels controls which models are tested. Possible
values are 'all', 'withmain', 'top', and 'bottom'. Setting
whichModels to 'all' will test all models that can be created by
including or not including a main effect or interaction. 'top' will test all
models that can be created by removing or leaving in a main effect or
interaction term from the full model. 'bottom' creates models by adding
single factors or interactions to the null model. 'withmain' will test all
models, with the constraint that if an interaction is included, the
corresponding main effects are also included.
For the rscaleFixed and rscaleRandom arguments, several named
values are recognized: "medium", "wide", and "ultrawide", corresponding to
\(r\) scale values of 1/2, \(\sqrt{2}/2\), and 1,
respectively. In addition, rscaleRandom can be set to the "nuisance",
which sets \(r=1\) (and is thus equivalent to "ultrawide"). The "nuisance"
setting is for medium-to-large-sized effects assumed to be in the data but
typically not of interest, such as variance due to participants.
Gelman, A. (2005) Analysis of Variance---why it is more important than ever. Annals of Statistics, 33, pp. 1-53.
Rouder, J. N., Morey, R. D., Speckman, P. L., Province, J. M., (2012) Default Bayes Factors for ANOVA Designs. Journal of Mathematical Psychology. 56. p. 356-374.
Zellner, A. and Siow, A., (1980) Posterior Odds Ratios for Selected Regression Hypotheses. In Bayesian Statistics: Proceedings of the First Interanational Meeting held in Valencia (Spain). Bernardo, J. M., Lindley, D. V., and Smith A. F. M. (eds), pp. 585-603. University of Valencia.
lmBF, for testing specific models, and
regressionBF for the function similar to anovaBF for
linear regression models.
# NOT RUN {
## Classical example, taken from t.test() example
## Student's sleep data
data(sleep)
plot(extra ~ group, data = sleep)
## traditional ANOVA gives a p value of 0.00283
summary(aov(extra ~ group + Error(ID/group), data = sleep))
## Gives a Bayes factor of about 11.6
## in favor of the alternative hypothesis
anovaBF(extra ~ group + ID, data = sleep, whichRandom = "ID",
progress=FALSE)
## Demonstrate top-down testing
data(puzzles)
result = anovaBF(RT ~ shape*color + ID, data = puzzles, whichRandom = "ID",
whichModels = 'top', progress=FALSE)
result
## In orthogonal designs, the top down Bayes factor can be
## interpreted as a test of the omitted effect
# }
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