anovaBF: Function to compute Bayes factors for ANOVA designs
In BayesFactor: Computation of Bayes factors for common designs

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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 = options()$BFprogress,
    rscaleFixed = "medium", rscaleRandom = "nuisance",
    multicore = FALSE, method = "auto", noSample = FALSE)

`formula`	a formula containing all factors to include in the analysis (see Examples)
`data`	a data frame containing data for all factors in the formula
`whichRandom`	a character vector specifying which factors are random
`whichModels`	which set of models to compare; see Details
`iterations`	How many Monte Carlo simulations to generate, if relevant
`progress`	if `TRUE`, show progress with a text progress bar
`rscaleFixed`	prior scale for standardized, reduced fixed effects. A number of preset values can be given as strings; see Details.
`rscaleRandom`	prior scale for standardized random effects
`multicore`	if `TRUE` use multiple cores through the `doMC` package. Unavailable on Windows.
`method`	approximation method, if needed. See `nWayAOV` for details.
`noSample`	if `TRUE`, do not sample, instead returning NA.

Models, priors, and methods of computation are provided in Rouder et al. (2012).

The ANOVA model for a vector of observations y is

y = μ + X_1 θ_1 + … + X_pθ_p +ε,

where θ_1,…,θ_p are vectors of main-effect and interaction effects, X_1,…,X_p are corresponding design matrices, and ε is a vector of zero-centered noise terms with variance σ^2. Zellner and Siow (1980) inspired g-priors are placed on effects, but with a separate g-prior parameter for each covariate:

θ_1~N(0,g_1σ^2), …, θ_p~N(0,g_p σ^2).

A Jeffries prior is placed on μ and σ^2. Independent scaled inverse-chi-square priors with one degree of freedom are placed on g_1,…,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 θ 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,…,g_p may be expressed in closed-form; the integration of g_1,…,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.

An object of class BFBayesFactor, containing the computed model comparisons

The function anovaBF will compute Bayes factors for all possible combinations of fixed factors and interactions, against the null hypothesis that all effects are 0. The total number of tests computed will be 2^(2^K - 1) for K fixed factors. This number increases very quickly with the number of factors. For instance, for a five-way ANOVA, the total number of tests exceeds two billion. Even though each test takes a fraction of a second, the time taken for all tests could exceed your lifetime. An option is included to prevent this: options('BFMaxModels'), which defaults to 50,000, is the maximum number of models that 'anovaBF' will analyze at once. This can be increased by increasing the option value.

It is possible to reduce the number of models tested by only testing the most complex model and every restriction that can be formed by removing one factor or interaction using the whichModels argument. Setting this argument to 'top' reduces the number of tests to 2^K-1, which is more manageable. The Bayes factor for each restriction against the most complex model can be interpreted as a test of the removed factor/interaction. Setting whichModels to 'withmain' will not reduce the number of tests as much as 'top' but the results may be more interpretable, since an interaction is only allowed when all interacting effects (main or interaction) are also included in the model.

Richard D. Morey (richarddmorey@gmail.com)

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.

## 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

Loading required package: coda
Loading required package: Matrix
************
Welcome to BayesFactor 0.9.12-2. If you have questions, please contact Richard Morey (richarddmorey@gmail.com).

Type BFManual() to open the manual.
************

Error: ID
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals  9  58.08   6.453               

Error: ID:group
          Df Sum Sq Mean Sq F value  Pr(>F)   
group      1 12.482  12.482    16.5 0.00283 **
Residuals  9  6.808   0.756                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Bayes factor analysis
--------------
[1] group + ID : 11.65004 <U+00B1>1.88%

Against denominator:
  extra ~ ID 
---
Bayes factor type: BFlinearModel, JZS

Bayes factor top-down analysis
--------------
When effect is omitted from shape + color + shape:color + ID , BF is...
[1] Omit color:shape : 2.723156  <U+00B1>3.94%
[2] Omit color       : 0.2292998 <U+00B1>2.97%
[3] Omit shape       : 0.2292272 <U+00B1>2.77%

Against denominator:
  RT ~ shape + color + shape:color + ID 
---
Bayes factor type: BFlinearModel, JZS