This function computes Bayes factors for all main-effects and interaction contrasts in an ANOVA design.
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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
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.
approximation method, if needed. See
callback function for third-party interfaces
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).
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
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
anovaBF computes Bayes factors for either all submodels or select
submodels missing a single main effect or covariate, depending on the
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
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.
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
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
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
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 (email@example.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.
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## 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 (firstname.lastname@example.org). 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 --------------  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...  Omit color:shape : 2.723156 <U+00B1>3.94%  Omit color : 0.2292998 <U+00B1>2.97%  Omit shape : 0.2292272 <U+00B1>2.77% Against denominator: RT ~ shape + color + shape:color + ID --- Bayes factor type: BFlinearModel, JZS
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