ttestBF: Function for Bayesian analysis of one- and two-sample 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, or samples from the posterior, for one- and two-sample designs.

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  ttestBF(x, y = NULL, formula = NULL, mu = 0,
    nullInterval = NULL, paired = FALSE, data = NULL,
    rscale = "medium", posterior = FALSE, ...)

`x`	a vector of observations for the first (or only) group
`y`	a vector of observations for the second group (or condition, for paired)
`formula`	for independent-group designs, a (optional) formula describing the model
`mu`	for one-sample and paired designs, the null value of the mean (or mean difference)
`nullInterval`	optional vector of length 2 containing lower and upper bounds of an interval hypothesis to test, in standardized units
`paired`	if `TRUE`, observations are paired
`data`	for use with `formula`, a data frame containing all the data
`rscale`	prior scale. A number of preset values can be given as strings; see Details.
`posterior`	if `TRUE`, return samples from the posterior instead of Bayes factor
`...`	further arguments to be passed to or from methods.

The Bayes factor provided by ttestBF tests the null hypothesis that the mean (or mean difference) of a normal population is mu0 (argument mu). Specifically, the Bayes factor compares two hypotheses: that the standardized effect size is 0, or that the standardized effect size is not 0. For one-sample tests, the standardized effect size is (mu-mu0)/sigma; for two sample tests, the standardized effect size is (mu2-mu1)/sigma.

A noninformative Jeffreys prior is placed on the variance of the normal population, while a Cauchy prior is placed on the standardized effect size. The rscale argument controls the scale of the prior distribution, with rscale=1 yielding a standard Cauchy prior. See the references below for more details.

For the rscale argument, several named values are recognized: "medium", "wide", and "ultrawide". For the one-sample test, these correspond to r scale values of 1/2, sqrt(2)/2, and 1, respectively. For the two-sample test, they correspond to sqrt(2)/2, 1, and sqrt(2) respectively.

The Bayes factor is computed via Gaussian quadrature.

If posterior is FALSE, an object of class BFBayesFactor containing the computed model comparisons is returned. If nullInterval is defined, then two Bayes factors will be computed: The Bayes factor for the interval against the null hypothesis that the standardized effect is 0, and the corresponding Bayes factor for the compliment of the interval.

If posterior is TRUE, an object of class BFmcmc, containing MCMC samples from the posterior is returned.

The default priors have scale has changed from 1 to √{2}/2 for the two-sample t test, and 1/2 for the one-sample t test. The factor of √{2} in the two-sample t test is to be consistent with Morey et al. (2011) and Rouder et al. (2012), and the factor of 1/2 in both is to better scale the expected effect sizes; the previous scaling put more weight on larger effect sizes. To obtain the same Bayes factors as Rouder et al. (2009), change the prior scale to 1.

Richard D. Morey (richarddmorey@gmail.com)

Morey, R. D., Rouder, J. N., Pratte, M. S., & Speckman, P. L. (2011). Using MCMC chain outputs to efficiently estimate Bayes factors. Journal of Mathematical Psychology, 55, 368-378

Morey, R. D. \& Rouder, J. N. (2011). Bayes Factor Approaches for Testing Interval Null Hypotheses. Psychological Methods, 16, 406-419

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16, 752-760

Perception and Cognition Lab (University of Missouri): Bayes factor calculators. http://pcl.missouri.edu/bayesfactor

integrate, t.test

## Sleep data from t test example
data(sleep)
plot(extra ~ group, data = sleep)

## paired t test
ttestBF(x = sleep$extra[sleep$group==1], y = sleep$extra[sleep$group==2], paired=TRUE)

## Sample from the corresponding posterior distribution
samples = ttestBF(x = sleep$extra[sleep$group==1], y = sleep$extra[sleep$group==2], paired=TRUE, posterior = TRUE, iterations = 1000)
plot(samples[,"mu"])

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

Type BFManual() to open the manual.
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Bayes factor analysis
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[1] Alt., r=0.707 : 17.25888 <U+00B1>0%

Against denominator:
  Null, mu = 0 
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Bayes factor type: BFoneSample, JZS