ttestBF {BayesFactor}

Function for Bayesian analysis of one- and two-sample designs

Description

This function computes Bayes factors, or samples from the posterior, for one- and two-sample designs.

Usage

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

Arguments

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

callback

callback function for third-party interfaces

...

further arguments to be passed to or from methods.

Details

The Bayes factor provided by ttestBF tests the null hypothesis that the mean (or mean difference) of a normal population is \mu_0 (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-\mu_0)/\sigma; for two sample tests, the standardized effect size is (\mu_2-\mu_1)/\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". These correspond to r scale values of \sqrt{2}/2, 1, and \sqrt{2} respectively.

The Bayes factor is computed via Gaussian quadrature.

Value

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.

Note

The default priors have changed from 1 to \sqrt{2}/2. The factor of \sqrt{2} 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.

Author(s)

Richard D. Morey ([e-mail])

References

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, 225-237

See Also

integrate, t.test

Examples

## 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"])