# ttestBF: Function for Bayesian analysis of one- and two-sample designs In BayesFactor: Computation of Bayes factors for common designs

## Description

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

## Usage

 ```1 2 3``` ``` ttestBF(x, y = NULL, formula = NULL, mu = 0, nullInterval = NULL, paired = FALSE, data = NULL, rscale = "medium", posterior = FALSE, ...) ```

## 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 `...` 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 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.

## 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 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.

## Author(s)

Richard D. Morey ([email protected])

## 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, 752-760

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

`integrate`, `t.test`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```## 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"]) ```

### Example output

```Loading required package: coda
************

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

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

BayesFactor documentation built on May 31, 2017, 4:17 a.m.