pingouin.bayesfactor_ttest

pingouin.bayesfactor_ttest(t, nx, ny=None, paired=False, alternative='two-sided', r=0.707)[source]

Bayes Factor of a T-test.

Parameters

tfloat: T-value of the T-test
nxint: Sample size of first group
nyint: Sample size of second group (only needed in case of an independent two-sample T-test)
pairedboolean: Specify whether the two observations are related (i.e. repeated measures) or independent.
alternativestring: Defines the alternative hypothesis, or tail of the test. Must be one of “two-sided” (default), “greater” or “less”.

Warning

One-sided Bayes Factor (BF) are simply obtained by doubling the two-sided BF, which is not the same behavior as R or JASP. Be extra careful when interpretating one-sided BF, and if you can, always double-check your results.
rfloat: Cauchy scale factor. Smaller values of r (e.g. 0.5), may be appropriate when small effect sizes are expected a priori; larger values of r are appropriate when large effect sizes are expected (Rouder et al 2009). The default is \(\sqrt{2} / 2 \approx 0.707\).

Returns

bffloat: Scaled Jeffrey-Zellner-Siow (JZS) Bayes Factor (BF10). The Bayes Factor quantifies the evidence in favour of the alternative hypothesis.

See also

ttest: T-test
pairwise_ttest: Pairwise T-tests
bayesfactor_pearson: Bayes Factor of a correlation
bayesfactor_binom: Bayes Factor of a binomial test

Notes

Adapted from a Matlab code found at https://github.com/anne-urai/Tools/tree/master/stats/BayesFactors

If you would like to compute the Bayes Factor directly from the raw data instead of from the T-value, use the pingouin.ttest() function.

The JZS Bayes Factor is approximated using the formula described in ref [1]:

\[\text{BF}_{10} = \frac{\int_{0}^{\infty}(1 + Ngr^2)^{-1/2} (1 + \frac{t^2}{v(1 + Ngr^2)})^{-(v+1) / 2}(2\pi)^{-1/2}g^ {-3/2}e^{-1/2g}}{(1 + \frac{t^2}{v})^{-(v+1) / 2}}\]

where \(t\) is the T-value, \(v\) the degrees of freedom, \(N\) the sample size, \(r\) the Cauchy scale factor (= prior on effect size) and \(g\) is is an auxiliary variable that is integrated out numerically.

Results have been validated against JASP and the BayesFactor R package.

References

1: Rouder, J.N., Speckman, P.L., Sun, D., Morey, R.D., Iverson, G., 2009. Bayesian t tests for accepting and rejecting the null hypothesis. Psychon. Bull. Rev. 16, 225–237. https://doi.org/10.3758/PBR.16.2.225

Examples

Bayes Factor of an independent two-sample T-test

>>> from pingouin import bayesfactor_ttest
>>> bf = bayesfactor_ttest(3.5, 20, 20)
>>> print("Bayes Factor: %.3f (two-sample independent)" % bf)
Bayes Factor: 26.743 (two-sample independent)

Bayes Factor of a paired two-sample T-test

>>> bf = bayesfactor_ttest(3.5, 20, 20, paired=True)
>>> print("Bayes Factor: %.3f (two-sample paired)" % bf)
Bayes Factor: 17.185 (two-sample paired)

Now specifying the direction of the test

>>> tval = -3.5
>>> bf_greater = bayesfactor_ttest(tval, 20, alternative='greater')
>>> bf_less = bayesfactor_ttest(tval, 20, alternative='less')
>>> print("BF10-greater: %.3f | BF10-less: %.3f" % (bf_greater, bf_less))
BF10-greater: 0.029 | BF10-less: 34.369