inst/help/BinomialTestBayesian.md

In jasp-stats/jaspFrequencies: Frequencies Module for JASP

Bayesian Binomial Test

The Bayesian binomial test allows you to test whether a proportion of a dichotomous variable is equal to a test value (presumed population value). The analysis returns a binomial test for each level of the dependent variable against all other levels, so it will also work for variables with more than two levels.

Assumptions

• The variable should be a dichotomous scale.
• Observations should be independent.

Input

• Test value: The proportion of the variable under the null hypothesis. By default, this is set to 0.5.

Alt. Hypothesis

• Direction
• ≠ Test value: Two-sided alternative hypothesis that the proportion is not equal to test value.
• > Test value: One-sided alternative hypothesis that the proportion is larger than the test value.

• < Test value: One-sided alternative hypothesis that the proportion is smaller than the test value.

• Prior: Beta parameters a and b for the prior distribution under the alternative hypothesis. By default they are set to '1' each. This corresponds to a uniform prior distribution. See Learn Bayes module for more information and alternatives.

Bayes Factor

• BF10: By selecting this option, the Bayes factor will show evidence for the alternative hypothesis relative to the null hypothesis. This option is selected by default.
• BF01 : By selecting this option, the Bayes factor will show evidence for the null hypothesis relative to the alternative hypothesis. This is equal to 1/BF10.
• Log(BF10) : By selecting this option, the natural logarithm of BF10, BFm, BFInclusion, BF10, U will be displayed in the output.

Plots

• Prior and posterior: Displays the prior and posterior density of the population proportion under the alternative hypothesis.
  • Additional info: Adds the Bayes factor computed with the user-defined prior; adds a probability wheel depicting the odds of the data under the null vs. alternative hypothesis; adds the median and the 95% credible interval of the posterior density of the effect size.
• Sequential analysis: Displays the development of the Bayes factor as the data come in using the user-defined prior.

Output

Bayesian Binomial Test

• Level: The two options of the dichotomous variable.
• Counts: the count of the instances at the certain level of the dichotomous variable.
• Total: the total number of observations.
• Proportion: calculated by counts/total.
• BF10 (or BF01): Bayes factor. If a one-sided test is requested:
• BF+0: Bayes factor that quantifies evidence for the one-sided alternative hypothesis that the population mean is larger than the test value, relative to the null hypothesis.
• BF-0: Bayes factor that quantifies evidence for the one-sided alternative hypothesis that the population mean is smaller than the test value, relative to the null hypothesis.
• BF0+: Bayes factor that quantifies evidence for the null hypothesis, relative to the one-sided alternative hypothesis that the population mean is larger than the test value.
• BF0-: Bayes factor that quantifies evidence for the null hypothesis, relative to the one-sided alternative hypothesis that that the population mean is smaller than the test value.

Plots

• Prior and posterior:
• Displays the prior (dashed line) and posterior (solid line) density of the population proportion under the alternative hypothesis; the gray circles represent the height of the prior and the posterior density at the test value. The horizontal solid line represents the width of the 95% credible interval of the posterior.
• Additional info: Displays the Bayes factor computed with the user-defined prior; displays a probability wheel depicting the odds of the data under the null vs. alternative hypothesis; displays the median and 95% credible interval of the posterior density.
• Sequential analysis:
• Displays the development of the Bayes factor as a function of the number of observations (n) using the user-defined prior; displays the Bayes factor computed with the user-defined prior; displays a probability wheel depicting the odds of the data under the null vs. alternative hypothesis; displays the median and 95% credible interval of the posterior density; shows the decisiveness of the evidence in terms of Jeffreys' (1961) evidence categories.

References

• Jeffreys, H. (1961). Theory of Probability. Oxford, Oxford University Press.
• O’Hagan, A., & Forster, J. (2004). Kendall’s advanced theory of statistics vol. 2B: Bayesian inference (2nd ed.). London: Arnold.
• Haldane, J. B. S. (1932). A note on inverse probability. Mathematical Proceedings of the Cambridge Philosophical Society, 28, 55-61.

R Packages

• ggplot2
• plotrix
• stats

jasp-stats/jaspFrequencies documentation built on April 5, 2025, 3:53 p.m.