binbf01: Binomial Bayes factor

Description

This function computes the Bayes factor for testing a binomial proportion \(p\) based on \(x\) observed successes out of \(n\) trials. Two types of tests are available:

Test of a point null hypothesis: The Bayes factor quantifies the evidence for \(H_0 \colon p = p_0\) against \(H_1 \colon p \neq p_0\). A beta prior is assigned to the proportion \(p\) under the alternative hypothesis \(H_1\).
Test of a directional null hypothesis: The Bayes factor quantifies the evidence for \(H_0 \colon p \leq p_0\) against \(H_1 \colon p > p_0\). A beta prior that is truncated to the range \([0, p_0]\) under the null \(H_0\) and to \((p_0, 1]\) under the alternative \(H_1\) is assigned to the proportion \(p\) under the corresponding hypothesis.

Usage

binbf01(
  x,
  n,
  p0 = 0.5,
  type = c("point", "direction"),
  a = 1,
  b = 1,
  log = FALSE
)

Value

Bayes factor in favor of the null hypothesis over the alternative (\(\text{BF}_{01}\) > 1 indicates evidence for the null hypothesis, whereas \(\text{BF}_{01}\) < 1 indicates evidence for the alternative)

Arguments

x: Number of successes
n: Number of trials
p0: Tested binomial proportion. Defaults to 0.5
type: Type of test. Can be "point" or "directional". Defaults to "point"
a: Number of successes parameter of the beta prior distribution. Defaults to 1
b: Number of failures parameter of the beta prior distribution. Defaults to 1
log: Logical indicating whether the natural logarithm of the Bayes factor should be returned. Defaults to FALSE

Author

Samuel Pawel

Examples

Run this code

## example on Mendelian inheritance from ?stats::binom.test
binbf01(x = 682, n = 925, p0 = 3/4, a = 1, b = 1, type = "point")
## 18.6 => strong evidence for the hypothesized p = 3/4 compared to other p

## with directional hypothesis
binbf01(x = 682, n = 925, p0 = 3/4, a = 1, b = 1, type = "direction")
## 1.5 => only anecdotal evidence for p <= 3/4 over p > 3/4

## Particle-counting experiment from Stone (1997) with point null
binbf01(x = 106298, n = 527135, p0 = 0.2, a = 1, b = 1, type = "point")
## 8.1 => moderate evidence for the alternative over the null

## Coin flip experiment from Bartos et al. (2023) with point null
binbf01(x = 178079, n = 350757 , p0 = 0.5, a = 5100, b = 4900, type = "point")
## => 1/1.72e+17 extreme evidence in favor of the alternative over the null