Exact Binomial Test

This is a Statistical Test for proportions that uses the Binomial Distribution as the null (sampling) distribution.

It doesn't use the Normal Approximation

  • because sometimes it's possible to use the Binomial model directly
  • or because it's not possible to use the Normal Model: some conditions are not met


Binomial Model

Recall the formula:

  • $P(\text{success}) = { n \choose k } p^k (1 - p)^{n - k}$
  • this is the null distribution of our test


Test

  • the tail area of the null distribution:
    • add up the probabilities (using the formula) for all $k$ that support the alternative hypothesis $H_A$
  • one-sided test - use single tail area
  • two-sided - compute single tail and double it


Examples

Example 1: Medical Consultant (One-Sample)

  • medical consultant helps patients
  • he claims that with his help the ratio of complications is lower than usually
    • (i.e. lower than 0.10)
  • is it true?


We want to test a hypothesis:

  • $H_0: p_A = 0.10$ - ratio of complications without a specialist
  • $H_A: p_A < 0.10$ - specialist helps, the complications ratio is lower than usual

Observed data:

  • 3 complications in 62 cases
  • $\hat{p} = 0.048$
  • is it only due to chance?


Normal Model


Apply the Binomial Model:

  • $p\text{-val} = \sum_{j = 0}^3 { n \choose j } p^j (1 - p)^{n - j} = 0.0015 + 0.01 + 0.034 + 0.0355 = 0.121$
  • we don't reject the $H_0$ at $\alpha = 0.05$

check! sim got 0.04