StatTools : Binomial Test Explained

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Binomial Test Program Page

Introduction Example 1 Example 2 References
The Binomial test evaluate the probability Pbin of seeing k or more extreme number of positive cases in a sample of n cases, if the reference proportion is Pref.

Please note : Binomial probability assumes that the observation is one off. In most quality assurance programs repeated observations are made, and the probability estimates must take into account variations from repeated measurements, and different statistics should be used.

Three parameters are involved :

  • Reference proportion (Pref) : also known as expected proportion or probability, is the expected, theoretical, or reference proportion of cases that are positive in the indexed outcome (deaths, cures, winners, losers, complications, etc). This is usually expressed as a number between 0 and 1, so that 0.25 represents 25%. The abbreviation Pref is used in these programs.
  • Sample size (n) : is the number of cases observed in the current study, and this is abbreviated to n.
  • Numbers positives (k) : is the number of cases found to be positive in the sample observed in the current study, and this is abbreviated to k.

The test result is the Binomial Probability, abbreviated to Pbin. Two such values are calculated, although usually one is required in a particular study.

  • Pbin<=k is the probability of observing k or less number of positive cases amongst a sample of n cases, if the true proportion is Pref
  • Pbin>=k is the probability of observing k or more number of positive cases amongst a sample of n cases, if the true proportion is Pref

Calculations use the Binomial Coefficient, which required calculation of Factorial numbers. The duration of calculations therefore increases exponentially with large numbers, and sample size in excess of 1000 may exceed the time limits allowed by the server and crash the program. Should this happens, an alternative algorithm using the approximate Normal transformation of the proportion, using the following algorithm.

  1. mean (μ) = Pref * n
  2. Standard Deviation (sd) = sqrt(n * Pref * (1 - Pref))
  3. y = k
    • if k<μ then y = k + 0.5
    • if k>μ then y = k - 0.5
  4. z = (y-μ)/sd
  5. Pbin = probability of z, calculated in the Probability of z Program Page