StatTools : Sample Size for Two Counts : Explain, Calculations, and Tables

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Sample Size Introduction and Explanation Page
Poisson Distribution and Procedures Explained Page

Introduction Calculations Tables References
Introduction Comparison of Different Methods of Calculations
General discussions on Poisson distribution and comparing two counts are provided in the Poisson Distribution and Procedures Explained Page.

Three methods of calculating sample size for comparing two counts are available.

  • The Poisson's Test comparing two counts was initially described by Przyborowski and Wilenski (see reference), and is known as the Conditional Test (the C Test). The test is based on the null hypothesis that the ratio of the two count rates (λ2 / λ1) is equal to 1.
  • Krishnamoorthy and Thomson (see reference) proposed an improvement on the C Test, where the null hypothesis is that the difference between the two count rates (λ2 - λ1) is equal to 0. Althought this test is more complex, the advantages are that it is both more robust and more powerful, so the sample size required is smaller than that of the C Test.
  • Whitehead (see reference), in his text book on unpaired sequential analysis, provided algorithms to determine sample sizes for non-sequential methods, and a method for comparing two counts was also described.

Sample size calculation for the C Test and the E Test are based on the Poisson distribution. Computation requires multiple nested loops of calculating the Binomial Coefficient, and the computation time increases exponentially as the sample size increases. Sample size of more than 100 takes a few seconds to compute, but sample size over 1000 may take up to 30 minutes to compute.

Whitehead's algorithm is based on a transformation, where the difference between the two count rates (λ2 - λ1) is assumed to be a mean of a Normally distributed variable. The calculation is therefore short, and the samle size calculated is smaller than that for both the C Test and the E Test. This makes the whitehead approach eadier to use, but the assumption of normal distribution becomes increadingly inappropriate as the sample size decreases.

Please Note : In StatTools, estimating sample size requirement for comparing two counts uses the one tail model. For a two tail model, do the same calculation using half the α value. For example, sample size for α=0.05 in a two tail model is the same as that for α=0.1 in the one tail model, everything else being the same.