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A Note on the Poisson's Binomial Distribution in Item Response Theory
Jorge González1, Marie Wiberg2, Alina A von Davier3
1Pontificia Universidad Católica de Chile, Santiago, Chile.
Applied Psychological Measurement
|June 9, 2018
Summary
The Poisson-binomial distribution is equivalent to the compound binomial distribution in item response theory. This finding offers an exact alternative to the traditional Lord and Wingersky approximation for calculating score distributions.
Area of Science:
- Statistics
- Psychometrics
- Item Response Theory
Background:
- The Poisson-binomial distribution models successes in independent, non-identically distributed binary trials.
- This distribution is relevant in item response theory where item probabilities vary.
- Compound binomial distribution is often used for calculating score probabilities.
Purpose of the Study:
- To demonstrate the equivalence between Poisson-binomial and compound binomial distributions.
- To identify computational algorithms for these distributions.
- To compare calculation methods in a simulation study.
Main Methods:
- Theoretical derivation of probability equivalence.
- Algorithm comparison, including the Lord and Wingersky algorithm.
- Simulation study to assess computational methods.
Main Results:
- Poisson-binomial and compound binomial distributions yield equivalent probabilities.
- A Poisson-binomial algorithm exactly matches the Lord and Wingersky algorithm.
- Simulation results validate the comparison of calculation methods.
Conclusions:
- The study establishes a significant equivalence in item response theory.
- An exact method for calculating score distributions is provided.
- This work offers a valuable alternative to existing approximations.
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