A Note on the Poisson's Binomial Distribution in Item Response Theory

The Poisson's binomial (PB) is the probability distribution of the number of successes in independent but not necessarily identically distributed binary trials. The independent non-identically distributed case emerges naturally in the field of item response theory, where answers to a set of binary items are conditionally independent given the level of ability, but with different probabilities of success. In many applications, the number of successes represents the score obtained by individuals, and the compound binomial (CB) distribution has been used to obtain score probabilities. It is shown here that the PB and the CB distributions lead to equivalent probabilities. Furthermore, one of the proposed algorithms to calculate the PB probabilities coincides exactly with the well-known Lord and Wingersky (LW) algorithm for CBs. Surprisingly, we could not find any reference in the psychometric literature pointing to this equivalence. In a simulation study, different methods to calculate the PB distribution are compared with the LW algorithm. Providing an exact alternative to the traditional LW approximation for obtaining score distributions is a contribution to the field.