Computing elementary symmetric functions and their derivatives: A didactic

The computation of elementary symmetric functions and their derivatives is an integral part of conditional maximum likelihood estimation of item parameters under the Rasch model. The conditional approach has the advantages of parameter estimates that are consistent (assuming the model is correct) and statistically rigorous goodness-of-fit tests. Despite these characteristics, the conditional approach has been limited by problems in computing the elementary symmetric functions. The introduction of recursive formulas for computing these functions and the availability of modern computers has largely mediated these problems; however, detailed documentation of how these formulas work is lacking. This paper describes how various recursion formulas work and how they are used to compute elementary symmetric functions and their derivatives. The availability of this information should promote a more thorough understanding of item parameter estimation in the Rasch model among both measurement specialists and practitioners.