Computing elementary symmetric functions and their derivatives: A didactic

The computation of elementary symmetric functionsand their derivatives is an integral part of conditionalmaximum likelihood estimation of item parameters underthe Rasch model. The conditional approach has theadvantages of parameter estimates that are consistent(assuming the model is correct) and statistically rigorousgoodness-of-fit tests. Despite these characteristics, theconditional approach has been limited by problems incomputing the elementary symmetric functions. The introductionof recursive formulas for computing thesefunctions and the availability of modem computers haslargely mediated these problems; however, detaileddocumentation of how these formulas work is lacking.This paper describes how various recursion formulaswork and how they are used to compute elementarysymmetric functions and their derivatives. The availabilityof this information should promote a more thoroughunderstanding of item parameter estimation in the Raschmodel among both measurement specialists andpractitioners. Index terms: algorithms, computationaltechniques, conditional maximum likelihood, elementarysymmetric functions, Rasch model.