Maximum-Likelihood Estimation

Authored by: Cees A. W. Glas

Handbook of Item Response Theory Volume Two Statistical Tools

Print publication date:  February  2016
Online publication date:  March  2017

Print ISBN: 9781466514324
eBook ISBN: 9781315373645
Adobe ISBN:

10.1201/b19166-15

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Abstract

Item response theory (IRT) models are used to explain P × I matrices of responses to test items, U, using person parameters, ?, and item parameters, ?. In general, every person and every item has at least one parameter. A naive approach to maximum-likelihood (ML) estimation would be to maximize the likelihood function jointly with respect to ? and ?, using a standard technique such as a Newton–Raphson algorithm. But there are two problems with this approach. The first problem is a practical one arising from the large number of parameters. Application of a Newton–Raphson algorithm needs the inversion of the matrix of second-order derivatives of the likelihood function with respect to all parameters, so for larger numbers of persons this approach quickly becomes infeasible. Still, this practical problem might be solvable. A second, more fundamental problem is related to the consistency of the parameter estimates. Neyman and Scott (1948) showed that if the number of parameters grows proportional with the number of observations, this can lead to inconsistent parameter estimates (see Chapter 9). In a basic application of an IRT model, the problem applies to the person parameters. If we try to increase the precision of the item parameter estimates by sampling more persons, we automatically increase the number of person parameters. Simulation studies by Wright and Panchapakesan (1969) and Fischer and Scheiblechner (1970) showed that inconsistencies in the estimates for IRT models can occur indeed.

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