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On the Relationships between Sum Score Based Estimation and Joint Maximum Likelihood Estimation

Published online by Cambridge University Press:  01 January 2025

Guido del Pino ,
Ernesto San Martín ,
Jorge González  and
Paul De Boeck
Guido del Pino
Affiliation:
Pontificia Universidad Católica de Chile
Ernesto San Martín*
Affiliation:
Pontificia Universidad Católica de Chile
Jorge González
Affiliation:
K.U. Leuven
Paul De Boeck
Affiliation:
K.U. Leuven
*
Requests for reprints should be sent to Ernesto San Martín, Department of Statistics, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile. E-mail: esanmart@mat.puc.cl
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Abstract

This paper analyzes the sum score based (SSB) formulation of the Rasch model, where items and sum scores of persons are considered as factors in a logit model. After reviewing the evolution leading to the equality between their maximum likelihood estimates, the SSB model is then discussed from the point of view of pseudo-likelihood and of misspecified models. This is then employed to provide new insights into the origin of the known inconsistency of the difficulty parameter estimates in the Rasch model. The main results consist of exact relationships between the estimated standard errors for both models; and, for the ability parameters, an upper bound for the estimated standard errors of the Rasch model in terms of those for the SSB model, which are more easily available.

Keywords

Rasch modelstandard errorinformation matrixpseudo-likelihood
Type
Theory and Methods
Information
Psychometrika , Volume 73 , Issue 1 , March 2008 , pp. 145 - 151
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

The authors acknowledge partial financial support from the FONDECYT Project No. 1060722 from the Chilean Government, and the BIL05/03 grant to P. De Boeck, E. Lesaffre and G. Molenberghs (Flanders) for a collaboration with G. del Pino, E. San Martín, F. Quintana and J. Manzi (Chile).

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