# Global Rasch Fit Statistic

Global Rasch Fit Statistic

Question: A Journal Editor insists I include a global statistic for fit of the Rasch model to my data. What do you recommend?

Answer: The Editor misunderstands the Rasch model, but this is not the moment to rectify that. Numerous global fit tests have been proposed reflecting the different ways in which the data can misfit the unattainable ideal of the Rasch model. Here's a practical approach. For each observation, there is a standardized residual and a model probability. So we can always compute usefully approximate chi-square statistics, regardless of missing data:

1. Pearson chi-square = sum of squared standardized residuals for all observations.

2. Log-likelihood chi-square = -2 * sum of the natural logarithms of the model probabilities for all observations. In practice, these values will differ. So we can choose the value better fitting our intentions, as is usually done in statistical modeling, or report both statistics. In both cases, the degrees of freedom for dichotomous data approximate:

d.f. = data point count - (person count + item count)

Omit items and persons with zero or perfect scores before doing these computations. For polytomies, also deduct from the d.f. the number of active categories (less 2) for each polytomous scale.

Since the expectation of a chi-square statistic is its d.f., you can obtain a more accurate estimate of the d.f. by simulating multiple sets of data with the same measurement structure as your data, and then using the average of their chi-square values as the reported d.f. for your chi-square.

Global Rasch Fit Statistic … Rasch Measurement Transactions, 2007, 21:2 p. 1103

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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