# Fit to Models: Rasch Model vs. Correlation Model

Viewed as a statistical device, the Rasch model is one of thousands in current use. One of those thousands most frequently employed is the Pearson Correlation Model.

The Correlation Model

The size of the Pearson product-moment correlation between two variables is frequently reported, sometimes accompanied by whether it is significantly different from 0.00. But rarely reported are:

1) whether the observed correlation departs insignificantly from 1.00, which is perfect correlation. But high correlations, regardless of their statistical significance, could be indicative of collinearity. Near-perfect correlation should be regarded with suspicion.

2) whether the observations violate the assumptions underlying the Correlation Model. Violations are rarely tested explicitly because the correlation model is too useful not to use. Pearson correlations are often reported for data which are known not to meet its assumptions.

The Rasch Model

The Rasch model is similarly too useful not to use. Further, near perfect fit to the Rasch Model should be regarded with suspicion. Empirical processes are uneven. The validity of scientific work has come into question when statistical findings appear to be too perfect.

Taking the same position with regards to the Rasch Model as we do for the Correlation Model, the crucial question is not "Is the correlation statistically 1.0", expressed as "Do the data fit the Rasch model statistically perfectly?" This question has been the focal point of most global fit analysis with the Rasch model. Instead the crucial question becomes "Is the correlation statistically different from 0.00", expressed as "Is there a Rasch dimension which is significantly larger than a point?"

The Rasch dimension reduces to the size of a point when the data are perfectly random. Jacob Cohen (1992) suggests that, for the ratio of explained variance to unexplained variance, 2% is a small effect size, 15% is a medium effect size, and 35% is a large effect size. Recast this as the percentage of total variance explained and 2% is a small effect size, 13% is a medium effect size, and 26% is a large effect size. For comparison, the variance explained by the Rasch measures for the Liking for Science data is 51% [revised, 2008] and for the Knox Cube Test data is 71% [revised, 2008]. Even the variance explained for a relatively central, poorly fitting, NSF survey data set is 30% [revised, 2008]. Rasch papers can routinely report effect statistics, which, if they were the findings of correlation studies, would produce great joy among social scientists.

John M. Linacre

Cohen J. (1992) A Power Primer, Psychological Bulletin, 112, 155-159.

Fit to Models: Rasch Model vs. Correlation Model. Linacre J.M. … Rasch Measurement Transactions, 2005, 19:3 p. 1029

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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