## Quantifying Item Dependency

Some items are more closely related than others. Comprehension items that address the same written text and arithmetic items requiring the same operation are examples. In Rasch analysis, this is no major source for concern provided there is no extreme systematic dependency between two or a few items. When such dependency does exist, combining the separate items into one partial-credit item may be useful. Low values of the Rasch INFIT statistic signal dependency. When a diagnostically specific investigation of inter-item dependency is required, Fisher's Z is useful.

1)Obtain the Rasch-based standardized residuals for each observation of Person n on item i:

zni = (Observedni - Expectedni)/Standard Errorni

2)Correlate the standardized residuals for all pairs of items i, j across all N persons:

rij = (sum(zni znj) - sum(zni)sum(znj)/N) / N SD(zni) SD(znj)

3)Use Fisher's Z to linearize Pearson correlations:

Zij = 1/2 Loge( (1+rij)/(1-rij) )

4)Draw a histogram of the Fisher Z values. Noticeable positive skew (see Figures) indicates the presence of sets of dependent items. A cluster analysis of the Z matrix will identify the sets. Outlying positive Z values flag pairs or sets of highly dependent items for further investigation.

Quantifying item dependency. Shen L. … Rasch Measurement Transactions, 1996, 10:1 p.485

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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
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