When does a Gap between Measures Matter?

When two item difficulty measures (or two person measures) are located along a latent variable, how big must the gap be for it to be important?

Gaps based on probabilities: for dichotomies, these correspond to what differential chance of success would matter. If a 60% chance of success is thought to be importantly different from a 50% chance, then the logit difference is 0.4 logits, so a gap of 0.4 logits matters. For polytomies, this calculation tends is more complex.

Gaps based on substance: these usually correspond to "what is the smallest difference that an informed observer would see to be definitely different"? In many educational situations a gap that matters is about 0.5 logits, roughly half a grade level at school.

Gaps based on statistical significance: these are computed from the standard errors of the individual measures. The more data usually the smaller the standard errors. So for .15 logits to represent a statistically significant gap (using a two-sided .05 t-test) between two measures, the individual measure standard errors must be about .05 logits, corresponding to about 250 dichotomous responses underlying each measure.
Algebraically, t = (M1 - M2) / sqrt(SE1**2 + SE2**2) where M1 is one measure with standard errorr SE1, and M2 is the other measure with standard error SE2.

Gaps based on effect-size: these are used in education, where it is felt that students whose abilities are 2 S.D.s above the sample mean ability are in a higher performing group.

For polytomies (rating scales, partial credit, etc.): The math is more complicated and probabilistic implications hard to explain, so it usually comes down to substance. Lai & Eton (2002, RMT 15:4, 850) report 0.5 logits to be a clinically meaningful gap for one instrument.

Linacre J.M. (2004) When does a gap between measures matter?, … Rasch Measurement Transactions, 18:3 p. 993

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
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