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

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Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
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