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
Please help with Standard Dataset 4: Andrich Rating Scale Model
|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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