# Predicting Reliabilities and Separations of Different Length Tests

The Spearman-Brown "prophecy" formula predicts test (sample) reliabilities for similar dichotomous tests of different lengths. When the sample reliability of a hypothetical test of M items is to be predicted from a similar test of K items:

or, in terms of the sample Wright's Separation index:

Note: in RUMM2020 documentation, the "Separation Index" is the Rasch reliability (R). Wright' Separation = √ ((R)/(1-R))

Thus, the sample Separation index for a "unit" test of 1 item is SepK/K, and its sample reliability is RK/(1+K(1-RK)). These formulae also hold for tests containing homogeneous polytomous items, and for the estimation of item reliability and separation indices.

Here is an extension for the sample reliability of a test of polytomous items with differing numbers of categories.

A polytomous item of m ordered categories contains m-1 dichotomous category boundaries. The number of items in the known test is K. Then the number of active categories in the known test is:

where mi is the number of categories for item i (no matter how it is modeled). Thus a test of K items contains CK - K dichotomous decisions.

Then, for a similar test of M items, its sample reliability RM is:

and its sample Separation index, SepM, is:

and the Separation index for a "unit" test comprising 1 category boundary is

SepK/(CK-K),

and its reliability is

RK/(1+(CK-K)(1-RK)).

John Michael Linacre

1. Reliability, separation, strata statistics, Fisher WP Jr. … 6:3 p.238
2. Reliability and separation nomograms, Linacre JM. … 1995, 9:2 p.421
3. Reliability and separation, Wright BD. … 1996, 9:4 p.472
4. Predicting Reliabilities and Separations of Different Length Tests, Linacre, J.M. … 2000, 14:3 p.767
5. Going beyond Unreliable Reliabilities, Mallinson T., Stelmack J. … 2001, 14:4 p.787-8
6. Separation, Reliability and Skewed Distributions: Statistically Different Levels of Performance, Wright B.D. … 2001, 14:4 p.786
7. Number of Person or Item Strata (4G+1)/3, Wright BD, Masters GN. … 2002, 16:3 p.888
8. Cash value of Reliability, WP Fisher … Rasch Measurement Transactions, 2008, 22:1 p. 1160

Predicting Reliabilities and Separations of Different Length Tests. Linacre, J.M. … Rasch Measurement Transactions, 2000, 14:3 p.767

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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