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:
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
and its reliability is
John Michael Linacre
Predicting Reliabilities and Separations of Different Length Tests. Linacre, J.M. Rasch Measurement Transactions, 2000, 14:3 p.767
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