Georg Rasch was concerned to construct measurement models with "parameters that are specific to the individuals" (1960, p. 9). An early extension of the Rasch model specified that the "individuals" were to be treated as random variables drawn from some distribution. This is now done routinely using the PROX algorithm and Marginal Maximum Likelihood Estimation MMLE.
Lalitha Sanathanan in "Some Properties of the Logistic Model for Dichotomous Response" (JASA, 69, 347, 744-749, 1974!) attempts this, but immediately encounters a major hurdle, the multiplicative form of the Rasch model then in common use. After Herculaean effort, she derives a simple approximation, but is forced to present it in additive form in order to make it tractable. Rewritten, the approximation is:
where N is the sample size, Si is the number of correct answers to item i, and Di is the difficulty of item i. α and β are sample dependent, but how they relate to the mean and standard deviation of the distribution is omitted from the paper. In fact, this formulation is equivalent to the PROX equation for a sample distributed N(μ,σ²), when
Sanathanan realizes that she has "shown how the parameters in the model can be calculated in a rough ready manner" (p. 749), but the utility of her insight was lost due to her abstruse math.
Sanathanan's plot of expected score against item difficulty for different sample distributions.
Random effects Rasch model. Sanathanan, L. Rasch Measurement Transactions, 2002, 16:2 p.881
|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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