Estimates and Estimators: connecting model to data
The essential attributes of a Rasch model are sufficient statistics and separable parameters, which allow, but don’t guarantee, specific objectivity. Well, actually sufficient statistics come pretty close if they really are sufficient to capture all the relevant information in the data. We will come back to this in the discussion of what Rasch called control of the model and most of us call goodness of fit. The current topic is a demonstration, more intuitive than mathematical, of how to manipulate the model to estimate item difficulties.
The process begins with the basic Rasch model for how likely the person wins when one person takes one dichotomous item: . . . The Disappearing Beta Trick
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2 thoughts on “IV. Doing the math (and a little algebra)”
A Little History for Credibility
Rasch himself in his seminal work (1960) used a pair-wise approach with the Poisson model both to obtain estimates and, equally important, to illuminate the method. Choppin (1968, 1983) and Andrich (1988) provided the details for the standard binomial form for dichotomous items. A close relative, pair-wise maximum likelihood, is discussed by Masters and Wright (1982) and Andrich and Luo (2003) gave detailed explanations for both Partial Credit and Rating Scale analyses and implemented in the RUMM software (Andrich, Lyne, Sheridan, and Luo (1997)). Another elaboration has been done by Garner and Englhard (2002).
The method discussed here is a least squares approach and while not the most efficient, in the statistical sense, it is easily implemented and does not require iterating.
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