In my last Note I reminded you of the fact that the Rasch model has simple sufficient statistics for its parameters. A favorable consequence of the presence of sufficient statistics is that conditional likelihood equations can be established that only have ability or difficulty parameters. This Note is to warn you not to over-interpret this property as "sample-independent estimation in the Rasch model."
For example, let's take the case of conditional maximum likelihood estimation CMLE of the ability parameters and assume that the Rasch model holds for a certain pool of items. An overly simple interpretation would be to state that from the above property it follows that "the same difficulty parameters can be estimated from any subset of examinees." This statement holds for any estimation problem! The pertinent question is not if we can estimate parameters from different samples, but how well we can estimate them. Statements that do not refer to any criterion of optimality holding for the estimators are therefore incomplete.
Such criteria of optimality are always formulated in terms of the distributions of estimators over repeated sampling. Now what properties of the sampling distribution remain invariant if we replicate measurement of the same examinee, but include different items in the test? Is it the full distribution of the ability estimator that does not change? Or just its expected value (mean) or variance?
We know that the variance of the estimator of the ability parameter depends on the item parameters. This property is welcome, for it allows us to construct an optimal test for a population of examinees. Thus, for different items we have different variances of our estimators. And hence even different distributions! But what about the mean? Does the mean not remain invariant? For tests of finite sizes, the estimators for the ability parameters, like most maximum likelihood estimators, are known to be biased. And the bias may be dependent on the values of the parameters in the sample - as is demonstrated, for example, by the non-zero probability that some of the items have to be removed from the test because all examinees have them correct. Hence, for different tests of finite length, we are likely to have estimators of the same ability with a different mean.
So what is left? It is only the property that if test length goes to infinity, each ability estimator will have the same expected value or mean, whatever the composition of the test. This property is known as consistency in statistics. It holds uniquely for the Rasch model among all response models with incidental parameters.
It is recommended to use this well-defined term, consistency, rather than the contradictio in terminis "sample-independent estimation."
Sample Independency in the Rasch Model?, W van der Linden … Rasch Measurement Transactions, 1993, 6:4 p. 247
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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 |
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