David Andrich | PubMed Health

PubMed2003

Conditional pairwise estimation in the Rasch model for ordered response categories using principal components.

In the Rasch model for items with more than two ordered response categories, the thresholds that define the successive categories are an integral part of the structure of each item in that the probability of the response in any category is a function of all thresholds, not just the thresholds between any two categories. This paper describes a method of estimation for the Rasch model that takes advantage of this structure. In particular, instead of estimating the thresholds directly, it estimates the principal components of the thresholds, from which threshold estimates are then recovered. The principal components are estimated using a pairwise maximum likelihood algorithm which specialises to the well known algorithm for dichotomous items. The method of estimation has three advantageous properties. First, by considering items in all possible pairs, sufficiency in the Rasch model is exploited with the person parameter conditioned out in estimating the item parameters, and by analogy to the pairwise algorithm for dichotomous items, the estimates appear to be consistent, though unlike for the dichotomous case, no formal proof has yet been provided. Second, the estimates of each item parameter is a function of frequencies in all categories of the item rather than just a function of frequencies of two adjacent categories. This stabilizes estimates in the presence of low frequency data. Third, the procedure accounts readily for missing data. All of these properties are important when the model is used for constructing variables from large scale data sets which must account for structurally missing data. A simulation study shows that the quality of the estimates is excellent.

Algorithms↗

PubMed2002

Understanding resistance to the data-model relationship in Rasch's paradigm: a reflection for the next generation.

The case for the Rasch models, that The comparison between two stimuli should be independent of which particular individuals were instrumental for the comparison; and vice versa (Rasch, 1961), does not depend on the models accounting for any data set. This has two distinctive consequences on the data-model relationship for the Rasch models. First, and this was recognized by Rasch, when there are deviations of one sort or another, it turns upside down the question of whether it is the model or the test that has gone wrong (Rasch, 1960). Second, because the invariance of comparisons among stimuli, and vice versa, is built into the model rather than being merely a requirement of data, further implications of this requirement can be derived mathematically. These implications, too, inevitably turn some questions, and their solutions, upside down. It is argued that having to look at these implications upside down produces substantial psychological and intellectual resistance amongst those schooled in looking at them in the traditional way. It is also argued that in turning the question upside down, Rasch had an insight that goes beyond the mathematical derivations, and that to sustain this insight requires a paradigm shift (Kuhn, 1970) in the data-model relationship. Using an illustrative example, it is suggested that to maintain this paradigm shift, even by those who research the Rasch models, requires the same uncompromising consistency and passion that Rasch displayed in maintaining faith in his insight.

Humans↗