Understanding Rasch measurement: estimation methods for Rasch measures

Rasch parameter estimation methods can be classified as non-interative and iterative. Non-iterative methods include the normal approximation algorithm (PROX) for complete dichotomous data. Iterative methods fall into 3 types. Datum-by-datum methods include Gaussian least-squares, minimum chi-square, and the pairwise (PAIR) method. Marginal methods without distributional assumptions include conditional maximum-likelihood estimation (CMLE), joint maximum-likelihood estimation (JMLE) and log-linear approaches. Marginal methods with distributional assumptions include marginal maximum-likelihood estimation (MMLE) and the normal approximation algorithm (PROX) for missing data. Estimates from all methods are characterized by standard errors and quality-control fit statistics. Standard errors can be local (defined relative to the measure of a particular item) or general (defined relative to the abstract origin of the scale). They can also be ideal (as though the data fit the model) or inflated by the misfit to the model present in the data. Five computer programs, implementing different estimation methods, produce statistically equivalent estimates. Nevertheless, comparing estimates from different programs requires care.