Essential Independence and Likelihood-Based Ability Estimation for Polytomous Items.

A definition of essential independence is proposed for sequences of polytomous items. For items which satisfy the assumption that the expected amount of credit awarded increases with examinee ability, a theory of essential unidimensionality is developed that closely parallels that of W. F. Stout (1987, 1990). Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait that can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), an M-estimator based on maximum likelihood may be shown to be consistent for theta under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is provided, and the standard error of the estimator is explored. These results suggest that the ability estimation methods that rely on the summation form of the log-likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed. A 33-item list of references is included. (Author/SLD)