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A simple scheme is proposed for smoothly approximating the ability distribution for relatively long tests, assuming that the item characteristic curves (ICCs) are known or well estimated. The scheme works for a general class of ICCs and is guaranteed to completely recover the theta distribution as the test length increases. The proposed method of estimating the ability distribution is robust to some violations of local independence. After an initial function inversion, the scheme can be inexpensively used to recover the theta distribution in each of several different administrations of the same test or several subpopulations in one test administration. Moreover, this approach could be used to recover the distribution of a dominant ability dimension when local independence fails. The scheme provides a starting place for diagnostics concerning assumptions about the shape of the theta distribution or ICCs of a particular test. Work is currently under way to further examine and refine these methods using essentially unidimensional simulation data and to apply the estimator to real tests. Kernel smoothing is also considered. A 16-item list of references, 10 tables, 8 graphs, and 2 appendixes that provide details of the simulation and proofs are included. (RLC)

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)