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https://hdl.handle.net/2142/71503
Description
Title
Statistical Aspects of a New Latent Trait Model
Author(s)
Junker, Brian William
Issue Date
1988
Department of Study
Statistics
Discipline
Statistics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Education, Tests and Measurements
Statistics
Psychology, Psychometrics
Abstract
Current achievement and aptitude test modeling--item response theory--is based on the overly-optimistic assumption of local independence: that examinee's responses to different test questions will be independent conditional on the latent trait (ability) being measured by the questions. A more realistic account is presented here based on Stout's (1988a, 1988b) notion of essential independence in which the average covariance between the examinee's responses is small but not zero.
Essential independence is seen to be more natural psychometrically and more amenable to statistical tests of model fit than local independence. A new theorem proved here shows that the principal difference between local and essential independence is conditional association, a property introduced by Holland & Rosenbaum (1986).
Estimation procedures may still be developed under local independence as long as they are subsequently examined and calibrated for practical use under essential independence. This transition from local to essential independence is illustrated with two useful estimation procedures.
First a computationally simple latent trait distribution estimator, motivated under local independence, is shown to be consistent for estimating the latent distribution in an examinee population under the most general essential independence model. Pilot simulations show that this estimator should be useful even when local independence holds.
Second we examine the behavior of the maximum likelihood estimator $\\vartheta\sb{\rm J}$, computed from the likelihood under local independence, when in fact only essential independence holds. Under a technical strengthening of essential independence which is psychometrically innocuous, we show that $\\vartheta\sb{\rm J}$ continues to be consistent for the latent trait. If we require the average inter-item covariance to go to zero like 1/(test length) and impose a global controlling condition on the questions such as $\varphi$-mixing or association, $\\vartheta\sb{\rm J}$ is asymptotically normal and efficient. The central role of "proportion correct" and its variants in driving the behavior of latent trait estimators is also illustrated.
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