The Item Log-Likelihood Surface for Two- and Three-Parameter Item Characteristic Curve Models

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Abstract
This article investigated the form of item log-likeli hood surface under two- and three-parameter logistic models. Graphs of the log-likelihood surfaces for items under two-parameter and three-parameter (with a fixed value of c) models were very similar, but were characterized by the presence of a ridge. These graphs suggest that the task of finding the maximum of the surface should be roughly equivalent under these two models when c is fixed in the three-parameter model. For two items, the item log-likelihood surface was plotted for several values of c to obtain the contour line of the maxima. For an item whose value of Lord's b — 2/ a index was less than the criterion value, the contour line was relatively flat. The item having an index value above the criterion value had a contour line with a very sharp peak. Thus, under a three-pa rameter model, finding the maximum of the item log- likelihood is more difficult when the criterion for Lord's index is not met. These results confirm that the LOGIST program procedures used to locate the maxi mum of the likelihood function are consistent with the form of the item log-likelihood surface. Index terms: estimation, item parameter; likelihood surfaces; LOGIST procedures; log-likelihood; maximum likelihood estimation.