Marginal Maximum Likelihood Estimation

Maximum Likelihood Estimation

Maximum likelihood estimation procedures rest on assumptions about the data distribution of the population from which the sample was drawn. For example, in IRT, it is generally assumed that the latent trait of interest, θ, is normally distributed, although it is sometimes possible to relax this assumption. Consider a reading test made up of 20 multiple-choice items, each of which has a specific level of difficulty. Furthermore, let’s assume that the difficulty values are known. Given this information, one can use the item response pattern to estimate an examinee’s reading ability. If Examinees A and B both answer 8 of the 20 items correctly, each will have a raw score of 8. However, if the 8 items answered correctly by Examinee A were more difficult than the 8 items answered correctly by Examinee B, then the value of ability level, θ, for Examinee A should be estimated as higher than that of Examinee B. The ability to account for differences in the difficulty of items answered correctly by respondents is one of the great strengths of using maximum likelihood estimation in the context of IRT parameter estimation.

Maximum likelihood estimation works by finding the values of the model parameters that maximize a likelihood function for the item responses. For the reading test example, the item response pattern is simply the combination of 1s and 0s that reflect the correct and incorrect responses to the items. Thus, for the 20-item reading assessment, one possible response pattern would be 11010011101110001110. In short, the maximum likelihood approach can be thought of as a sophisticated search algorithm that seeks to find the combination of item and person parameters that as closely as possible reproduces the various item response patterns in the data.

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