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After marginal maximum likelihood estimates, and , of , the matrix of effects, and , the residual covariance matrix are computed, five sets of distributional draws (plausible or imputed values; Rubin 1987), denoted by _{m} (m = 1, 2, 3, 4, 5), for all sampled students are drawn in the following three-step process.

1) A vector _{m} is drawn randomly from the distribution of as estimated using the population-structure model, conditional on the data, the matrix (the marginal maximum likelihood estimate from the population-structure model, assumed to be fixed), and , the Item Response Theory (IRT) parameter estimates (assumed to be fixed). The distribution

from which _{m} is drawn is estimated when the population-structure models are estimated.

2) Conditional on the generated value _{m} and the fixed value , the estimated mean _{rm} and the estimated variance _{rm} of student r are computed from the distribution

using the EM algorithm (see Thomas 1993a).

3) A multivariate plausible value, _{rm}, is drawn independently from a multivariate normal distribution with mean _{rm} and variance _{rm}. In other words, for each student r in the student sample,

where _{rm} is a vector of length k, the number of subscales for the content area.

These three steps are repeated five times producing five sets of distributional draws (m = 1, 2, 3, 4, 5) that are attached to the records of all sampled respondents. Each plausible value is drawn from a distribution that differs in its mean, _{rm} and its variance _{rm}. Each set of plausible values is based on a different estimate of , but the same estimate of .

Last updated 01 February 2008 (GF)