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 .