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Creation of Plausible Values

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

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

The probability of matrix gamma given matrices x, y, alpha hat, and sigma hat

from which gamma hatm is drawn is estimated when the population-structure models are estimated.

2) Conditional on the generated value gamma hatm and the fixed value sigma hat, the estimated mean theta tilderm and the estimated variance tau tildarm of student r are computed from the distribution

The probability of matrix gamma given matrices x, y, alpha hat, and sigma hat

using the EM algorithm (see Thomas 1993a).

3) A multivariate plausible value, theta hatrm, is drawn independently from a multivariate normal distribution with mean theta tildarm and variance tau tildarm. In other words, for each student r in the student sample,

The vector theta hat sub r m is distributed as multivariate normal with a mnean of theta bar sub r m and standard deviation of sigma sub r m

where theta hatrm 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, theta tildarm and its variance tau tildarm. Each set of plausible values is based on a different estimate of matrix gamma, but the same estimate of matrix sigma.

Last updated 01 February 2008 (GF)

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