NAEP Analysis and Scaling → Estimation of Population and Student Group Distributions → Using Population-Structure Model Parameters to Create Plausible Values → Creation of Plausible Values

NAEP Technical DocumentationCreation of Plausible Values

After obtaining marginal maximum likelihood estimates of parameters, and Σ, of in the population structure model (denoted by and ), twenty sets of plausible values, denoted by θ_m (m = 1,2,…,20), for all sampled respondents are drawn in the following three-step process.

1. A vector or a matrix _m is drawn from the distribution of as estimated using the population-structure model,

    

   where x, is the item response matrix, y is a matrix containing group membership information, is the matrix for item parameter estimates, and is the estimated covariance matrix in the population structure model.

2. Conditional on the generated value _m, and the fixed value , and the fixed value , the mean vector and the covariance matrix for the predictive conditional distribution of respondent r are computed from the predictive conditional distribution

   

3. Finally, a multivariate plausible value is drawn from a multivariate normal distribution with mean vector and covariance matrix

   

These three steps are repeated twenty times producing twenty sets of plausible values for all sampled respondents.

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