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Creation of Plausible Values

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

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

    Follows a distribution conditional on item responses, background variables as well as related parameters. 

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

  2. Conditional on the generated value gamma hatm, and the fixed value sigma hat, and the fixed value a hat, the mean vector The mean vector and the covariance matrix The variance-covariance matrix of the distribution of the ability vector for person r at mth jackknife sample. for the predictive conditional distribution of respondent r are computed from the predictive conditional distribution

    It is a predictive conditional probability of latent abilities for person r at mth jackknife sample, given item responses, background variables, as well as the related parameters for mth jackknife sample.

  3. Finally, a multivariate plausible value The ability vector for person r at mth jackknife sample. is drawn from a multivariate normal distribution with mean vector The mean vector and covariance matrix The variance-covariance matrix of the distribution of the ability vector for person r at mth jackknife sample.

    The distribution of ability vector for person r at mth jackknife sample.

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


Last updated 02 June 2016 (GF)