, Volume 80, Issue 2, pp 535–559 | Cite as

Lord–Wingersky Algorithm Version 2.0 for Hierarchical Item Factor Models with Applications in Test Scoring, Scale Alignment, and Model Fit Testing



Lord and Wingersky’s (Appl Psychol Meas 8:453–461, 1984) recursive algorithm for creating summed score based likelihoods and posteriors has a proven track record in unidimensional item response theory (IRT) applications. Extending the recursive algorithm to handle multidimensionality is relatively simple, especially with fixed quadrature because the recursions can be defined on a grid formed by direct products of quadrature points. However, the increase in computational burden remains exponential in the number of dimensions, making the implementation of the recursive algorithm cumbersome for truly high-dimensional models. In this paper, a dimension reduction method that is specific to the Lord–Wingersky recursions is developed. This method can take advantage of the restrictions implied by hierarchical item factor models, e.g., the bifactor model, the testlet model, or the two-tier model, such that a version of the Lord–Wingersky recursive algorithm can operate on a dramatically reduced set of quadrature points. For instance, in a bifactor model, the dimension of integration is always equal to 2, regardless of the number of factors. The new algorithm not only provides an effective mechanism to produce summed score to IRT scaled score translation tables properly adjusted for residual dependence, but leads to new applications in test scoring, linking, and model fit checking as well. Simulated and empirical examples are used to illustrate the new applications.


multidimensional item response theory bifactor model  testlet linking scale alignment test equating goodness-of-fit testing summed score 



Part of this research is supported by the Institute of Education Sciences (R305B080016 and R305D100039) and the National Institute on Drug Abuse (R01DA026943 and R01DA030466). The views expressed here belong to the author and do not reflect the views or policies of the funding agencies. The author is grateful to Dr. David Thissen and members of the UCLA psychometric lab (in particular Carl Falk, Jane Li, and Ji Seung Yang) for comments on an earlier draft.


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Copyright information

© The Psychometric Society 2014

Authors and Affiliations

  1. 1.CRESSTUniversity of CaliforniaLos AngelesUSA

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