MARGINAL MAXIMUM-LIKELIHOOD-ESTIMATION FOR A PSYCHOMETRIC MODEL OF DISCONTINUOUS DEVELOPMENT

Item response theory models posit latent variables to account for regu larities in students' performances on test items. Wilson's ''Saltus'' model extends the ideas of IRT to development that occurs in stages, w here expected changes can be discontinuous, show different patterns fo r different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psy chological development and Siegler's rule-based learning. This paper d erives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing app roximation based on the EM algorithm. For individual examinees, empiri cal Bayes probabilities of learning-stage are given, along with profic iency parameter estimates conditional on stage membership. The MML sol ution is illustrated with simulated data and an example from the domai n of mixed number subtraction.