Reweighting Data in the Spirit of Tukey: Using Bayesian Posterior Probabilities as Rasch Residuals for Studying...

A new variant of the iterative "data = fit + residual" data-analytical approach described by Mosteller and Tukey is proposed and implemented in the context of item response theory psychometric models. Posterior probabilities from a Bayesian mixture model of a Rasch item response theory model and an unscalable latent class are expressed as weights for the original data. The data are weighted by the units' posterior probabilities for the unscalable class and used for further exploration of structures. Factor analysis models are compared with the original data and data as reweighted by the posterior probabilities for the unscalable class. In comparing two weighted data sets, Rasch-weighted data and data considered unscalable, differences were evident. Pattern types are detected for the Rasch baseline with patterns that are different patterns from random or systematic contamination. Rasch baseline patterns are strongest near item difficulties closest to the mean generating value of θs. Patterns in baseline conditions are weaker as they depart from an item difficulty of zero and move toward extreme values. Random contamination patterns are typically flat and near zero regardless of item difficulty. Systematic contamination using reversed Rasch-generated data produced alternate patterns to the Rasch baseline condition and in some conditions showed an opposite effect from Rasch patterns. Differences could be detected within residually weighted data between the Rasch-generated subtest and contaminated subtest. Rasch subtest often had Rasch patterns while contaminated subtest had random/flat or systematic/reversed pattern.