Nonparametric IRT: Scoring functions and ordinal parameter estimation of isotonic probabilistic models (ISOP)

The most popular unidimensional psychological test evaluation rule is the trivial scoring function: To the answers of each item of a test item scores of 0,1,...m points are awarded and the unweighted or simple or total sum of the item scores gives the test score (also: Likert score). This total score has desirable stochastic ordering properties like monotone likelihood ratio (MLR), stochastic ordering of the sum score by the latent dimension (SOM) and stochastic ordering of the latent dimension by the sum score (SOL) for several models. SOL cannot be shown for nonparametric graded response models in general. The new stochastic ordering property of monotone likelihood order (MLO) postulates that the likelihood that the largeness order of the observed responses and the true order (order of true subject parameters) are similar, is larger than their disagreement (their discordance). It is shown that the ISOP model (nonparametric isotonic rating scale model) has the MLO property. If the responses to the items are evaluated by the (modified) percentile score and if the total test score is the average item percentile score, then a maximum likelihood estimate of the ordinal subject parameter is achieved.