Modeling data from water-level tasks: a test theoretical analysis.

The water-level tasks were invented by Piaget to diagnose the level of mental development of spatial abilities, especially behavior of liquids. It has become usual practice to dichotomise water-level responses by the subjects into right vs wrong using a certain tolerance limit (departure from the horizontal measured in degrees) and to fit a mixture of binomials to the raw scores resulting from a series of water-level tasks. The present study questions this procedure. Based on a series of 12 water-level tasks (round bottles at 12 different angles of orientation) presented to 431 subjects, both children and adults, females and males, the following results were obtained. First, considering the task difficulties (proportions of correct answers) and the scores, the effect of age was significant, but that for sex was not. Second, a mixture of binomials were shown to be inappropriate due to their ignoring the heterogeneity of the task difficulties, whereas latent class models, due to their taking this heterogeneity as well as that of the subjects into account, were successful in rendering a complete and well-rounded description of the data as observed for the eight water-level tasks showing bottles with orientations corresponding to 1, 2, 4, 5, 7, 8, 10, and 11 o'clock. Third, the Rasch model was valid as justification for the simple interpretation that subjects and tasks have a unidimensional scale in common with each other and that the raw score is the sufficient statistic for the subjects' performance. Fourth, employing the linear logistic test model, the task difficulty parameters according to the Rasch model could be attributed to a single parameter associated with the angle of inclination of the bottle. In all, the results do not give a final answer to the question of whether the concept of types (classes) or that of a continuous trait is the better of the two, but they do give rise to some warnings against the use of a mixture of binomials for modeling data on water-level tasks.