Analyzing nonadditive conjoint structures: compounding events by Rasch model probabilities

The following study proposes a Rasch method to measure variables of nonadditive conjoint structures, where dichotomous response combinations are evaluated. In this framework, both the number of endorsed items and their latent positions are considered. This is different from the cumulative response process (measurable by the Rasch model), where the probability of a positive response to an item with measure delta iota is considered a monotonic increasing function of the person's measure beta nu. This is also unlike the unfolding framework, where the probability of a positive response is maximum when beta nu = delta iota, and monotonically decreases as magnitude of beta nu-delta iota approaches infinity. The method involves four steps. In Step 1, items are scaled by the Rasch model for paired comparisons to produce a variable definition. These scale values serve as a basis for Steps 2 and 4. In Step 2, the nonadditive conjoint system is restructured to additive. The quantitative hypothesis of the restructured data is tested by the axioms of conjoint measurement theory in Step 3. This data is then analyzed by the Rasch rating scale model in Step 4 to evaluate individual response combinations, using the Step 1 item calibrations as anchors. The method was applied to simulated person responses of the Schedule of Recent Events (Holmes and Rahe, 1967). The results suggest that the method is useful and effective. It scales items with a robust method of paired comparisons, ensures additivity and quantification of the conjoint person-item matrix, produces a reasonable ordering of person measures from the perspective of individual response combinations, and provides satisfactory person and item separation (i.e., reliability). Furthermore, the restructured data reproduces SRE item scale values obtained by paired comparisons in Step 1.