Purpose: Mokken scale analysis (MSA) is an attractive scaling procedure for ordinal data. MSA is frequently used in health-related quality of life research. Two of MSA's prime features are the scalability coefficients and the automated item selection procedure (AISP). The AISP partitions a (large) set of items into scales based on the observed item scores; the resulting scales can be used as measurement instruments. There exist two issues in MSA: First, point estimates, standard errors, and test statistics for scalability coefficients are inappropriate for clustered item scores, which are omnipresent in quality of life research data. Second, the AISP insufficiently takes sampling fluctuation of Mokken’s scalability coefficients into account. Methods: We solved both issues by providing point estimates and standard errors for the scalability coefficients for clustered data and by implementing a Wald-based significance test in the AISP algorithm, resulting in a test-guided AISP (T-AISP), that is available for both nonclustered and clustered test scores. Results: We integrated the T-AISP into a two-step, test-guided MSA for scale construction, to guide the analysis for nonclustered and clustered data. The first step is performing a T-AISP and select the final scale(s). For clustered data, within-group dependency is investigated on the final scale(s). In the second step, the strength of the scale(s) is determined and further analyses are performed. The procedure was demonstrated on clustered item scores obtained from administering a questionnaire on quality of life in schools to 639 students nested in 30 classrooms. Conclusions: We developed a two-step, test-guided MSA for scale construction that takes into account sample fluctuation of all scalability coefficients and that can be applied to item scores obtained by a nonclustered or clustered sampling design.

Two-level Mokken scale analysis is a generalization of Mokken scale analysis for multi-rater data. The bias of estimated scalability coefficients for two-level Mokken scale analysis, the bias of their estimated standard errors, and the coverage of the confidence intervals has been investigated, under various testing conditions. It was found that the estimated scalability coefficients were unbiased in all tested conditions. For estimating standard errors, the delta method and the cluster bootstrap were compared. The cluster bootstrap structurally underestimated the standard errors of the scalability coefficients, with low coverage values. Except for unequal numbers of raters across subjects and small sets of items, the delta method standard error estimates had negligible bias and good coverage. Post hoc simulations showed that the cluster bootstrap does not correctly reproduce the sampling distribution of the scalability coefficients, and an adapted procedure was suggested. In addition, the delta method standard errors can be slightly improved if the harmonic mean is used for unequal numbers of raters per subject rather than the arithmetic mean.

Currently, two-level Mokken scale analysis for clustered test data is being developed. This paper contributes to this development by providing model-fit procedures for two-level Mokken scale analysis. New theoretical insights suggested that the existing model-fit procedure from traditional (one-level) Mokken scale analyses can be used for investigating model fit at both level 1 (respondent level) and level 2 (cluster level) of two-level Mokken scale analysis. However, the traditional model-fit procedure requires some modifications before it can be used at level 2. In this paper, we made these modifications and investigated the resulting model-fit procedure. For two model assumptions, monotonicity and invariant item ordering, we investigated the false-positive count and the sensitivity count of the level 2 model-fit procedure, with respect to the number of model violations detected, and the number of detected model violations deemed statistically significant. For monotonicity, the detection of model violations was satisfactory, but the significance test lacked power. For invariant item ordering, both aspects were satisfactory.

Educational and psychological tests with an ordered item structure enable efficient test administration procedures and allow for intuitive score interpretation and monitoring. The effectiveness of the measurement instrument relies to a large extent on the validated strength of its ordering structure. We define three increasingly strict types of ordering for the ordering structure of a measurement instrument with clustered items: a weak and a strong invariant cluster ordering and a clustered invariant item ordering. Following a nonparametric item response theory (IRT) approach, we proposed a procedure to evaluate the ordering structure of a clustered item set along this three-fold continuum of order invariance. The basis of the procedure is (a) the local assessment of pairwise conditional expectations at both cluster and item level and (b) the global assessment of the number of Guttman errors through new generalizations of the H -coefficient for this item-cluster context. The procedure, readily implemented in R, is illustrated and applied to an empirical example. Suggestions for test practice, further methodological developments, and future research are discussed.