Over the past decade, Mokken scale analysis (MSA) has rapidly grown in popularity among researchers from many different research areas. This tutorial provides researchers with a set of techniques and a procedure for their application, such that the construction of scales that have superior measurement properties is further optimized, taking full advantage of the properties of MSA. First, we define the conceptual context of MSA, discuss the two item response theory (IRT) models that constitute the basis of MSA, and discuss how these models differ from other IRT models. Second, we discuss dos and don'ts for MSA; the don'ts include misunderstandings we have frequently encountered with researchers in our three decades of experience with real-data MSA. Third, we discuss a methodology for MSA on real data that consist of a sample of persons who have provided scores on a set of items that, depending on the composition of the item set, constitute the basis for one or more scales, and we use the methodology to analyse an example real-data set.

Investigating an invariant item ordering for polytomously scored itemsLigtvoet, R.; van der Ark, L.A.; Te Marvelde, J.M.; Sijtsma, K. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim.

This article first discusses a statistical test for investigating whether or not the pattern of missing scores in a respondent-by-item data matrix is random. Since this is an asymptotic test, we investigate whether it is useful in small but realistic sample sizes. Then, we discuss two known simple imputation methods, person mean (PM) and two-way (TW) imputation, and we propose two new imputation methods, response-function (RF) and mean response-function (MRF) imputation. These methods are based on few assumptions about the data structure. An empirical data example with simulated missing item scores shows that the new method RF was superior to the methods PM, TW, and MRF in recovering from incomplete data several statistical properties of the original complete data. Methods TW and RF are useful both when item score missingness is ignorable and nonignorable.

We propose using latent class analysis as an alternative to loglinear analysis for the multiple imputation of incomplete categorical data. Similar to log-linear models, latent class models can be used to describe complex association structures between the variables used in the imputation model. However, unlike loglinear models, latent class models can be used to build large imputation models containing more than a few categorical variables. To obtain imputations reflecting uncertainty about the unknown model parameters, we use a nonparametric bootstrap procedure as an alternative to the more common full Bayesian approach. The proposed multiple imputation method, which is implemented in Latent GOLD software for latent class analysis, is illustrated with two examples. In a simulated data example, we compare the new method to well-established methods such as maximum likelihoodWe would like to thank Paul Allison and Jay Magidson, as well as the editor and the three anonymous reviewers, for their comments, which very much helped to improve this article. We would also like to thank Greg Richards for providing the data of the ATLAS Cultural Tourism Research Project, 2003. *Tilburg University † Leiden University 369 370 VERMUNT, VAN GINKEL, VAN DER ARK, AND SIJTSMA estimation with incomplete data and multiple imputation using a saturated log-linear model. This example shows that the proposed method yields unbiased parameter estimates and standard errors. The second example concerns an application using a typical social sciences data set. It contains 79 variables that are all included in the imputation model. The proposed method is especially useful for such large data sets because standard methods for dealing with missing data in categorical variables break down when the number of variables is so large.

We explain why invariant item ordering (IIO) is an important property in non-cognitive measurement and we discuss that IIO cannot be easily generalized from dichotomous data to polytomous data, as some authors seem to suggest. Methods are discussed to investigate IIO for polytomous items and an empirical example shows how these methods can be used in practice.

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Test norms enable determining the position of an individual test taker in the group. The most frequently used approach to obtain test norms is traditional norming. Regression-based norming may be more efficient than traditional norming and is rapidly growing in popularity, but little is known about its technical properties. A simulation study was conducted to compare the sample size requirements for traditional and regression-based norming by examining the 95% interpercentile ranges for percentile estimates as a function of sample size, norming method, size of covariate effects on the test score, test length, and number of answer categories in an item. Provided the assumptions of the linear regression model hold in the data, for a subdivision of the total group into eight equal-size subgroups, we found that regression-based norming requires samples 2.5 to 5.5 times smaller than traditional norming. Sample size requirements are presented for each norming method, test length, and number of answer categories. We emphasize that additional research is needed to establish sample size requirements when the assumptions of the linear regression model are violated.

Good scales are required for assessment in clinical practice and the present paper shows how a relatively recently developed method for analysing Mokken scales can contribute to this. The two scales used as examples for analysis are highly clinically relevant.