Ordered-categorical Data Research Articles

General Factor Mean Difference Estimation in Bifactor Models with Ordinal Data

ABSTRACT A simulation study was conducted to explore robustness of general factor mean difference estimation in bifactor measurement models with ordered-categorical measures. Common analysis misspecifications in which generated bifactor data were fitted using a unidimensional model and/or ordered-categorical data were treated as continuous were compared under generation conditions varying by sample size, number of response categories, effect size of the general factor mean difference, and loadings on the specific factors. Fitting bifactor data using unidimensional models resulted in estimation bias in the general factor mean difference, with magnitude largely determined by degree of unidimensionality and size of the general factor mean difference. Although bifactor models produced less estimation bias, estimates were less precise. Modeling the data as categorical and employing the WLSMV estimator provided somewhat more power, whereas modeling the data as continuous and applying MLR produced relatively less estimation bias when unidimensional models were specified for strongly bifactor data.

Using Pooled Heteroskedastic Ordered Probit Models to Improve Small-Sample Estimates of Latent Test Score Distributions

This article describes an extension to the use of heteroskedastic ordered probit (HETOP) models to estimate latent distributional parameters from grouped, ordered-categorical data by pooling across multiple waves of data. We illustrate the method with aggregate proficiency data reporting the number of students in schools or districts scoring in each of a small number of ordered “proficiency” levels. HETOP models can be used to estimate means and standard deviations of the underlying (latent) test score distributions but may yield biased or very imprecise estimates when group sample sizes are small. A simulation study demonstrates that the pooled HETOP models described here can reduce the bias and sampling error of standard deviation estimates when group sample sizes are small. Analyses of real test score data demonstrate the use of the models and suggest the pooled models are likely to improve estimates in applied contexts.

Practical Significance of Longitudinal Measurement Invariance Violations in the Dutch-Flemish PROMIS Item Banks for Depression and Anxiety: An Illustration With Ordered-Categorical Data.

We investigated longitudinal measurement invariance in the Dutch-Flemish PROMIS adult v1.0 item banks for Depression and Anxiety using two clinical samples with mood and anxiety disorders (n = 640 and n = 528, respectively). Factor analysis was used to evaluate whether the item banks were sufficiently unidimensional at two test-occasions and whether the measured constructs remained the same over time. The results indicated that the item banks were sufficiently unidimensional, but the thresholds and residual variances of the constructs changed over time. However, using tentative rules of thumb, these invariance violations did not substantially affect the endorsement of a specific response category of a specific item at a specific test-occasion. Furthermore, the impact on the mean latent change scores of the item banks remained below the proposed cutoff value for substantial bias. These findings suggest that the invariance violations lacked practical significance for test-users, meaning that the item banks provide sufficiently invariant latent factor scores for use in clinical practice.

Response-adaptive treatment allocation for clinical studies with ordinal responses.

Ordinal responses are common in clinical studies. Although the proportional odds model is a popular option for analyzing ordered-categorical data, it cannot control the type I error rate when the proportional odds assumption fails to hold. The latent Weibull model was recently shown to be a superior candidate for modeling ordinal data, with remarkably better performance than the latent normal model when the data are highly skewed. In clinical trials with ordinal responses, a balanced design is common, with equal sample allocation for each treatment. However, a more ethical approach is to adopt a response-adaptive allocation scheme in which more patients receive the better treatment. In this paper, we propose the use of the doubly adaptive biased coin design to generate treatment allocations that benefit the trial participants. The proposed treatment allocation scheme not only allows more patients to receive the better treatment, it also maintains compatible test power for the comparison of treatment efficiencies. A clinical example is used to illustrate the proposed procedure.

Random Permutation Testing Applied to Measurement Invariance Testing with Ordered-Categorical Indicators

We describe and evaluate a random permutation test of measurement invariance with ordered-categorical data. To calculate a p-value for the observed (∆)χ2, an empirical reference distribution is built by repeatedly shuffling the grouping variable, then saving the χ2 from a configural model, or the ∆χ2 between configural and scalar-invariance models, fitted to each permuted dataset. The current gold standard in this context is a robust mean- and variance-adjusted ∆χ2 test proposed by Satorra (2000), which yields inflated Type I errors, particularly when thresholds are asymmetric, unless samples sizes are quite large (Bandalos, 2014; Sass et al., 2014). In a Monte Carlo simulation, we compare permutation to three implementations of Satorra’s robust χ2 across a variety of conditions evaluating configural and scalar invariance. Results suggest permutation can better control Type I error rates while providing comparable power under conditions that the standard robust test yields inflated errors.

Enhancing cost savings through early involvement of purchasing professionals in sourcing projects: Bayesian estimation of a structural equation model

As in many organizations the role of the purchasing department is shifting from a clerical buying function to an internally integrated, strategic business function, purchasing professionals are increasingly involved in sourcing projects. Building on Social Exchange Theory, this study explores the effects of early purchasing involvement on internal customer satisfaction and cost savings in sourcing projects, as well as how this involvement is bounded by the strategic impact of the sourcing project. To this end, the authors gathered data on 644 sourcing projects from a database of a large, private financial services company. A research model deploying the research hypotheses was analyzed using Bayesian structural equation modeling. This technique introduced and highlighted as a valuable and more correct estimation procedure in hypothesis testing research involving ordered-categorical data. The results confirm that early purchasing involvement enhances the cost savings outcome in sourcing projects. However, this involvement has remained underutilized for projects with high strategic impact. This study empirically establishes the importance of purchasing professionals for realizing cost savings in sourcing projects.

Bifactorial structure of Locus of Control cross-culturally invariant across Spain, Chile and United Kingdom

<p>El locus de control (LOC) es una variable frecuentemente evaluada debido a que juega un importante papel en diferentes contextos. Sin embargo, no existe un acuerdo unánime sobre cuántas dimensiones componen la estructura factorial del locus de control. El objetivo de esta investigación fue añadir nuevas evidencias de validez transcultural en relación con la invarianza bifactorial del LOC. El test fue aplicado a un total de 1781 participantes procedentes de España (697), Chile (890) y Reino Unido (194). El estudio de la invarianza factorial entre los grupos se realizó utilizando modelos de análisis factorial confirmatorio multigrupo para datos categóricamente ordenados. La evaluación progresiva de la invarianza factorial confirma que las cargas factoriales, los umbrables, y las varianzas de los errores son invariantes a través de los grupos. Finalmente, no se encontraron diferencias transculturales relevantes en LOC entre España, Chile y Reino Unido (PS < .50).</p>

Trust in American Government: Longitudinal Measurement Equivalence in the ANES, 1964–2008

For over 50 years (1958–2012) the American National Election Studies (ANES) survey has been measuring citizens’ evaluations of the trustworthiness of the “government in Washington”—an indicator that has been widely used to monitor the dynamics of political trust in the US over time. However, a critical assumption in using attitudinal constructs for longitudinal research is that the meaning-and-interpretation of such items should be comparable across groups of respondents at any one point in time and across samples over time. Using multigroup confirmatory factor analysis for ordered-categorical data, we test the measurement equivalence assumption with data collected by the ANES from 1964 to 2008. The results confirm that the ANES’ political trust scale has the same basic factorial structure over time. But for two key items, several threshold parameters were found to be different across time points, indicating that the meaning-and-interpretation of these questions, and especially the question about whether the government in Washington wastes money that people pay in taxes, varies significantly over time.

A Note on Power and Sample Size Calculations for the Kruskal–Wallis Test for Ordered Categorical Data

Although the Kruskal–Wallis test has been widely used to analyze ordered categorical data, power and sample size methods for this test have been investigated to a much lesser extent when the underlying multinomial distributions are unknown. This article generalizes the power and sample size procedures proposed by Fan et al. (2011) for continuous data to ordered categorical data, when estimates from a pilot study are used in the place of knowledge of the true underlying distribution. Simulations show that the proposed power and sample size formulas perform well. A myelin oligodendrocyte glycoprotein (MOG) induced experimental autoimmunce encephalomyelitis (EAE) mouse study is used to demonstrate the application of the methods.

Item factor analysis: Current approaches and future directions.

The rationale underlying factor analysis applies to continuous and categorical variables alike; however, the models and estimation methods for continuous (i.e., interval or ratio scale) data are not appropriate for item-level data that are categorical in nature. The authors provide a targeted review and synthesis of the item factor analysis (IFA) estimation literature for ordered-categorical data (e.g., Likert-type response scales) with specific attention paid to the problems of estimating models with many items and many factors. Popular IFA models and estimation methods found in the structural equation modeling and item response theory literatures are presented. Following this presentation, recent developments in the estimation of IFA parameters (e.g., Markov chain Monte Carlo) are discussed. The authors conclude with considerations for future research on IFA, simulated examples, and advice for applied researchers.

Permutation Anderson–Darling Type and Moment-Based Test Statistics for Univariate Ordered Categorical Data

In this article we consider permutation methods for univariate testing on ordered variables based on the nonparametric combination of permutation dependent tests (Pesarin, 2001). Several solutions have been proposed to cope with univariate testing problems on ordered categorical data, most of which are based on the restricted maximum likelihood ratio test. These solutions are generally criticized, because the degree of accuracy of their asymptotic null and alternative distributions is difficult to assess and to characterize. By working within the nonparametric combination of dependent permutation tests (NPC), it is possible to find exact solutions to that kind of problems.We examine some of these solutions and discuss one more new exact solution based on simultaneous analysis of a finite set of sampling moments of ranks, or general scores, assigned to ordered classes and processed by the NPC method. A simulation study analyzes the empirical power of the considered NPC solutions compared with the Wilcoxon test with ties correction. The new permutation test based on Fisher's combination of sampling moments shows a good behavior in power in all considered situations both for balanced and unbalanced sample sizes.

Assessing Factorial Invariance in Ordered-Categorical Measures

The factor analysis of ordered-categorical measures has been described in the literature on factor analysis, but the extension of the analysis to the multiple-population case is less well-known. For example, a comprehensive statement of identification conditions for the multiplepopulation case seems absent in the literature. We review this multiple-population extension here, with an emphasis on model specification and identification. The use of the method in the study of factorial invariance is described. New results on identification are given for a variety of factor structures and types of measures. Two widely-available software packages, LISREL 8.52 (Joreskog & Sorbom, 1996) and Mplus 2.12 (Muthen & Muthen, 1998), are applied in simulated data to illustrate the method. The two programs are shown to have different model specifications for this method, leading to different fit results in some cases. The final section discusses some remaining problems facing researchers who wish to study factorial invariance in ordered-categorical data.

Characterization and Optimization of a Wave-Soldering Process

Abstract A case study for improving the quality of a wave-soldering process that produced printed circuit boards (PCB's) is presented. A mixed-level fractional factorial design was implemented in a high-volume production system during normal operational hours. The observed ordered-categorical data from the bottom-side soldered leads were weighted to formulate the average, spatial uniformity, and dispersion process performance measurements. For lead classes like the integrated circuit and printed grid array, a polynomial model was established using the least squares method with weights provided by a dispersion function. The main-effect and interaction model terms were selected by forward and all-subsets regressions. Production quotas, topside defects, presoldering board temperature, and variance models were used to set the constraints for simultaneously optimizing predictions from the average and the uniformity models of all leads. A nonlinear optimization routine was used to determine the best and most robust settings for the continuous and discrete process variables. Results from a confirmatory experiment showed an improvement of mean soldering quality by 33% and of uniformity by 39%.

Characterization and Optimization of a Wave-Soldering Process

Abstract A case study for improving the quality of a wave-soldering process that produced printed circuit boards (PCB's) is presented. A mixed-level fractional factorial design was implemented in a high-volume production system during normal operational hours. The observed ordered-categorical data from the bottom-side soldered leads were weighted to formulate the average, spatial uniformity, and dispersion process performance measurements. For lead classes like the integrated circuit and printed grid array, a polynomial model was established using the least squares method with weights provided by a dispersion function. The main-effect and interaction model terms were selected by forward and all-subsets regressions. Production quotas, topside defects, presoldering board temperature, and variance models were used to set the constraints for simultaneously optimizing predictions from the average and the uniformity models of all leads. A nonlinear optimization routine was used to determine the best and most robust settings for the continuous and discrete process variables. Results from a confirmatory experiment showed an improvement of mean soldering quality by 33% and of uniformity by 39%.