Generalized latent variable models with non-linear effects

Asymptotic Posterior Normality of Multivariate Latent Traits in an IRT Model

The asymptotic posterior normality (APN) of the latent variable vector in an item response theory (IRT) model is a crucial argument in IRT modeling approaches. In case of a single latent trait and under general assumptions, Chang and Stout (Psychometrika, 58(1):37-52, 1993) proved the APN for a broad class of latent trait models for binary items. Under the same setup, they also showed the consistency of the latent trait's maximum likelihood estimator (MLE). Since then, several modeling approaches have been developed that consider multivariate latent traits and assume their APN, a conjecture which has not been proved so far. We fill this theoretical gap by extending the results of Chang and Stout for multivariate latent traits. Further, we discuss the existence and consistency of MLEs, maximum a-posteriori and expected a-posteriori estimators for the latent traits under the same broad class of latent trait models.

The Fisher information function and scoring in binary ideal point item response models: a cautionary tale

This article examines the Fisher information functions, I(θ) , and explores implications for scoring of binary ideal point item response models. These models typically appear to have I(θ) that are bimodal and identically equal to 0 at the ideal point. The article shows that this is an inherent property of ideal point IRT models, which either have this property or are indeterminate and thus violate the likelihood regularity conditions. For some models, the indeterminacy can be resolved, generating an effectively unimodal I(θ) , albeit with violated regularity conditions. In other cases, I(θ) diverges. All reasonable ideal point IRT models exhibit this behaviour. Users should exercise caution when relying on asymptotics, particularly for shorter assessments. Use of simulated plausible values or prediction from a fully Bayesian estimation is recommended for scoring.

Factor copula models for mixed data

We develop factor copula models to analyse the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and nonlinear dependence. They can be explained as conditional independence models with latent variables that do not necessarily have an additive latent structure. We focus on important issues of interest to the social data analyst, such as model selection and goodness of fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by reanalysing three mixed response data sets. Our studies suggest that there can be a substantial improvement over the standard factor model for mixed data and make the argument for moving to factor copula models.

Psychometric properties the of Brazilian Portuguese version of Snaith-Hamilton Pleasure Scale (SHAPS)

INTRODUCTION

Anhedonia is defined as the reduced ability to feel pleasure and is a core symptom of various psychiatric disorders such as depression and schizophrenia. The Snaith-Hamilton Pleasure Scale (SHAPS) was developed to assess the presence of anhedonia. The objective of this study was to assess the psychometric properties of the Brazilian Portuguese version of the SHAPS.

METHODS

In this study, the SHAPS (14 items) was translated into Brazilian Portuguese and validated using data obtained from 228 subjects within a clinical sample. Psychometric properties were assessed using item response theory (logistic models) and classical test theory (Cronbach's alpha). We checked for external validity using a non-parametric correlation with an independent scale: Hospital Anxiety and Depression Scale - Depression subscale (HAD-D).

RESULTS

The SHAPS presented good internal consistency, with a Cronbach's α coefficient of 0.759 and adequacy to an IRT 1 parameter logistic (Rasch) model. The SHAPS presented significant correlation with the external measure HAD-D, with Spearman's ρ = 0.249 (S = 1368914; p < 0.001).

CONCLUSION

These results suggest that the Brazilian Portuguese version of the SHAPS is a reliable and valid instrument to assess hedonic tone.

A two-stage estimation procedure for non-linear structural equation models

Applications of structural equation models (SEMs) are often restricted to linear associations between variables. Maximum likelihood (ML) estimation in non-linear models may be complex and require numerical integration. Furthermore, ML inference is sensitive to distributional assumptions. In this article, we introduce a simple two-stage estimation technique for estimation of non-linear associations between latent variables. Here both steps are based on fitting linear SEMs: first a linear model is fitted to data on the latent predictor and terms describing the non-linear effect are predicted by their conditional means. In the second step, the predictions are included in a linear model for the latent outcome variable. We show that this procedure is consistent and identifies its asymptotic distribution. We also illustrate how this framework easily allows the association between latent variables to be modeled using restricted cubic splines, and we develop a modified estimator which is robust to non-normality of the latent predictor. In a simulation study, we compare the proposed method to MLE and alternative two-stage estimation techniques.

Latent Variable Interactions With Ordered-Categorical Indicators: Comparisons of Unconstrained Product Indicator and Latent Moderated Structural Equations Approaches

Methods to handle ordered-categorical indicators in latent variable interactions have been developed, yet they have not been widely applied. This article compares the performance of two popular latent variable interaction modeling approaches in handling ordered-categorical indicators: unconstrained product indicator (UPI) and latent moderated structural equations (LMS). We conducted a simulation study across sample sizes, indicators' distributions and category conditions. We also studied four strategies to create sets of product indicators for UPI. Results supported using a parceling strategy to create product indicators in the UPI approach or using the LMS approach when the categorical indicators are symmetrically distributed. We applied these models to study the interaction effect between third- to fifth-grade students' social skills improvement and teacher-student closeness on their state English language arts test scores.

Modelling non-normal data: The relationship between the skew-normal factor model and the quadratic factor model

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew-normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew-normal factor are equivalent to those under a quadratic model up to third-order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew-normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.

An IRT Modeling Approach for Assessing Item and Person Discrimination in Binary Personality Responses

Conventional item response theory (IRT) modeling of personality responses considers two item characteristics-location and discrimination-but only one person characteristic-location or trait level. An IRT modeling approach that jointly considers item and person discriminations, however, is thought to be more realistic and appropriate in this domain and has several potential advantages. This article develops a model of this type for unidimensional binary responses together with procedures for estimating item and person parameters and assessing model appropriateness (including person fit). A series of preliminary simulations suggests that the approach is feasible, and a real-data example illustrates the potential advantages with respect to the standard two-parameter model. Limitations of the proposal and further work are also discussed.

Partially Compensatory Multidimensional Item Response Theory Models: Two Alternate Model Forms

Partially compensatory models may capture the cognitive skills needed to answer test items more realistically than compensatory models, but estimating the model parameters may be a challenge. Data were simulated to follow two different partially compensatory models, a model with an interaction term and a product model. The model parameters were then estimated for both models and for the compensatory model. Either the model used to simulate the data or the compensatory model generally had the best fit, as indexed by information criteria. Interfactor correlations were estimated well by both the correct model and the compensatory model. The predicted response probabilities were most accurate from the model used to simulate the data. Regarding item parameters, root mean square errors seemed reasonable for the interaction model but were quite large for some items for the product model. Thetas were recovered similarly by all models, regardless of the model used to simulate the data.

Predicting allergic disease at age four using an atopy predisposition score at age two: the application of item response theory

When defining allergic outcomes in epidemiology studies, results of the skin prick test (SPT) panel are often dichotomized as positive/negative or categorized based on the number of positive responses. Item response theory (IRT) models, however, may prove to be a better alternative with the ability to generate scores that account for both type and number of positive SPTs. IRT was applied to SPT responses administered to 537 children at age two to determine predictability of allergic disease at age four. The children received SPTs to 15 aeroallergens and two foods. Atopy predisposition scores were obtained from the IRT model using the posterior distribution of the latent trait, atopy. These scores were used to predict persistent wheeze, rhino-conjunctivitis, and eczema at age four. Results were compared to the dichotomized and categorical (positive to ≥2, positive to one, vs. negative to all allergens) SPT variables. At age two, 39% of children had at least one positive SPT. All three allergic disease outcomes were significantly associated with IRT atopy scores: persistent wheeze odds ratio (OR) = 1.7 (95% confidence interval (CI): 1.2, 2.3); rhino-conjunctivitis OR = 1.7 (95% CI: 1.2, 2.3); eczema OR = 1.6 (95% CI: 1.2, 2.3). In contrast, rhino-conjunctivitis was the only outcome significantly associated with the dichotomized SPT variable with an OR = 1.9 (95% CI: 1.2, 3.0). For the categorical SPT variable, all three allergic symptoms were significantly associated with positive to ≥2 allergens compared to negative to all, but no difference was observed between those with positive to one compared to negative to all. The IRT model proved to be an informative methodology to assess the predictability of early SPT responses and identify the allergens most associated with atopy predisposition.

Mathematische Kompetenzen in der Grundschule

Zusammenfassung. Der vorliegende Beitrag geht der Frage nach, inwieweit sich unterschiedliche inhaltsbezogene und prozessbezogene mathematische Kompetenzen analytisch trennen lassen. Stichproben von N = 10328 und N = 6638 Schülerinnen und Schülern aus 3. und 4. Grundschulklassen bearbeiteten umfangreiche Itempools, die sich fünf inhaltsbezogenen Kompetenzen (Zahlen und Operationen, Raum und Form, Muster und Struktur, Größen und Messen sowie Daten, Häufigkeit und Wahrscheinlichkeit) und sechs prozessbezogenen Kompetenzen (Grundfertigkeiten, Problemlösen, Kommunizieren, Argumentieren, Modellieren sowie Darstellen) zuordnen lassen. Dimensionsanalysen belegen, dass ein Modell mit fünf inhaltsbezogenen Faktoren die Daten am besten abbildet. Die inhaltsbezogenen Skalen erweisen sich als hoch reliabel und Korrelationen mit anderen Instrumenten (z.B. DEMAT 3 und 4) belegen die hohe Validität. Analysen zum Zusammenhang mit Tests zur Erfassung kognitiver Grundfähigkeiten zeigen, dass mathematische Kompetenzen und kognitive Grundfähigkeiten jeweils distinkte Faktoren darstellen. Die Befunde werden im Hinblick auf die Frage diskutiert, welche Konstrukte mit Schulleistungstests erfasst werden.