Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T_rml proposed by Satorra and Bentler was recommended in the literature of SEM. However, T_rml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T_rml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T_rml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.     

Two-Way Regularized Fuzzy Clustering of Multiple Correspondence Analysis

Multiple correspondence analysis (MCA) is a useful tool for investigating the interrelationships among dummy-coded categorical variables. MCA has been combined with clustering methods to examine whether there exist heterogeneous subclusters of a population, which exhibit cluster-level heterogeneity. These combined approaches aim to classify either observations only (one-way clustering of MCA) or both observations and variable categories (two-way clustering of MCA). The latter approach is favored because its solutions are easier to interpret by providing explicitly which subgroup of observations is associated with which subset of variable categories. Nonetheless, the two-way approach has been built on hard classification that assumes observations and/or variable categories to belong to only one cluster. To relax this assumption, we propose two-way fuzzy clustering of MCA. Specifically, we combine MCA with fuzzy k-means simultaneously to classify a subgroup of observations and a subset of variable categories into a common cluster, while allowing both observations and variable categories to belong partially to multiple clusters. Importantly, we adopt regularized fuzzy k-means, thereby enabling us to decide the degree of fuzziness in cluster memberships automatically. We evaluate the performance of the proposed approach through the analysis of simulated and real data, in comparison with existing two-way clustering approaches.     

A Comparison of Methods for Creating Multiple Imputations of Nominal Variables

Many variables that are analyzed by social scientists are nominal in nature. When missing data occur on these variables, optimal recovery of the analysis model's parameters is a challenging endeavor. One of the most popular methods to deal with missing nominal data is multiple imputation (MI). This study evaluated the capabilities of five MI methods that can be used to treat incomplete nominal variables: multiple imputation with chained equations (MICE) using polytomous regression as the elementary imputation method; MICE based on classification and regression trees (CART); MICE based on nested logistic regressions; the ranking procedure described by Allison (2002 Allison, P. D. (2002). Missing data. Thousand Oaks, CA: Sage Publications. https://doi.org/10.4135/9780857020994.n4[Crossref] , [Google Scholar]); and a joint modeling approach based on the general location model. We first motivate our inquiry with an applied example and then present the results of a Monte Carlo simulation study that compared the performance of the five imputation methods under conditions of varying sample size, percentage of missing data, and number of nominal response categories. We found that MICE with polytomous regression was the strongest performer while the Allison (2002 Allison, P. D. (2002). Missing data. Thousand Oaks, CA: Sage Publications. https://doi.org/10.4135/9780857020994.n4[Crossref] , [Google Scholar]) ranking procedure and MICE with CART performed poorly in most conditions.     

Clustering and switching in unconstrained,phonemic and semantic verbal fluency: the role of age and school type

We investigated the role of age and school type on clustering and switching in verbal fluency tasks (VFTs) with Brazilian children. The children were administered unconstrained, phonemic and semantic VFTs with a duration of 150 or 120 s, respectively. Both age and school type influenced all variables and in terms of performance over time. Older children and private school students outperformed the remainder of the sample, with the first 30 s of each VFT usually being the most productive. Although the size of the clusters produced did not differ between groups, the types of clusters did show some variations, with semantic clusters being the most frequent. Our results revealed strong correlations between switching ability and word production in all three VFTs. In conclusion, the executive functions known as planning and cognitive flexibility play a crucial role in word production by organising and facilitating the recall of lexical information from memory.     

Generalized multilevel structural equation modeling

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.gllamm can be downloaded from http://www.gllamm.org. The paper was written while Sophia Rabe-Hesketh was employed at and Anders Skrondal was visiting the Department of Biostatistics and Computing, Institute of Psychiatry, King's College London.     

Psychometrics in Psychological Research: Role Model or Partner in Science?

This is a reaction to Borsboom's (2006) discussion paper on the issue that psychology takes so little notice of the modern developments in psychometrics, in particular, latent variable methods. Contrary to Borsboom, it is argued that latent variables are summaries of interesting data properties, that construct validation should involve studying nomological networks, that psychological research slowly but definitely will incorporate latent variable methods, and that the role of psychometrics in psychology is that of partner, not role model. Requests for reprints should be sent to Klaas Sijtsma, Department of Methodology and Statistics, FSW, Tilburg University, PO Box 90153, 5000 LE, Tilburg, The Netherlands.     

On a test of dimensionality in redundancy analysis

Lazraq and Cléroux (Psychometrika, 2002, 411–419) proposed a test for identifying the number of significant components in redundancy analysis. This test, however, is ill-conceived. A major problem is that it regards each redundancy component as if it were a single observed predictor variable, which cannot be justified except for the rare situations in which there is only one predictor variable. Consequently, the proposed test leads to drastically biased results, particularly when the number of predictor variables is large, and it cannot be recommended for use. This is shown both theoretically and by Monte Carlo studies.The work reported in this paper was supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada.     

Age-related effects of practice experience on collective behaviours of football players in small-sided games

The purpose of this study was to examine whether offensive and defensive collective behaviours emerging in six-a-side football games (GK+5 vs. 5+GK) varied according to age-related practice experience of young, male players (U16, U17 and U19 yrs). Players’ were not instructed to implement specific tactical plans and their movement trajectories (2D analyses) were recorded using 10 GPS units. Four common measures of team dispersion investigated in previous research (surface area, stretch index, length and width of a team) were used to analyse team performance behaviours. After recording these collective variables, we used sample entropy (SampEn) and cross-sample entropy (Cross-SampEn) measures to assess the regularity and synchronization of participant actions in teams. Results demonstrated clear age-related variations in effects on the collective performance measures analysed. In attacking phases, older and more experienced players occupied a greater surface area and displayed higher values of team width and stretch index. In defensive phases, significant differences were observed in team length and stretch index. Cross-SampEn analysis demonstrated a greater synchronization between offensive and defensive surface areas and team width in older age groups (U17 and U19 yrs). Data suggest how coaches can manipulate practice task constraints to enhance development of team tactical performance behaviours in developing footballers between 16 and 19 yrs of age.