Carmen Ximénez

A Monte Carlo Study of Recovery of Weak Factor Loadings in Confirmatory Factor Analysis

The recovery of weak factors has been extensively studied in the context of exploratory factor analysis. This article presents the results of a Monte Carlo simulation study of recovery of weak factor loadings in confirmatory factor analysis under conditions of estimation method (maximum likelihood vs. unweighted least squares), sample size, loading size, factor correlation, and model specification (correct vs. incorrect). The effects of these variables on goodness of fit and convergence are also examined. Results show that recovery of weak factor loadings, goodness of fit, and convergence are improved when factors are correlated and models are correctly specified. Additionally, unweighted least squares produces more convergent solutions and successfully recovers the weak factor loadings in some instances where maximum likelihood fails. The implications of these findings are discussed and compared to previous research.

Optimization of sample size in controlled experiments: The CLAST rule

Sequential rules are explored in the context of null hypothesis significance testing. Several studies have demonstrated that the fixed-sample stopping rule, in which the sample size used by researchers is determined in advance, is less practical and less efficient than sequential stopping rules. It is proposed that a sequential stopping rule called CLAST (composite limited adaptive sequential test) is a superior variant of COAST (composite open adaptive sequential test), a sequential rule proposed by Frick (1998). Simulation studies are conducted to test the efficiency of the proposed rule in terms of sample size and power. Two statistical tests are used: the one-tailed t test of mean differences with two matched samples, and the chi-square independence test for twofold contingency tables. The results show that the CLAST rule is more efficient than the COAST rule and reflects more realistically the practice of experimental psychology researchers.

Recovery of weak factor loadings in confirmatory factor analysis under conditions of model misspecification.

This article presents the results of two Monte Carlo simulation studies of the recovery of weak factor loadings, in the context of confirmatory factor analysis, for models that do not exactly hold in the population. This issue has not been examined in previous research. Model error was introduced using a procedure that allows for specifying a covariance structure with a specified discrepancy in the population. The effects of sample size, estimation method (maximum likelihood vs. unweighted least squares), and factor correlation were also considered. The first simulation study examined recovery for models correctly specified with the known number of factors, and the second investigated recovery for models incorrectly specified by underfactoring. The results showed that recovery was not affected by model discrepancy for the correctly specified models but was affected for the incorrectly specified models. Recovery improved in both studies when factors were correlated, and unweighted least squares performed better than maximum likelihood in recovering the weak factor loadings.

Effect of Variable and Subject Sampling on Recovery of Weak Factors In CFA

Abstract. Two general issues central to the design of a study are subject sampling and variable sampling. Previous research has examined their effects on factor pattern recovery in the context of exploratory factor analysis. The present paper focuses on recovery of weak factors and reports two simulation studies in the context of confirmatory factor analysis. Conditions investigated include the estimation method (ML vs. ULS), sample size (100, 300, and 500), number of variables per factor (3, 4, or 5), loading size in the weak factor (.25 or .35), and factor correlation (null vs. moderate). Results show that both subject and variable sample size affect the recovery of weak factors, particularly if factors are not correlated. A small but consistent pattern of differences between methods occurs, which favors the use of ULS. Additionally, the frequency of nonconvergent and improper solutions is also affected by the same variables.

Extending the CLAST sequential rule to one-way ANOVA under group sampling.

Several studies have demonstrated that the fixed-sample stopping rule (FSR), in which the sample size is determined in advance, is less practical and efficient than are sequential-stopping rules. The composite limited adaptive sequential test (CLAST) is one such sequential-stopping rule. Previous research has shown that CLAST is more efficient in terms of sample size and power than are the FSR and other sequential rules and that it reflects more realistically the practice of experimental psychology researchers. The CLAST rule has been applied only to thet test of mean differences with two matched samples and to the chi-square independence test for twofold contingency tables. The present work extends previous research on the efficiency of CLAST to multiple group statistical tests. Simulation studies were conducted to test the efficiency of the CLAST rule for the one-way ANOVA for fixed effects models. The ANOVA general test and two linear contrasts of multiple comparisons among treatment means are considered. The article also introduces four rules for allocatingN observations toJ groups under the general null hypothesis and three allocation rules for the linear contrasts. Results show that the CLAST rule is generally more efficient than the FSR in terms of sample size and power for one-way ANOVA tests. However, the allocation rules vary in their optimality and have a differential impact on sample size and power. Thus, selecting an allocation rule depends on the cost of sampling and the intended precision.

Factorial Invariance in a Repeated Measures Design: An Application to the Study of Person-Organization Fit

An important methodological concern of any research based on a person-environment (P-E) fit approach is the operationalization of the fit, which imposes some measurement requirements that are rarely empirically tested with statistical methods. Among them, the assessment of the P and E components along commensurate dimensions is possibly the most cited one. This paper proposes to test the equivalence across the P and E measures by analyzing the measurement invariance of a multi-group confirmatory factor analysis model. From a methodological point of view, the distinct aspect of this approach within the context of P-E fit research is that measurement invariance is assessed in a repeated measures design. An example illustrating the procedure in a person-organization (P-O) fit dataset is provided. Measurement invariance was tested at five different hierarchical levels: (1) configural, (2) first-order factor loadings, (3) second-order factor loadings, (4) residual variances of observed variables, and (5) disturbances of first-order factors. The results supported the measurement invariance across the P and O measures at the third level. The implications of these findings for P-E fit studies are discussed.

Recovery of Weak Factor Loadings When Adding the Mean Structure in Confirmatory Factor Analysis: A Simulation Study.

This article extends previous research on the recovery of weak factor loadings in confirmatory factor analysis (CFA) by exploring the effects of adding the mean structure. This issue has not been examined in previous research. This study is based on the framework of Yung and Bentler (1999) and aims to examine the conditions that affect the recovery of weak factor loadings when the model includes the mean structure, compared to analyzing the covariance structure alone. A simulation study was conducted in which several constraints were defined for one-, two-, and three-factor models. Results show that adding the mean structure improves the recovery of weak factor loadings and reduces the asymptotic variances for the factor loadings, particularly for the models with a smaller number of factors and a small sample size. Therefore, under certain circumstances, modeling the means should be seriously considered for covariance models containing weak factor loadings.

Bayesian Dimensionality Assessment for the Multidimensional Nominal Response Model

This article introduces Bayesian estimation and evaluation procedures for the multidimensional nominal response model. The utility of this model is to perform a nominal factor analysis of items that consist of a finite number of unordered response categories. The key aspect of the model, in comparison with traditional factorial model, is that there is a slope for each response category on the latent dimensions, instead of having slopes associated to the items. The extended parameterization of the multidimensional nominal response model requires large samples for estimation. When sample size is of a moderate or small size, some of these parameters may be weakly empirically identifiable and the estimation algorithm may run into difficulties. We propose a Bayesian MCMC inferential algorithm to estimate the parameters and the number of dimensions underlying the multidimensional nominal response model. Two Bayesian approaches to model evaluation were compared: discrepancy statistics (DIC, WAICC, and LOO) that provide an indication of the relative merit of different models, and the standardized generalized discrepancy measure that requires resampling data and is computationally more involved. A simulation study was conducted to compare these two approaches, and the results show that the standardized generalized discrepancy measure can be used to reliably estimate the dimensionality of the model whereas the discrepancy statistics are questionable. The paper also includes an example with real data in the context of learning styles, in which the model is used to conduct an exploratory factor analysis of nominal data.

Estimación bayesiana de un modelo psicométrico multinivel con efectos aleatorios

The present study examines the problem of the development of psychometric models for multi-level designs, that aim to compare the medium level of subjects from different groups organized in levels hierarchically defined. A psychometric multilevel model based on the Item Response Theory (IRT) and a Bayesian procedure to obtain estimations in hierarchical models of IRT are presented. The model refers to dichotomous data and a one-dimensional latent trait, and put emphasis on the hierarchical aspect of the analysis. In addition to formally introducing the model, an illustration of the application of the procedure is presented by an example that includes empirical data referred to a test of mathematical knowledge that was applied to a sample of 1,000 Spanish students organized in schools from three different regions. The results provide information about each student, school, and region. Additionally, the syntax code used in the Bayesian estimation with the OpenBUGS and Stan programs is included in order to provide the reader with a tool that can be adjusted to his/her own research problem. Finally, the implications of the use of multilevel models and future research directions are discussed.