Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

The structure of Bryant's Empathy Index for children: a cross-validation study

The present study examined the structure of Bryant's Empathy Index (BEI) using different samples for conducting exploratory and confirmatory analyses. The BEI was administered to a sample of 2714 children (mean age 11.12, SD = 1.59). Exploratory and confirmatory factor analyses showed a three-factor structure: Feelings of Sadness, Understanding Feelings and Tearful Reaction. The results revealed both the multidimensionality of the instrument and appropriate fit indices for the model proposed. Although these results were very similar to those reported in other studies with a Spanish population, the analyses were conducted in a more robust way: with a larger sample and using polychoric correlations and cross validation estimation.     

Validation of the Chinese version of the Life Orientation Test with a robust weighted least squares approach

Of the several measures of optimism presently available in the literature, the Life Orientation Test (LOT; Scheier & Carver, 1985) has been the most widely used in empirical research. This article explores, confirms, and cross-validates the factor structure of the Chinese version of the LOT with ordinal data by using robust weighted least squares (robust WLS) estimation within the Taiwanese cultural context. Results of exploratory and confirmatory factor analyses using 3 different samples (Ntotal = 1,119) show that the factor structure of the Chinese version of the LOT is better conceptualized as a correlated 2-factor model than a single-factor model. The composite reliability was 0.7 for the "disagreement on optimism" factor and 0.74 for the "agreement on optimism" factor. In addition, comparison results of the 2 estimators using empirical data and simulation data suggest that robust WLS is less biased than maximum likelihood (ML) for estimating factor loadings and interfactor correlations in the factor analytic model of the Chinese version of the LOT. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Comparison of alternative estimation methods in confirmatory factor analyses of the General Health Questionnaire

This paper examines the implications of violating assumptions concerning the continuity and distributional properties of data in establishing measurement models in social science research. The General Health Questionnaire-12 uses an ordinal response scale. Responses to the GHQ-12 from 201 Hong Kong immigrants on arrival in Australia showed that the data were not normally distributed. A series of confirmatory factor analyses using either a Pearson product-moment or a polychoric correlation input matrix and employing either maximum likelihood, weighted least squares or diagonally weighted least squares estimation methods were conducted on the data. The parameter estimates and goodness-of-fit statistics provided support for using polychoric correlations and diagonally weighted least squares estimation when analyzing ordinal, nonnormal data.     

Contributions to Estimation of Polychoric Correlations

This research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator.     

Comparing meta-analytic moderator estimation techniques under realistic conditions

One of the most problematic issues in contemporary meta-analysis is the estimation and interpretation of moderating effects. Monte Carlo analyses are developed in this article that compare bivariate correlations, ordinary least squares and weighted least squares (WLS) multiple regression, and hierarchical subgroup (HS) analysis for assessing the influence of continuous moderators under conditions of multicollinearity and skewed distribution of study sample sizes (heteroscedasticity). The results show that only WLS is largely unaffected by multicollinearity and heteroscedasticity, whereas the other techniques are substantially weakened. Of note, HS, one of the most popular methods, typically provides the most inaccurate results, whereas WLS, one of the least popular methods, typically provides the most accurate results.     

Robustness of Parameter Estimation to Assumptions of Normality in the Multidimensional Graded Response Model

A central assumption that is implicit in estimating item parameters in item response theory (IRT) models is the normality of the latent trait distribution, whereas a similar assumption made in categorical confirmatory factor analysis (CCFA) models is the multivariate normality of the latent response variables. Violation of the normality assumption can lead to biased parameter estimates. Although previous studies have focused primarily on unidimensional IRT models, this study extended the literature by considering a multidimensional IRT model for polytomous responses, namely the multidimensional graded response model. Moreover, this study is one of few studies that specifically compared the performance of full-information maximum likelihood (FIML) estimation versus robust weighted least squares (WLS) estimation when the normality assumption is violated. The research also manipulated the number of nonnormal latent trait dimensions. Results showed that FIML consistently outperformed WLS when there were one or multiple skewed latent trait distributions. More interestingly, the bias of the discrimination parameters was non-ignorable only when the corresponding factor was skewed. Having other skewed factors did not further exacerbate the bias, whereas biases of boundary parameters increased as more nonnormal factors were added. The item parameter standard errors recovered well with both estimation algorithms regardless of the number of nonnormal dimensions.     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.     

Limited information estimation and testing of discretized multivariate normal structural models

Discretized multivariate normal structural models are often estimated using multistage estimation procedures. The asymptotic properties of parameter estimates, standard errors, and tests of structural restrictions on thresholds and polychoric correlations are well known. It was not clear how to assess the overall discrepancy between the contingency table and the model for these estimators. It is shown that the overall discrepancy can be decomposed into a distributional discrepancy and a structural discrepancy. A test of the overall model specification is proposed, as well as a test of the distributional specification (i.e., discretized multivariate normality). Also, the small sample performance of overall, distributional, and structural tests, as well as of parameter estimates and standard errors is investigated under conditions of correct model specification and also under mild structural and/or distributional misspecification. It is found that relatively small samples are needed for parameter estimates, standard errors, and structural tests. Larger samples are needed for the distributional and overall tests. Furthermore, parameter estimates, standard errors, and structural tests are surprisingly robust to distributional misspecification. This research was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grants BSO2000-0661 and BSO2003-08507 of the Spanish Ministry of Science and Technology.     

Estimating the polychoric correlation from misclassified data

Many variables that are used in social and behavioural science research are ordinal categorical or polytomous variables. When more than one polytomous variable is involved in an analysis, observations are classified in a contingency table, and a commonly used statistic for describing the association between two variables is the polychoric correlation. This paper investigates the estimation of the polychoric correlation when the data set consists of misclassified observations. Two approaches for estimating the polychoric correlation have been developed. One assumes that the probabilities in relation to misclassification are known, and the other uses a double sampling scheme to obtain information on misclassification. A parameter estimation procedure is developed, and statistical properties for the estimates are discussed. The practicability and applicability of the proposed approaches are illustrated by analysing data sets that are based on real and generated data. Excel programmes with visual basic for application (VBA) have been developed to compute the estimate of the polychoric correlation and its standard error. The use of the structural equation modelling programme Mx to find parameter estimates in the double sampling scheme is discussed.     

A short note on the estimation of the asymptotic covariance matrix for polychoric correlations

By using a Taylor expansion of the equations that define the two step estimator for polychoric correlations, the asymptotic covariance matrix for the estimated correlations can be derived in a simple and straightforward way.     

A Monte Carlo study comparing PIV,ULS and DWLS in the estimation of dichotomous confirmatory factor analysis

We conducted a Monte Carlo study to investigate the performance of the polychoric instrumental variable estimator (PIV) in comparison to unweighted least squares (ULS) and diagonally weighted least squares (DWLS) in the estimation of a confirmatory factor analysis model with dichotomous indicators. The simulation involved 144 conditions (1,000 replications per condition) that were defined by a combination of (a) two types of latent factor models, (b) four sample sizes (100, 250, 500, 1,000), (c) three factor loadings (low, moderate, strong), (d) three levels of non‐normality (normal, moderately, and extremely non‐normal), and (e) whether the factor model was correctly specified or misspecified. The results showed that when the model was correctly specified, PIV produced estimates that were as accurate as ULS and DWLS. Furthermore, the simulation showed that PIV was more robust to structural misspecifications than ULS and DWLS.     

Latent structure and measurement invariance of the Hospital Anxiety and Depression Scale in cancer outpatients

Background/ObjectiveThe aim of the present study was to compare competing psychometric models and analyze measurement invariance of the Hospital Anxiety and Depression Scale (HADS) in cancer outpatients.MethodThe sample included 3,260 cancer outpatients. Latent structure of the HADS was analyzed using confirmatory factor analysis (CFA) with robust maximum likelihood estimation (MLR). Measurement invariance was tested for age, time of response, gender, and cancer type by comparing nested multigroup CFA models with parameter restrictions.ResultsExcept for the one-factor solutions, all models showed acceptable model fit and measurement invariance. The model with the best fit was the originally proposed two-factor model with exclusion of two items. The one-factor solutions showed inacceptable model fit and were not invariant for age and gender.ConclusionsThe HADS has a robust two-factor structure in cancer outpatients. We recommend excluding item 7 and 10 when screening for anxiety and depression.     

Revisiting the Empirical Distinction Between Hedonic and Eudaimonic Aspects of Well-Being Using Exploratory Structural Equation Modeling

The distinction between hedonic (i.e., subjective well-being) and eudaimonic (i.e., psycho-social functioning) components of well-being is questioned by some researchers on the grounds that these two aspects of well-being are highly correlated. However, I argue that previous research has relied on confirmatory factor analysis (CFA), which is likely to overestimate interfactor correlations, because cross-loadings are constrained to be zero in CFA. In contrast, the new method of exploratory structural equation modeling (ESEM) does not constrain cross-ladings to zero, which results in more accurate factor intercorrelations. The present study used ESEM to reinvestigate the relationship between hedonic and eudaimonic aspects of well-being in a sample of 3986 American adults. The results showed that the ESEM model fitted the data better than the CFA model. As expected, interfactor correlations obtained with ESEM were substantially smaller than those obtained with CFA, indicating greater factor distinctiveness. These results suggest that hedonic and eudaimonic factors are correlated yet largely independent from each other. The results also suggest that ESEM is a more appropriate method than CFA in the study of multi-dimensional constructs, such as mental well-being.     

Confirmatory factor analysis on separability of planning and insight constructs

This study aimed to investigate the separability of planning, a form of noninsight problem solving, from insight problem solving by means of using confirmatory factor analysis (CFA). The relationships of these two types of problem-solving tasks with meta-cognitive awareness were also assessed. Participants performed a set of planning tasks, a set of insight tasks and a self-report inventory on metacognitive ability. The CFA results revealed that planning and insight problem solving were closely related constructs and were not clearly separable. Model comparisons indicated that the fit of the alternative one-factor model was slightly better than the fit of the two-factor model. The correlational results showed that both planning task performance and insight problem-solving performance had no correlations with the metacognitive knowledge or the metacognitive regulation components of the metacognitive awareness inventory.     

Dimensionality assessment of ordered polytomous items with parallel analysis

Parallel analysis (PA) is an often-recommended approach for assessment of the dimensionality of a variable set. PA is known in different variants, which may yield different dimensionality indications. In this article, the authors considered the most appropriate PA procedure to assess the number of common factors underlying ordered polytomously scored variables. They proposed minimum rank factor analysis (MRFA) as an extraction method, rather than the currently applied principal component analysis (PCA) and principal axes factoring. A simulation study, based on data with major and minor factors, showed that all procedures consistently point at the number of major common factors. A polychoric-based PA slightly outperformed a Pearson-based PA, but convergence problems may hamper its empirical application. In empirical practice, PA-MRFA with a 95% threshold based on polychoric correlations or, in case of nonconvergence, Pearson correlations with mean thresholds appear to be a good choice for identification of the number of common factors. PA-MRFA is a common-factor-based method and performed best in the simulation experiment. PA based on PCA with a 95% threshold is second best, as this method showed good performances in the empirically relevant conditions of the simulation experiment.     

More efficient parameter estimates for factor analysis of ordinal variables by ridge generalized least squares

Data in psychology are often collected using Likert‐type scales, and it has been shown that factor analysis of Likert‐type data is better performed on the polychoric correlation matrix than on the product‐moment covariance matrix, especially when the distributions of the observed variables are skewed. In theory, factor analysis of the polychoric correlation matrix is best conducted using generalized least squares with an asymptotically correct weight matrix (AGLS). However, simulation studies showed that both least squares (LS) and diagonally weighted least squares (DWLS) perform better than AGLS, and thus LS or DWLS is routinely used in practice. In either LS or DWLS, the associations among the polychoric correlation coefficients are completely ignored. To mend such a gap between statistical theory and empirical work, this paper proposes new methods, called ridge GLS, for factor analysis of ordinal data. Monte Carlo results show that, for a wide range of sample sizes, ridge GLS methods yield uniformly more accurate parameter estimates than existing methods (LS, DWLS, AGLS). A real‐data example indicates that estimates by ridge GLS are 9–20% more efficient than those by existing methods. Rescaled and adjusted test statistics as well as sandwich‐type standard errors following the ridge GLS methods also perform reasonably well.     

A Reexamination of the Structure of the Inward Outward Upward Prayer Scale

The dimensional structure of the Inward Outward Upward Prayer Scale was examined using exploratory factor analytic and confirmatory factor analytic (CFA) methods. A total of 703 adult Jewish pray-ers participated in the study. CFA indicated a satisfactory fit for the first-order eight-factor structure after excluding two cross-loaded items. The eight prayer subscales demonstrated acceptable levels of reliability, and the correlations between the subscales were similar to those reported in past research. CFA on second-order models indicated that a two-dimensional modality model (cognitive prayer/emotional prayer) had the best fit and was clearly superior to the directionality and intentionality models presented in past research. The implications of these findings for future research on the psychology of prayer are discussed.     

Measuring stress in clinical and nonclinical subjects using a German adaptation of the Perceived Stress Scale

Background/Objective: Stress is perceived differently across individuals, which might be particularly true for nonclinical and clinical subjects. For this reason, we tested a German adaption of the 10-item Perceived Stress Scale (PSS-10) for model fit and measurement invariance in a big nonclinical and clinical sample. Method: We (1) conducted multiple confirmatory factor analysis (CFA) in 1,248 nonclinical subjects and 575 outpatients, (2) measurement invariance with multigroup CFA, (3) assessed correlations with relevant constructs and (4) calculated internal consistencies for overall stress and the subscales Helplessness and Self-efficacy. Results: In both samples, CFA revealed a robust two-factorial structure with an excellent model fit. Group comparisons revealed strict measurement invariance. Correlations with associated measures support validity. Internal consistencies were good to very good. Conclusions: We show highly satisfactory psychometric properties of the German PSS-10 for nonclinical and clinical individuals. Measurement invariance analyses demonstrated that varying stress levels of people with a different mental health status are due to true interindividual differences.     

Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small-sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples.