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

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  • This article was downloaded by: [Ume University Library]On: 15 November 2014, At: 01:31Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

    Structural Equation Modeling: AMultidisciplinary JournalPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/hsem20

    A Monte Carlo Study ofRecovery of Weak FactorLoadings in ConfirmatoryFactor AnalysisCarmen XimnezPublished online: 19 Nov 2009.

    To cite this article: Carmen Ximnez (2006) A Monte Carlo Study of Recovery of WeakFactor Loadings in Confirmatory Factor Analysis, Structural Equation Modeling: AMultidisciplinary Journal, 13:4, 587-614, DOI: 10.1207/s15328007sem1304_5

    To link to this article: http://dx.doi.org/10.1207/s15328007sem1304_5

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    http://www.tandfonline.com/page/terms-and-conditions

  • A Monte Carlo Study of Recoveryof Weak Factor Loadings in

    Confirmatory Factor Analysis

    Carmen XimnezAutonoma University of Madrid, Spain

    The recovery of weak factors has been extensively studied in the context of exploratoryfactoranalysis.Thisarticlepresents the resultsofaMonteCarlo simulationstudyof re-coveryofweakfactor loadings inconfirmatory factoranalysisunderconditionsofesti-mationmethod(maximumlikelihoodvs.unweighted least squares), samplesize, load-ing size, factor correlation, and model specification (correct vs. incorrect). The effectsof these variables on goodness of fit and convergence are also examined. Results showthat recovery of weak factor loadings, goodness of fit, and convergence are improvedwhen factors are correlated and models are correctly specified. Additionally, un-weighted least squares produces more convergent solutions and successfully recoversthe weak factor loadings in some instances where maximum likelihood fails. The im-plications of these findings are discussed and compared to previous research.

    The development of confirmatory factor analysis (CFA) has provided considerablemeans for theory construction and evaluation (Browne, 1984). The CFA model(Jreskog & Srbom, 1981) can be given as:

    x = + (1)

    where x is a vector of p observed variables, is a vector of q factors such that q < p, is a p q matrix of factor loadings, and is a vector of p measurement error vari-ables. It is assumed that E(x) = E() = E() = 0 and that E() = 0. The covariancematrix for x, denoted by is:

    = + (2)where is the q q covariance matrix of and the p p covariance matrix of. For convenience, it is usually assumed that = and that is diagonal.

    STRUCTURAL EQUATION MODELING, 13(4), 587614Copyright 2006, Lawrence Erlbaum Associates, Inc.

    Correspondence should be addressed to Carmen Ximnez, Universidad Autonoma de Madrid,Departamento de Psicologia Social y Metodologia, Cantoblanco s/n, 28049 Madrid, Spain. E-mail:carmen.ximenez@uam.es

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  • When a CFA is conducted, the researcher must decide model specification,identification, parameter estimation method, and assessment of model fit (Bollen,1989). In practical applications, researchers often face the problem of finding fac-torial structures containing one or more weak factors. A weak factor is a factor thatshows relatively little influence on the set of measured variables or is defined bysmall loading sizes. The recovery of weak factors has been extensively studied inthe context of exploratory factor analysis (EFA). The majority of studies examinethe effect of sample size, model error, and estimation method for correctly speci-fied models. This article extends the previous study of variables that affect the re-covery of weak factor loadings to the context of CFA. We present the results of alarge Monte Carlo simulation study of recovery of weak factor loadings in CFAunder conditions of estimation method, sample size, loading size, model specifica-tion (correct vs. incorrect), and factor correlations.

    PAST RESEARCH

    Within the context of EFA, Briggs and MacCallum (2003) examined the perfor-mance of maximum likelihood (ML) and unweighted least squares (ULS) estima-tion methods to recover a known factor structure with relatively weak factors. Theyintroduced two types of error (model and sampling error) separately and in combi-nation. Results of a simulation study indicated that in situations with a moderateamount of error, ML often failed to recover the weak factor, whereas ULS suc-ceeded. With small sample sizes (e.g., N = 100), ML failed more often than didULS. This failure was associated with the occurrence of Heywood cases.

    Other studies have examined the factor pattern recovery in EFA under conditionsof sample size, number of factors, number of indicators per factor, and level ofcommunalities. For example, MacCallum, Widaman, Zhang, and Hong (1999) andMacCallum, Widaman, Preacher, and Hong (2001) found that the factor pattern re-covery of ML solutions is better as sample size and number of indicators per factorincrease, and as number of factors decreases. They also found that as communalitiesbecome lower, achieving good recovery is more affected by sample size and factordetermination. Velicer and Fava (1998) conducted a study that included similar con-ditions to the studies by MacCallum et al. (2001; MacCallum et al., 1999) and alsoexamined the loadingsizeeffect.They found that recovery improvesas levelof load-ing size, sample size, and number of indicators per factor increase.

    Within the context of CFA, there are some studies evaluating those same effectsfor the ML estimation method (Anderson & Gerbing, 1984; Boomsma, 1982;Gerbing & Anderson, 1985). Similar to the studies in the context of EFA, theyfound that with lower loading sizes the recovery improves as sample size and num-ber of indicators per factor increase. The only studies that compare estimationmethods in the context of CFA are by Olsson, Troye, and Howell (1999) andOlsson, Foss, Troye, and Howell (2000), which refer to ML, generalized least

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  • squares (GLS), and weighted least squares (WLS) solutions. They evaluated theeffect of estimation method, model misspecification, and sample size on the recov-ery of underlying structure (which they called the theoretical fit) and the goodnessof fit (which they called the empirical fit). Their results suggested better theoreticalfit by ML but at the cost of lower empirical fit. In addition, they found thatmisspecification exerted the largest effect on both theoretical and empirical fit andconcluded that the larger the degree of misspecification, the higher the discrepancybetween the methods. These studies raised the possibility that the type of con-straints could affect factor recovery in CFA.

    THIS STUDY

    This work extends past research examining the recovery of weak factors and fac-tor pattern recovery in the context of CFA using Monte Carlo simulation. Focusis on the recovery of weak factor loadings for a known factor structure with arelatively weak factor. This study extends past research in several ways. First,the weak factor is defined with loading sizes as those commonly found by re-searchers in practice. Previous studies define the weak factor with loading sizesbetween .40 and .60, whereas in practice, as Briggs and MacCallum (2003)noted, a researcher would most likely consider a weak factor to be one withloadings of .20 and .30 (p. 53). Second, models with correlated factors are re-ferred to. The majority of previous studies refer to factor structures with or-thogonal factors but do not examine factor recovery with correlated factors. It ispossible that the recovery of weak factor loadings is affected if factors are corre-lated. The magnitude of this effect over a range of conditions is one topic of in-vestigation in this study. Third, we examine the recovery of weak factor loadingsunder misspecification conditions. The majority of previous studies are based onthe assumption that models are correct for the study population. This approach isof limited value as it ignores the fact that, in practice, models are incorrect(MacCallum, 2003). A common misspecification condition found in practice isthe specification of an incorrect number of factors. For example, a researchercould specify a larger number of factors in the model or a smaller one. This kindof misspecification implies that the variables load on different factors in the the-oretical (or true) model and in the estimated (or fitted) one. That is, if the modelis correctly specified, the population model and the estimated one have the samefactor loading pattern and the same number of factors, whereas if the model ismisspecified by altering the number of factors, the estimated model modifies thepattern of both the matrix (as the dimensionality of the matrix and some struc-tural zeros change) and the vector (as the number of factors is modified). It isunclear whether the estimated loadings are similar to the theoretical ones underthese circumstances. Previous studies in the context of EFA (e.g., Fava &Velicer, 1992, 1996) indicate that retention of an incorrect number of factors, es-

    RECOVERY OF WEAK FACTOR LOADINGS IN CFA 589

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  • pecially retention of too few factors, can cause major distortion of loading pat-terns. Therefore, it is expected that the CFA models misspecified by altering thenumber of factors not only show a poorer goodness of fit but also a poorer re-covery of weak factor loadings. The aim in this article is to investigate the mag-nitude of these effects. Fourth, the effect of the estimation method on the recov-ery of weak factor loadings is studied, referring to ML and ULS estimationmethods. Given the assumptions that the observations of x are independent andhave a multivariate normal distribution, the ML method is the most commonlyused estimation procedure, as ML estimators have the desirable asymptoticproperties of being unbiased, consistent, efficient, and normally distributed. TheULS method is commonly used when the normality assumption is not met. Themain difference between both methods is in the discrepancy functions and in theassumptions regarding error (in ML all error is sampling error and in ULS it ismodel error). Unless the model holds exactly for the population, they provide es-timates that are at least slightly different (see Bollen, 1989, pp. 104113, for amore detailed description of the ML and ULS estimation methods). Previous re-search in the context of EFA has found that ULS is superior to ML on the recov-ery of weak factors. Thus, it is expected that this finding generalizes to the con-firmatory case. Fifth, the goodness of fit of the model and the occurrence ofnonconvergent solutions and Heywood cases under the study conditions are alsoexamined. The following sections describe the main characteristics of the simu-lation design, present the results, and discuss their implications as compared toprevious research.

    MONTE CARLO SIMULATION STUDY

    The six-step approach for Monte Carlo simulation designs in structural equationmodels recommended by Skrondal (2000) and the Paxton, Curran, Bollen, Kirby,and Fen (2001) guidelines are used to present the design of the simulation study.

    Step 1: Statement of the Research Problem

    This study explores the effects of estimation method, sample size, loading size,factor correlation, and model specification (correct vs. incorrect) on the recoveryof weak factor loadings in the context of CFA. The effects of these variables on thegoodness of fit of the model and the occurrence of nonconvergent solutions andHeywood cases are also examined.

    Step 2: Experimental Plan

    The design was developed to address a reasonably diverse set of factor models andmodel characteristics, as to represent the range of values typically encountered inpractice. The general approach used in this study involved the following steps. First,

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  • population correlation matrices were defined as having known factor structures, un-der the assumption that the factor model holds exactly in the population. The popula-tion factor structures (or generating models) were defined on the basis of theMacCallum and Tucker (1991) model, which includes 12 measured normal vari-ables and three factors, the third factor being relatively weak. Two other factor struc-tures were defined with the same 12 measured variables. The first structure con-tained two factors, the second factor being the weak factor, whereas the secondstructurecontainedonlyonefactor,being theweakfactor.Asingle-factormodelwasincluded to examine the recovery of a single weak factor, whereas models with twoor three factors would be encountered more often in practice and allow study of howthe weak factor loadings are recovered in the presence of stronger factors. The weakfactors in these structures had loadings of .50 or below, to distinguish them from ma-jor factors, which had loadings of .70 or above. Next, sample correlation matriceswere generated from those populations, using various levels of sample size, loadingsize, and factor correlation. The sample correlation matrices were then factor ana-lyzed by ML and ULS estimation methods. Parameters were estimated for modelscorrectly and incorrectly specified. T...

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