University of Groningen

The robustness of estimation methods for structure analysis. Hoogland, Jeffrey Johannes

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Publication date: 1999

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Citation for published version (APA): Hoogland, J. J. (1999). The robustness of estimation methods for covariance structure analysis. s.n.

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Summary f, 87,93, LL8, l2+r25, i, 114123,132, L37-L44 This thesis investigates the robustnessof estimation methods in covariancestructure anal- ysis (CSA). By of CSA structural relations between hypothetical constructs can be -87, 93,97,99-100, studied. When covariancestructure analysis is applied, in theory a number of assumptions must hold. The central question is: how robust are the of model parameters, 42-50,65-66, 69.70, the estimators of the standard errors, and the goodness-of-fitstatistic for the model, when il128,134, 137-144 one or more of the underlying assumptionsa^re violated? In Chapter 1 the covariancestructure model and a number of estimation methods are described,such as the Maximum Likelihood (ML), the GeneralizedLeast Squa,res(GLS), 86,106 the Elliptical ReweightedLeast Squares(ERLS), and the Asymptotically Distribution Flee 0, 14&1rg (ADF) estimation method. This chapter explains how estimates for the model parameters, 5G51,58,68,107-113, estimates for the standard errors, and the chi-squaretest for the model fit can be obtained. Thereafter, five robustnessquestions are discussed,namely robustness against: 1) the use of small samples; 2) violation of the postulated distribution of the observedvariables; i, 111 3) the analysis of a correlation matrix instead of a covariancematrix; 5 4) model misspecification; 5) non-linear relations between latent va"riables. The robustness of an against violations of underlying assumptions can be determined empirically by means of a Monte Carlo study. Robustnessstudies with respect L2ïL28, L44 to CSA a.re published regularly. In Chapter 2 the results of 25 robustness studies are reviewed. These studies have investigated the effect of small sample size or distributional violations on the behavior of an estimator. They are described systematically and com- pa.redby means of characteristicsrelated to the population model, the simulated , the estimation methods,the replications,or the researchsummary . The conclusions of robustness studies frequently seem to contradict. An important cause for contradicting conclusionsis differencesin assessmentcriteria of robustness stud- ies. Becausethese criteria should be no causefor differencesin conclusions,general assess- ment criteria are defined. They are applied to the presented results in robustness studies whenever possible. The conclusionsof the author(s) are therefore not taken for granted. By means of a meta.analysis the findings of robustnessstudies are summaxized. The char- acteristics serve as explanatory variables for an analysis concerning the of estimators of the model pa,rameters,estimators of the standard errors, and the chi-square statistic for the model fit. In general, population models in robustnessstudies are rather small, that is, they have

-- 158 Summaty few degreesof freedom compared with models in applied research. On the basis of the results of the meta.analysis a Monte Carlo simulation study is conducted to fill such gaps in the robustnessresearch and to refine the conclusionsof the meta-analysis. In Chapter 3 the design of this simulation study is discussed,which consists of five population models, eleven continuous distributions of the observedva.riables, four sample sizes (200, 400, 800 and 1600),and four estimationmethods (ML, GLS, ERLS, and ADF). For eachcell of the Monte Carlo design 300 replications a,reanalyzed. A is included in the analysis if it provides a convergent and proper solution for each of the four estimation methods. There a.reseveral corrections for estimators of the standard errors and the chi-square statistic. Corrections of maximum likelihood estimators investigated in the simulation study a,rethe ScaledML X'statistic (Satorra & Bentler, 1994) and the Robust ML esti- mator of the standa"rderrors (Browne, 1984;Bentler & Dijkstra, 1985). The behavior of the Yuan-Bentler correctedADF / statistic is also studied (Yuan & Bentler, 1994). To simplify the assessmentof the performance an estimator, criteria a,redefined that dichotomize this performance a.sacceptable or unacceptable. Given that the performance of an estimator generally becomes better as the sample size increases,it is essential to know for which sample size this performance becomesacceptable. In a number of ca.ses the behavior of an estimator is unacceptableirrespective of the sample size. The results for one of the five population models are discussedextensively in Chapter 4. The behavior of an estimator is studied per para,uretertype, that is, for the factor load- ings, the path coefficients, the factor correlations, and the of the measurement errors, because of possible differencesbetween para,metertypes. In general, the overall bias of the GLS and ADF parameter estimators is mainly due to underestimation of the va,riancesof the measurementerrors. When the observedvariables have positive , the overall bias of the ML, GLS, a.ndERLS estimator of the standard errors is mainly due to underestimation of the standard errors of the estimated varia,ncesof the measurement errors. In Chapter 4 a number of regressionmodels that predict aspectsof the behavior of an estimator a^represented. The predictors are related to the skewnessand the kurtosis of the observed variables, and the sample size. The regressioncoefficients are estimated by means of the simulated data. In Chapter 5 the influence of the population model on the behavior of an estimator is examined. The examined model effects are related to the size of the factor loadings, the number of factors, the number of indicators per factor, and the existence of structural relations between factors. The design ofthe Monte Carlo study is chosenso that the effect of a specific model cha.racteristiccan be investigated by comparing the simulation results for two different models. Chapter 5 also investigates whether general conclusions across all models ca,nbe formulated. wmary Summaly 159 s of the An important finding applicable to each population model is that the ML estimator ah gaps of the model para,rreters is almost unbiased when the sa,mplesize is at least 200. In the rapter3 case of a small sa,urplethe GLS para,ureterestimator has a much larger bias than the ML models, pa,ra,ureter estimator when the model has at least twelve observed va"riables. The bias {00,800 of the ADF para,meterestimator increaseswhen the kurtosis increa^ses.With a positive rll of the kurtoeis the bias of the ADF parameter estimator is larger than that of the GLS parameter analysis estimator, irrespective of sa,mplesize. bhods. The ML and GLS estimators of the standard errors a,rebiased when the averagekurtosis ri-square of the observed rariables deviates from zero. The standard errors a,reunderestimated in nulation the case of a positive average kurtosis and overestimated in the case of negative average ML esti- kurtosis. The ADF estimation method gives a large underestimation of the standard errors nvior of when the sa,rrple size is small relative to the number of observed variables in the model. )4). The Robust ML standaxd error estimator has a smaller bias than the other estimators wheu the average kurtosis is at least 2.0 and the sa,mplesize is at nedthat least 400. The chi-square statistic is on average smaller when the factor loadings population brmance in the are relatively small, that is, when the observed variables are unreliable measures the entialto of underlying factors. The ML statistic is not robust against a large ,the of cases f model is then rejected too often. The Scaled ML X2 statistic grves a better description of the model fit in cases of large non-normality. The ADF statistic tends to reject models npter 4. f with many degreesof freedom too often, especially when the sample size is small. The ;or load- Yuan-Bentler corrected ADF statistic is more robust against small sa,mplesizes than urement I the ADF statistic. The behavior of these two goodness-of-fit statistics is insensitive l overall f to the distribution of the observedva^riables when the sa,urplesize is la,rgerelative to the n of the number of degreesof freedom. curtosis, An iurportant goal of both the meta-analysis and the Monte Carlo study is to find nly due guidelines for applied resea,rchersrega,rding choice of estimation method. These guidelines rement depend on specific properties ofthe postulated model and the sa;nple data. In general, they indicate how la,rge the sa,rrple size must be to obtain almost unbiased para,rreter estimates, lr of an almost unbiased estimates of the standard errors, or an acceptable rejection rate of the tmis of chi-square statistic at lhe 5Tolevel when the model is correctly specified. These guidelines Éed by are spelled out in Chapter 6. In addition to the required sa.mplesize, the expected sign of the bias of an estimator is imator supplied when the sarnple size is too small. Several tables and formulas are provided with rdiags, which the required sa;nple size and the sign of possible bias can be determined. The final rctural chapter also discussesa number ofunder-exposedsubjects, such as the effect ofcategorizing : effect observed ra,riables and model misspecification on the behavior of an estimator. reaults acro&9