r57
Subject index
Summary
f, 87,93,LL8,l2+r25,
i, 114123,132,L37-L44
-87,93,97,99-100,
This thesis investigatesthe robustnessof estimation methods in covariancestructure analysis (CSA). By means of CSA structural relations between hypothetical constructs can be
42-50,
65-66,69.70,
studied. When covariancestructure analysis is applied, in theory a number of assumptions
must hold. The central question is: how robust are the estimators of model parameters,
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^reviolated?
86,106
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),
the Elliptical ReweightedLeast Squares(ERLS), and the Asymptotically Distribution Flee
0, 14&1rg
5 G 5 1 , 5 8 , 6180, 7 - 1 1 3 ,
i, 111
5
L2ïL28, L44
(ADF) estimation method. This chapter explains how estimates for the model parameters,
estimates for the standard errors, and the chi-squaretest statistic 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;
3) the analysis of a correlation matrix instead of a covariancematrix;
4) model misspecification;
5) non-linear relations between latent va"riables.
The robustness of an estimator against violations of underlying assumptions can be
determined empirically by means of a Monte Carlo study. Robustnessstudies with respect
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 compa.redby means of characteristicsrelated to the population model, the simulated data, the
estimation methods, the replications,or the researchsummary statistics.
The conclusions of robustness studies frequently seem to contradict. An important
cause for contradicting conclusionsis differencesin assessmentcriteria of robustness studies. Becausethese criteria should be no causefor differencesin conclusions,general assessment 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 characteristics serve as explanatory variables for an analysis concerning the bias 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
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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 robustnessresearchand 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 replication 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
Corrections of maximum likelihood estimators investigated in the simulation
the
study a,re
ScaledML X'statistic (Satorra & Bentler, 1994) and the Robust ML esti-
statistic.
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.sacceptableor 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 loadings, the path coefficients, the factor correlations, and the variances 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 kurtosis,
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.
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An important finding applicable to each population model is that the ML estimator
of the model para,rreters is almost unbiased when the sa,mplesize is at least 200. In the
case of a small sa,urplethe GLS para,ureterestimator has a much larger bias than the ML
pa,ra,ureterestimator when the model has at least twelve observed va"riables. The bias
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159
of the ADF para,meterestimator increaseswhen the kurtosis increa^ses.With a positive
kurtoeis the bias of the ADF parameter estimator is larger than that of the GLS parameter
estimator, irrespective of sa,mplesize.
The ML and GLS estimators of the standard errors a,rebiasedwhen the averagekurtosis
of the observed rariables deviates from zero. The standard errors a,reunderestimated in
the case of a positive average kurtosis and overestimated in the case of negative average
kurtosis. The ADF estimation method gives a large underestimation of the standard errors
when the sa,rrple size is small relative to the number of observed variables in the model.
The Robust ML standaxd error estimator has a smaller bias than the other standard error
estimators wheu the average kurtosis is at least 2.0 and the sa,mplesize is at least 400.
The chi-square statistic is on averagesmaller when the factor loadings in the population
are relatively small, that is, when the observed variables are unreliable measures of the
underlying factors. The ML f statistic is not robust against a large skewness,the 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 f statistic tends to reject models
with many degreesof freedom too often, especially when the sample size is small. The
Yuan-Bentler corrected ADF I statistic is more robust against small sa,mplesizes than
the ADF f statistic. The behavior of these two goodness-of-fit statistics is insensitive
to the distribution of the observedva^riableswhen the sa,urplesize is la,rgerelative to the
number of degreesof freedom.
An iurportant goal of both the meta-analysis and the Monte Carlo study is to find
guidelines for applied resea,rchersrega,rding choice of estimation method. These guidelines
depend on specific properties ofthe postulated model and the sa;npledata. In general, they
indicate how la,rge the sa,rrple size must be to obtain almost unbiased para,rreter estimates,
almost unbiased estimates of the standard errors, or an acceptable rejection rate of the
chi-square statistic at lhe 5Tolevel when the model is correctly specified. These guidelines
are spelled out in Chapter 6.
In addition to the required sa.mplesize, the expected sign of the bias of an estimator is
supplied when the sarnple size is too small. Several tables and formulas are provided with
which the required sa;nple size and the sign of possible bias can be determined. The final
chapter also discussesa number ofunder-exposedsubjects, such as the effect ofcategorizing
observed ra,riables and model misspecification on the behavior of an estimator.