Ulf Henning Olsson

Structural Equation Modeling 26(5) Doi: https://doi.org/10.1080/10705511.2019.1586545

The asymptotically distribution-free (ADF) test statistic depends on very mild distributional assumptions and is theoretically superior to many other so-called robust tests available in structural equation modeling. The ADF test, however, often leads to model overrejection even at modest sample sizes. To overcome its poor small-sample performance, a family of robust test statistics obtained by modifying the ADF statistics was recently proposed. This study investigates by simulation the performance of the new modified test statistics. The results revealed that although a few of the test statistics adequately controlled Type I error rates in each of the examined conditions, most performed quite poorly. This result underscores the importance of choosing a modified test statistic that performs well for specific examined conditions. A parametric bootstrap method is proposed for identifying such a best-performing modified test statistic. Through further simulation it is shown that the proposed bootstrap approach performs well

Structural Equation Modeling s. 1-15 Doi: https://doi.org/10.1080/10705511.2019.1600408

Robust standard errors are of central importance in confirmatory factor models. In calculating these statistics a central ingredient is the inverse of the asymptotic covariance matrix of second-order moments calculated under the assumption of normality. Currently, two ways of estimating this matrix are employed in software packages. One approach uses the sample covariance matrix, the other the model-implied covariance matrix. Previous research based on a small confirmatory factor model demonstrated that the latter approach yielded a slight improvement in standard error performance. The present study argues theoretically that the discrepancy between the two approaches increases in models where there are few model parameters relative to p(p+1)/2, where p is the number of observed variables. We present simulation results that support this claim, in both small and large correctly specified models, across a large variety of non-normal conditions. We recommend the model-implied covariance matrix for robust standard error computation.

Multivariate Behavioral Research 51(2-3) s. 207-219 Doi: https://doi.org/10.1080/00273171.2015.1133274

We present and investigate a simple way to generate non-normal data using linear combinations of independent generator (IG) variables. The simulated data have prespecified univariate skewness and kurtosis, and a given covariance matrix. In contrast to the widely used Vale-Maurelli (VM) transform, the obtained data is shown to have a non-Gaussian copula. Analytically, we obtain asymptotic robustness conditions for the IG distribution. Empirically, we show that popular test statistics in covariance analysis tend to reject true models more often under the IG transform than under the VM transform. This implies that overly optimistic evaluations of stimators and fit statistics in covariance sstructure analysis may be tempered by including the IG transform for non-normal data generation. We provide an implementation of the IG transform in the R environment.

Springer

Multivariate Behavioral Research 50(5) s. 533-543 Doi: https://doi.org/10.1080/00273171.2015.1036964

Journal of educational and behavioral statistics 37(3) s. 367-386 Doi: https://doi.org/10.3102/1076998611411920

Quality & Quantity: International Journal of Methodology 46(5) s. 1547-1570 Doi: https://doi.org/10.1007/s11135-011-9466-5

Yet another paper on fit measures? To our knowledge, very few papers discuss how fit measures are affected by error variance in the Data Generating Process (DGP). The present paper deals with this. Based upon an extensive simulation study, this paper shows that the effects of increased error variance differ significantly for various fit measures. In addition to error variance the effects depend on sample size and severity of misspecification. The findings confirm the general notion that good fit as measured by the chi-square, RMSEA and GFI etc. does not necessarily mean that the model is correctly specified and reliable. One finding is that the chi square test may give support to misspecified models in situations with a high level of error variance in the DGP, for small sample sizes. Another finding is that the chi-square test looses power also for large sample sizes when the model is negligible misspecified. Other results include incremental fit indices as NFI and RFI which prove to be more informative indicators under these circumstances. At the end of the paper we formulate some guidelines for use of different fit measures.

British Journal of Mathematical & Statistical Psychology 65(1) s. 1-18 Doi: https://doi.org/10.1111/j.2044-8317.2010.02010.x

Computational Statistics & Data Analysis 55(7) s. 2263-2275 Doi: https://doi.org/10.1016/j.csda.2011.01.012

The Satorra Bentler (SB) and the Browne ADF chi-square statistics are used for testing structural equation models with non-normal data. The relationships between the SB and ADF statistics and kurtosis are developed and it is shown that the weighted deviations of the "population" true second-order moments and the tted second-order moments for these statistics tend to decrease with increasing kurtosis if the model does not hold. The results predict that high kurtosis can lead to loss of power. The results are obtained without simulation.

Safety Science 49(3) s. 531-541 Doi: https://doi.org/10.1016/j.ssci.2010.11.007

Event Management 14 s. 193-204

Baltic Journal of Management 5(1) s. 28-50 Doi: https://doi.org/10.1108/17465261011016540

Engineering Economics 21(1) s. 53-59

Quality & Quantity: International Journal of Methodology 40

This paper discusses consequences of violating the normal distribution assumption imbedded in Structural Equation Modeling (SEM). Based on real data from a large sample customer satisfaction survey we follow the procedures as suggested in leading textbooks. We document consequences of this practice and discuss its impact on decision making in marketing.

Psychology & Marketing 22(3) s. 247-269

Sociological Methods & Research 32(4) s. 453-500

Over the years several discrepancy,functions have been introduced both in the literature and in the software of Structural Equation Modeling (SEM). The test statistics for the discrepancy functions associated with Maximum Likelihood (ML), Generalized Least Squares (GLS), and Normal Theory Weighted Least Squares (NWLS) are all asymptotically equivalent. These test statistics are all approximately distributed as central chi-square under correct model specification and if the observed variables are multivariate normally distributed. However, it is known that the distribution of these test statistics will not approximate a central Chi-square distribution for models containing specification error, but is more likely to follow a non-central Chi-square distribution (Browne 1984). This study investigates the empirical distributions of the ML and NWLS discrepancy functions. The study includes 13 different factor models with different types and degrees of specification error It is found, except for small samples, that the empirical distribution of the ML-test statistic outperforms the empirical distribution of the NWLS-test statistic in terms of approximation to the theoretical non-central Chi-square distribution. Furthermore, in some cases, it turned out that the non-central Chi-square approximation was not appropriate even for models that contained minor and moderate degrees of specification error. Abstract: Over the years several discrepancy,functions have been introduced both in the literature and in the softwareof Structural Equation Modeling (SEM). The test statistics for the discrepancy functions associated with Maximum Likelihood (ML), Generalized Least Squares (GLS), and Normal Theory Weighted Least Squares (NWLS) are all asymptotically equivalent. These test statistics are all approximately distributed ascentral chi-square under correct model specification and if the observed variables are multivariate normally distributed.However, it is known that the distribution of these test statistics will not approximate a central Chi-square distribution for models containing specification error, but is more likely to follow a non-central Chi-square distribution (Browne 1984). This study investigates the empirical distributions of the ML and NWLS discrepancy functions. The study includes 13 different factor models with different types and degrees of specification error It is found, except for small samples, thattheempirical distribution of the ML-test statistic outperforms the empirical distribution of the NWLS-test statistic in terms of approximation to the theoretical non-central Chi-square distribution. Furthermore, in some cases, itturned out that the non-central Chi-square approximation was not appropriate even for models that contained minor and moderate degrees of specification error.

British Journal of Mathematical & Statistical Psychology 56(2) s. 289-303

Journal of International Business Studies 33(1) s. 149-167 Doi: https://doi.org/10.1057/palgrave.jibs.8491009

This study was particularly designed to respond to Jaffe and Nebenzahl's (1984) suggestion to test whether and how entity-based and attribute-based semantic differential scaling formats could predict purchasing behaviour within a country-of-origin setting. The study includes recent research on survey context effects in order to explain and extend our analysis, and uses attitude consistency theory as a conceptual framework. A comparison of the fit statistics between the two scaling formats on each entity (country) indicates that the attribute-based format estimates a higher attitude response consistency and predictive validity by giving better predictors of global country image (evaluation) and buying frequency of products from the different countries investigated. This study uses data collected from a probability sample of industrial buyers within the seafood industry.

Structural equation model : present and future : a festschrift in honor of Karl Jöreskog/Robert Cudeck, Stephen du Toit, Dag Sörbom (eds.) s. 169-194

IEEE Transactions on Software Engineering 27(11) s. 999-1013

Structural equation model : present and future : a festschrift in honor of Karl Jöreskog/Robert Cudeck, Stephen du Toit, Dag Sörbom (eds.) s. 169-194

This paper focuses on the effect of model size on the performance of various fit indices in structural equation models. In particular, the paper contrasts the performance of the root mean square error of approximation index (RMSEA) and the goodness-of-fit index (GFI). RMSEA contains a parsimony adjustment whereas GFI does not. Parsimony correction has been found to work well when comparing alterntive models for the same set of variables (Mulaik, et al., 1989; Williams & Holohan, 1994). However, parsimony ratios also appear to show better overall fit when more observed and latent variables are added to the model. To investigate effects of model size, we examine the fit of models that vary with respect to the number of included variables. Different types and degrees of specification errors are also examined. The findings suggest that RMSEA tends to favor models that include more variables and constructs over models that are simpler. Similar patterns were found for other indices correcting for parsimony. GFI was found to decrease with increasing model size for models containing specification errors that involved the same method factor(s) across submodels. The results suggest that in addition to parsimony, i.e., the number of fixed parameters in relation to the number of free parameters, researchers should also consider model size, i.e.,the number of variables when assessing model fit.

Structural Equation Modeling 7(4) s. 557-595

Structural Equation Modeling: The Present and Future. A Festchrift in honor of Karl G. Jøreskog

Multivariate Behavioral Research 34(1) s. 31-59

The Cornell Hotel and Restaurant Administration Quarterly s. 72-81