ABSTRACT: SEM has used higher-order moments to create robust estimators and/or robust asymptotic covariance matrix of estimators in nonnormal populations. The usage of higher-order moments does not solve any inherent problems of SEM such as equivalent models and inability of model assessment of saturate models. Recent research by Kano, Shimizu and our collaborators has shown that more ACTIVE use of higher-order moments as in independent component analysis (ICA) can solve many of these problems. In this talk, we first present basic ideas of such use of nonnormality and show how SEM people can apply the fundamental theorem by Comon (1994) to structural equation modeling including statistical causal analysis. Some comprehensive examples are provided. Some coincidence between ICA and SEM about independence assumption is pointed out. More technical results and applications will be given in later talks of this session.

ABSTRACT: Path analysis is often applied to observational data to study causal structures. The path analysis is an extension of regression analysis where many endogenous and exogenous variables can be analyzed simultaneously. Now path analysis is incorporated with factor analysis and allows latent variables in the model. The new framework is called structural equation modeling (SEM) and is a powerful tool of causal analysis. However, SEM is of confirmatory nature and researchers have to model true causal relationships based on background knowledge. Lack of background knowledge often involves many difficulties such as inability of determining direction of a path and a serious bias of an estimate caused by unnoticed confounding variables. These limitations mainly come from normal assumption in the SEM. Shimizu and Kano (2003) and Kano and Shimizu (2003) relaxed the restriction and showed that use of nonnormality extends conventional SEM and makes it possible to examine the following three models i) the determination of a direction of a path; ii) detection and adjustment of unobserved confounding variables; iii) a variant of bi-directed causal model. In this paper, we combine these three models and develop a new statistical method for exploratory causal inference using nonnormality of observed variables.

ABSTRACT: Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical methodology was developed to analyze such data. Most of the procedures for analyzing multilevel data are derived from maximum likelihood based on the normal distribution assumption. Standard errors for parameter estimates in these procedures are obtained from the corresponding information matrix. Because practical data typically contain heterogeneous marginal skewnesses and kurtoses, it is interesting to know how nonnormally distributed data affect the standard errors of parameter estimates in a two-level structural equation model. Specifically, we study how skewness and kurtosis in one level affect standard errors of parameter estimates within its level and outside its level. We also show that, parallel to asymptotic robustness theory in conventional factor analysis, conditions exist for asymptotic robustness of standard errors in a multilevel factor analysis model.

ABSTRACT: This paper deals with the bias correction of AIC for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown nonnormal distribution. We propose a corrected version of AIC which is partially constructed by predicted residuals and adjusted to the exact unbiased estimator of the risk when the candidate model includes the true model. It is pointed out that the influences of nonnormality in the bias of our criterion are smaller than the ones in AIC and TIC. We verify that our criterion is better than the AIC, TIC and EIC by conducting numerical experiments.

ABSTRACT: The asymptotic robustness of the normal theory asymptotic biases of the least squares estimators of the parameters in covariance structures against the violation of normality is shown, which is obtained under the conditions required for the asymptotic robustness for the normal theory standard errors and the usual chi-square statistic. The asymptotic robustness holds not only for the estimators of the parameters whose normal theory asymptotic standard errors are asymptotically robust, but also for the non-robust ones. The Wishart maximum likelihood estimators are also shown to have the asymptotic robustness. A numerical illustration for the factor analysis model shows that the empirical biases of robust estimators under nonnormality are close to their corresponding normal theory asymptotic biases.