Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and covariance structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unstructured mean and covariance estimates obtained by the EMalgorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assumptions. We also propose a minimum chi-square approach and show that the estimator obtained by This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017. We gratefully acknowledge the constructive feedback of the editors and two referees.*University of North Texas †University of California, Los Angeles 165 this approach is asymptotically at least as efficient as the two likelihood-based estimators for either normal or nonnormal data. The major contribution of this paper is that for each estimator, we give a test statistic whose asymptotic distribution is chisquare as long as the underlying sampling distribution enjoys finite fourth-order moments. We also give a characterization for each of the two likelihood ratio test statistics when the underlying distribution is nonnormal. Modifications to the likelihood ratio statistics are also given. Our working assumption is that the missing data mechanism is missing completely at random. Examples and Monte Carlo studies indicate that, for commonly encountered nonnormal distributions, the procedures developed in this paper are quite reliable even for samples with missing data that are missing at random.

Nonnormality of univariate data has been extensively examined previously (Blanca et al., Methodology Behavioral and Social Sciences, 9(2), 78-84, 2013; Miceeri, Psychological Bulletin, 105(1), 156, 1989). However, less is known of the potential nonnormality of multivariate data although multivariate analysis is commonly used in psychological and educational research. Using univariate and multivariate skewness and kurtosis as measures of nonnormality, this study examined 1,567 univariate distriubtions and 254 multivariate distributions collected from authors of articles published in Psychological Science and the American Education Research Journal. We found that 74 % of univariate distributions and 68 % multivariate distributions deviated from normal distributions. In a simulation study using typical values of skewness and kurtosis that we collected, we found that the resulting type I error rates were 17 % in a t-test and 30 % in a factor analysis under some conditions. Hence, we argue that it is time to routinely report skewness and kurtosis along with other summary statistics such as means and variances. To facilitate future report of skewness and kurtosis, we provide a tutorial on how to compute univariate and multivariate skewness and kurtosis by SAS, SPSS, R and a newly developed Web application. Almost all commonly used statistical methods in psychology and other social sciences are based on the assumption that the collected data are normally distributed. For example, t-and F-distributions for mean comparison, Fisher Ztransformation for inferring correlation, and standard errors and confidence intervals in multivariate statistics are all based on the normality assumption (Tabachnick & Fidell, 2012). Researchers rely on these methods to accurately portray the effects under investigation, but may not be aware that their data do not meet the normality assumption behind these tests or what repercussions they face when the assumption is violated. From a methodological perspective, if quantitative researchers know the type and severity of nonnormality that researchers are facing, they can examine the robustness of normal-based methods as well as develop new methods that are better suited for the analysis of nonnormal data. It is thus critical to understand whether practical data satisfy the normality assumption and if not, how severe the nonnormality is, what type of nonnormality it is, what the consequences are, and what can be done about it. : European Journal of Research Methods for theTo understand normality or nonnormality, we need to first define a measure of it. Micceri (1989) evaluated deviations from normality based on arbitrary cut-offs of various measures of nonnormality, including asymmetry, tail weight, outliers, and modality. He found that all 440 large-sample achievement and psychometric measures distributions were nonnormal, 90 % of which had sample sizes larger than 450. More recently, Blanca et al. (2013) evaluated nonnormality using the skewness and kurtosis 1 of 693 1 Without specifi...

Structural equation modeling is a well-known technique for studying relationships among multivariate data. In practice, high dimensional nonnormal data with small to medium sample sizes are very common, and large sample theory, on which almost all modeling statistics are based, cannot be invoked for model evaluation with test statistics. The most natural method for nonnormal data, the asymptotically distribution free procedure, is not defined when the sample size is less than the number of nonduplicated elements in the sample covariance. Since normal theory maximum likelihood estimation remains defined for intermediate to small sample size, it may be invoked but with the probable consequence of distorted performance in model evaluation. This article studies the small sample behavior of several test statistics that are based on maximum likelihood estimator, but are designed to perform better with nonnormal data. We aim to identify statistics that work reasonably well for a range of small sample sizes and distribution conditions. Monte Carlo results indicate that Yuan and Bentler's recently proposed F-statistic performs satisfactorily.

Model evaluation is one of the most important aspects of structural equation modeling (SEM). Many model fit indices have been developed. It is not an exaggeration to say that nearly every publication using the SEM methodology has reported at least one fit index. Most fit indices are defined through test statistics. Studies and interpretation of fit indices commonly assume that the test statistics follow either a central chi-square distribution or a noncentral chi-square distribution. Because few statistics in practice follow a chi-square distribution, we study properties of the commonly used fit indices when dropping the chi-square distribution assumptions. The study identifies two sensible statistics for evaluating fit indices involving degrees of freedom. We also propose linearly approximating the distribution of a fit index/statistic by a known distribution or the distribution of the same fit index/statistic under a set of different conditions. The conditions include the sample size, the distribution of the data as well as the base-statistic. Results indicate that, for commonly used fit indices evaluated at sensible statistics, both the slope and the intercept in the linear relationship change substantially when conditions change. A fit index that changes the least might be due to an artificial factor. Thus, the value of a fit index is not just a measure of model fit but also of other uncontrollable factors. A discussion with conclusions is given on how to properly use fit indices.In social and behavioral sciences, interesting attributes such as stress, social support, socio-economic status cannot be observed directly. They are measured by multiple indicators that are subject to measurement errors. By segregating measurement errors from the true scores of attributes, structural equation modeling (SEM), especially its special case of covariance structure analysis, provides a methodology for modeling the latent variables directly. Although there are many MULTIVARIATE BEHAVIORAL RESEARCH, 40(1),

This study proposes a rational expectations equilibrium model of crises and contagion in an economy with information asymmetry and borrowing constraints. Consistent with empirical observations, the model finds: (1) Crises can be caused by small shocks to fundamentals; (2) market return distributions are asymmetric; and (3) correlations among asset returns tend to increase during crashes. The model also predicts: (1) Crises and contagion are likely to occur after small shocks in the intermediate price region; (2) the skewness of asset price distributions increases with information asymmetry and borrowing constraints; and (3) crises can spread through investor borrowing constraints. Copyright 2005 by The American Finance Association.