In recent years, several methods have been proposed for the discovery of causal structure from non-experimental data. Such methods make various assumptions on the data generating process to facilitate its identification from purely observational data. Continuing this line of research, we show how to discover the complete causal structure of continuous-valued data, under the assumptions that (a) the data generating process is linear, (b) there are no unobserved confounders, and (c) disturbance variables have non-Gaussian distributions of non-zero variances. The solution relies on the use of the statistical method known as independent component analysis, and does not require any pre-specified time-ordering of the variables. We provide a complete Matlab package for performing this LiNGAM analysis (short for Linear Non-Gaussian Acyclic Model), and demonstrate the effectiveness of the method using artificially generated data and real-world data.

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model.

The most widely used multivariate statistical models in the social and behavioral sciences involve linear structural relations among observed and latent variables. In practice, these variables are generally nonnormally distributed, and hence classical multivariate analysis, based on multinormal error-free variables having no simultaneous interrelations, is not adequate to deal with such data. Since structural relations among variables imply a structure for the multivariate product moments of the variables, general methods for the analysis of mean and covariance structures have been proposed to estimate and test particular model structures. Unfortunately, extant statistical tests, such as the likelihood ratio test (LRT) and a test based on asymptotically distribution free (ADF) covariance structure analysis, have been found to be virtually useless in practical model evaluation at finite sample sizes with nonnormal data. For example, in one condition of a simulation on confirmatory facto...

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M 0 implies on a less restricted one M 1 .IfT 0 and T 1 denote the goodness-of-fit test statistics associated to M 0 and M 1 , respectively, then typically the difference T d = T 0 ; T 1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M 0 and M 1 . As in the case of the goodness-of-fit test, it is of interest to scale the statistic T d in order to improveitschi-square approximation in realistic, i.e., nonasymptotic and nonn...

According to Kenny and McCoach (2003), chi-square tests of structural equation models produce inflated Type I error rates when the degrees of freedom increase. So far, the amount of this bias in large models has not been quantified. In a Monte Carlo study of confirmatory factor models with a range of 48 to 960 degrees of freedom it was found that the traditional maximum likelihood ratio statistic, TML, overestimates nominal Type I error rates up to 70 % under conditions of multivariate normality. Some alternative statistics for the correction of model-size effects were also investigated: the scaled Satorra–Bentler statistic, TSC; the adjusted Satorra–

In this paper we extend the standard approach of correlation structure analysis in order to reduce the dimension of highdimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptical copulae a ’correlation-like ’ structure remains but different margins and non-existence of moments are possible. Moreover, elliptical copulae allow also for a ’copula structure analysis ’ of dependence in extremes. After introducing the new concepts and deriving some theoretical results we observe in a simulation study the performance of the estimators: the theoretical asymptotic behavior of the statistics can be observed even for a sample of only 100 observations. Finally, we test our method on real financial data and explain differences between our copula based approach and the classical approach. Our new method yields a considerable dimension reduction also in non-linear models.

Somer obustness questions in str uctur al equation modeling (SEM) ar intr duced. Factor that a#ect the occuruv ce of nonconver gence and impr: er solutions arr/7 ewed in detail.

Li’s pHd method uses an asymptotic chi-squared test statistic to evaluate a hypothesized dimensionality of a reduceddimension space in a largely nonparametric setting. This statistic is based on an assumed normal distribution of the predictors. When the distributional assumption is violated, a mixture chi-squared test proposed by Cook is theoretically more appropriate. However, both tests may not perform well with small or intermediate sized nonnormal samples. We propose two corrections to Li’s statistic to enable the chi-squared approximation to be more accurate in such samples. The corrections are based on the mean and variance of the statistic of Cook’s mixture distribution. The performance of Li’s, Cook’s, and the two new statistics are compared in some small simulation studies. Results show that one of the new tests performs about as well as Cook’s, while the other performs better than the previously proposed tests.

Covariance structure analysis plays an important role in social and behavioral sciences to evaluate hypothesized influences among unmeasured latent and observed variables. Existing methods for analyzing these data rely on unstructured sample means and covariances estimated under normality, and evaluate a proposed structural model using statistical theory based on normal theory MLE and generalized least squares (GLS) with a weight matrix obtained from inverting a matrix based on sample fourth moments and covariances. Since the influence functions associated with these methods are quadratic, a few outliers can make these classical procedures a total failure. Considering that data collected in social and behavioral sciences are not so accurate, some robust methods are necessary in estimation and testing. Even though the theory for robustly estimating multivariate location and scatter has been developed extensively, very little has been accomplished in robust mean and covariance structure ...

This study examined how three item response models performed when they were applied to data collected from a conventionally developed Likert-type personality scale. Each model examined is based on a different response assumption: the multiple linear factor analysis model (continuous responses), the TOBIT factor analysis model (censored responses), and the multidimensional graded response model. The item and examinee parameters of the models were estimated using different discrepancy functions and current software implementations. Comparisons were made in terms of the goodness-of-fit of the model, parameter estimates, and criterion-related validity. Results showed that the models ’ response assumptions were reasonably tenable and that the