## Computational methods (24)

*Chair: Yoshio Takane, Wednesday 22nd July, 15.25 - 16.45, Uppercroft, School of Pythagoras.*** **

**Sunho Jung** and Yoshio Takane,* Department of Psychology*, *McGill University, Montreal, Canada*. Regularized two-stage least squares estimation for structural equation models with latent variables. (123)

**Tatsuo Otsu**, *The National Center for University Entrance Examinations, Tokyo, Japan*. Classifying statistically equivalent graphical models by using a logic programming language. (192)

**Ting Hsiang Lin**, Department of Statistics , National Taipei University, Taiwan. Exploring factors affecting parameter estimates in mixture confirmatory factor analysis. (256)

ABSTRACTS

**Regularized two-stage least squares estimation for structural equation models with latent variables**. (123)

*Sunho Jung and Yoshio Takane
*In structural equation models with latent variables, maximum likelihood (ML) estimation is the most predominantly used estimation method. However, the ML method fails to provide accurate solutions in a number of situations including those involving small sample sizes, nonnormality, and model misspecification. To overcome these difficulties, regularized two-stage least squares estimation methods are proposed that incorporate a ridge type of regularization in the estimation of structural equation models with latent variables. Two simulation studies and two empirical applications demonstrate that the proposed methods provide a promising alternative to both the maximum likelihood and (non-regularized) two-stage least squares estimation methods. An optimal value of regularization parameter is found by the

*K*-fold cross validation technique, and a nonparametric bootstrap method is used to evaluate the stability of solutions.

**Classifying statistically equivalent graphical models by using a logic programming language**. (192)

*Tatsuo Otsu*

Recent advances in statistical methods have increased the need for complex statistical modeling. Although some recent statistical software products can handle complex object structures, their capabilities in regards to symbolic manipulation are rather limited. Here, I will demonstrate the capabilities of Prolog and Constraint Logic Programming (CLP), which are logic based programming languages for symbolic manipulation, as tools for exploring the structures of graphical models. A graph G (DAG or chain graph) specifies a class of probability distributions. In these models, nodes represent random variables. A probability distribution that satisfies the conditions derived from the graph is referred to as a G-Markov distribution. The correspondence between the graph and the class of distributions might not be unique.; there can be different graph structures that specify the same distribution class (Frydenberg, 1990; Spirtes et al., 1993/2001; MacCallum et al., 1993; Mayekawa, 1994; Andersson et al., 1997). Logic programming languages have functions for universal pattern matching and backtracking by indeterministic control mechanisms. Using these functions, we can easily classify and enumerate statistically equivalent classes. The programming details will be shown in the presentation.

**Exploring factors affecting parameter estimates in mixture confirmatory factor analysis**. (256)

*Ting Hsiang Lin*

Latent variable model (LVM) uses factor analysis to extract factors from data, and the model further explores the correlation and causal relationship of the latent variables. LVM is routinely applied in continuous variables, but the real data often contain categorical or ordinal variables. The recent developed mixture modeling collocating both continuous and categorical variables resolve the problem. We investigated a derivative of the mixture modeling, a mixture confirmatory factor analysis that includes a first-order factor analysis and a second-order latent class analysis. We conducted a simulation study and five effects were examined: sample size, factor loadings, number of first-order factors, correlation among first-order factors, and number of indicators to number of factors ratio (NI/NF ratio). We calculated bias and root mean square error, and conducted a multiple regression analysis to study the impact. The results showed when sample sizes, factor loadings, number of first-order factors, correlation among the factors, and NI/NF ratio increase, the bias and RMSE decrease. Using bias and RMSE as dependent variables and the five effects as independent variables in a regression analysis, we found sample size has the most significant effect on the bias and RMSE, while the number of first-order factors is the least significant.