Titles and Abstracts of Keynote Lectures

Peter M. Bentler(University of California,Los Angeles),Kevin H. Kim(University of Pittsburgh),and Ke-Hai Yuan (Notre Dame University)

Testing Homogeneity of Covariances with Infrequent Missing Data Patterns

ABSTRACT: When the data for factor analysis or structural equation modeling consists of many subsamples with possibly different characteristics, one may question whether all samples are from a single population with a common mean vector μ and common covariance matrix Σ. It may not make sense to entertain a single structural model in heterogeneous samples, although if only the means are not homogeneous, a single covariance structure would still be appropriate. This problem of homogeneity of means and covariances arises in the context of missing data, where different sets of subjects exhibit different patterns of the presence or absence of scores. When there are few patterns, the multiple group methodology of Allison (1987) and Muthén, Kaplan, and Hollis (1987) can be used to evaluate homogeneity of means and covariances. This methodology becomes impractical to implement when the number of patterns is large. Building on the work of Little (1988) and Tang and Bentler (1998), Kim and Bentler (2002) developed a generalized least squares (GLS) test of homogeneity of means (a minor variant of Little's test), evaluating whether various sample means z could be considered as samples from a common population with a common mean vector μ ; a GLS test of homogeneity of covariance matrices, testing whether various sample covariance matrices S could be considered as samples from a common population with a common covariance matrix Σ, and a combined GLS test of homogeneity of means and covariance matrices, that tests both hypotheses simultaneously. In limited sampling studies, they found these tests to perform reasonably well.
With multivariate normal data, these tests are tests of missing completely at random (MCAR). We study the test of homogeneity of covariance matrices under an extreme MCAR condition. We find that its performance can be disrupted when there are a large number of data patterns that are exhibited by only one or very few subjects. We study this issue theoretically and practically. Theoretically, we find that sample covariance matrices for such patterns may not be defined or may be very badly estimated, and that asymptotic statistical theory can break down. Practically, we find that deleting patterns with one or few observations can lead to substantially improved performance of the test statistic.

Jos M.F.ten Berge(University of Groningen, The Netherlands)

Factor analysis, reliability and unidimensionality.

ABSTRACT: For many decades, factor analysis has upheld intimate relationships with classical test theory. It is not just that practitioners use factor analysis to determine scales which are required to be reliable. The very mathematical model underlying factor analysis has the same structure as classical test theory. Both in factor analysis and in test theory, a convex set of "feasible tautologies" has been defined. In factor analysis, the set contains all nonnegative diagonal matrices Ψ of unique variances which entail a reduced covariance matrix Σ-Ψ that has no negative eigenvalues. In reliability theory, the set contains all nonnegative diagonal matrices of error variances which entail a true score covariance matrix with no negative eigenvalues. Each point in the set now defines a possible error variance matrix, and, by implication, a possible value of the reliability. In this paper, I will first discuss the set of tautologies from a reliability point of view, by reviewing the "greatest lower bound" (glb) to reliability (Jackson & Agunwamba, 1977). Next, the set will be considered from a factor analysis point of view. Minimum Trace Factor Analysis (Bentler, 1972; Shapiro, 1982) and Minimum Rank Factor Analysis (Ten Berge & Kiers, 1991) will be reviewed. The latter method minimizes, for a fixed number of factors r, the sum of the m-r smallest eigenvalues. A new type of least squares factor analysis, which is also in the set of feasible tautologies, is presented. It minimizes the sum of squares of the m-r smallest eigenvalues. Finally, the concepts of test unidimensionality and test reliability will be discussed in terms of the set of feasible tautologies. It will be argued that the glb is to be preferred to reliability measures based on a one-factor hypothesis.

Aapo Hyvarinen (University of Helsinki)

Why would factors or components be non-normal and square-correlated?

ABSTRACT:In independent component analysis, the components or factors are assumed to be non-normal (non-Gaussian) and independent. In recent extensions of independent component analysis, the components are assumed to be dependent in the sense of having correlations of squares. The assumptions of non-normality and square-correlation are rather different from what is usually assumed in factor analysis and related methods. Here, I review a collection of related results that may explain why many data sets would exhibit non-normality in contrast to the central limit theorem. This framework is based on assuming that the data comes from mixtures of gaussians, in particular scale mixtures. What is remarkable is that the same framework easily leads to square-correlations, and in time-series, to an ARCH-like dependency over time.

Sik-Yum Lee and Xin-Yuan Song (The Chinese University of Hong Kong)

Maximum Likelihood Analysis of Structural Equation Models with some types of Non-normal data.

ABSTRACT: We consider maximum likelihood (ML) analysis of non-normal data in the framework of structural equation modeling. The non-normality under consideration is due to the discreteness of the data, which are ordered categorical, and the non-linearity of the latent variables. To improve the applicability for practical problems, the model framework can accommodate two-level data that can be missing at random (MAR).
Based on the idea of data augmentation, ML estimate of the unknown parameters in the model is obtained by the Monte Carlo EM algorithm. The E-step is approximated by a sufficiently large sample of observations that are simulated from the appropriate conditional distribution via the Gibbs sampler together with the Metropolis-Hastings algorithm. Convergence of the algorithm is monitored by the method that is given in Shi ad Copas (2001), and standard errors are computing on the basis of the Louis formulae. The Bayesian information criterion (BIC) will be used for model comparison. A computational procedure that is based on path sampling is utilized to evaluate the observed data likelihood value in the BIC.
An illustrative example is provided. The data were obtained from the "Accelerated Schools for Quality Education Project", that was conducted by the Faculty of Education and Hong Kong Institute of Educational Research, The Chinese University of Hong Kong. The proposed model is used to assess the causal relations of "the school value inventory" and "teachers' empowerment" to teachers' "job satisfaction".

Ab Mooijaart(Leiden University)

Factor Analysis by Using Higher Order Moments

ABSTRACT: In Factor Analysis (FA) mostly covariances or correlations are fitted only. These coefficients are called second order mixed moments because they deal with cross-products of two variables. An obvious explanation for using second order moments only is that under the assumption of normality of the observed variables all the information about the data is contained in the first and second order moments. Mostly the first order moments (the means) are not fitted because they are supposed to be zero. However, if the observed variables are not normally distributed higher order moments (like cross-products of three variables) contain additional information. For instance, the skewnesses of the variables contain also information about the underlying structure. In this paper we extend the common formulation of FA in terms of second order moments to higher order moments. Both the one and more factor model will be discussed. A central consequence of our method is that in the case of FA with more than one factor there is a unique solution for the parameters, i.e. the so-called rotation problem in FA can be solved uniquely. An extension is that the relation between the latent and observed variables may be nonlinear. Two models will be discussed in particular: the polynomial factor model and a factor model with an interaction factor. The emphasis will be on models which use higher order moments for estimating the parameters. The parameters will be estimated by a GLS estimation procedure. The results of a small Monte Carlo study will be given and an example of a real data set will be discussed.

Albert Satorra(Universitat Pompeu Fabra. Barcelona), Willem E. Saris(University of Amsterdam. Holland)

Multitrait Multimethod Models with Data Incomplete by Design

ABSTRACT: In 1959 Campbell and Fiske suggested the multitrait�@ultimethod (MTMM) design for evaluating the validity of measurement instruments. Since then, several factor analysis models have been proposed for MTMM designs. Among them is the confirmatory factor analysis model for MTMM data (Althauser, Herberlein, and Scott 1971; Alwin 1974; Werts and Linn 1970). An alternative parameterization of this model proposed by Saris and Andrews (1991) is known as the true score (TS) model, while the correlated uniqueness model was put forward by Kenny (1976), Marsh (1989), and Marsh and Bailey (1991). Rather different models with what are called multiplicative method effects were suggested by Campbell and Connell (1967), Browne (1984), and Cudeck (1988).
Although the MTMM approach is accepted as a useful tool and is widely used, much attention has been given to its frequent problems of nonconvergence, underidentification or improper solutions for the confirmatory factor analysis model (Andrews 1984; Bagozzi and Yi 1991; Brannick and Spector 1990; Kenny and Kashy 1992; Marsh and Bailey 1991; Saris 1990). Grayson and Marsh (1994) showed that confirmatory factor analysis models with correlated method factors are usually underidentified, which may explain why these problems occur. Eid (2000) discussed these problems again and suggested an alternative model with one factor fewer than usual. Conversely, models with correlated traits uncorrelated methods (CTUM), which should not have the same problem, exist. This solution was also suggested by Andrews (1984) and Saris (1990). A recent study confirmed that a model equivalent to the CTUM model does indeed suffer from few problems (Corten et al.2002).
A more severe drawback of the standard MTMM approach is that at least three methods must be included to prevent even more severe problems of empirical underidentification (Kenny 1976); that is, every respondent is confronted with questions on the same trait three times. This poses quite a high burden for the respondent, and may also introduce memory effects that distort the validity of the model.
We believe this problem of three repeated measures threatens the MTMM approach more seriously than the technical problems of nonconvergence and improper solutions. Therefore, in recent work, Saris, Satorra and Coenders (Sociological Methodology, in press) propose new designs for MTMM studies that reduce the response burden by exploiting the feature of multiple group structural equation modeling (SEM). They propose to use multiple group data with different combinations of traits and methods being asked across groups. Parameter of the full MTMM model are not identified by the data of any single group, but the combination of all the groups in the same analysis yields identification of the model.
In the mentioned paper, it is shown how the new design enables researchers to evaluate measurement reliability and validity by means of the MTMM design, while reducing critically the response burden. New issues however arise from the use of MTMM design in the context of multiple group data, where the manifest variables in each group vary by design.
In the present paper we give the theory of estimation and testing for a general type of MTMM model design (that differs slightly from the one of Saris, Satorra and Coenders ) for multiple group data for multiple group data with incomplete data by design. Issues of identification as well of inference for non-normal data are developed. Efficiency of the different type of designs (the two or three group design, versus single group design ) is also discussed.

Kazuo Shigemasu(The University of Tokyo), Tkakahiro Hoshino(The Institute of Statistical Mathematics)

Improvements of LVM techniques by Bayesian Hierarchical Modeling

ABSTRACT: From the Bayesian viewpoint, the Latent Variables Modeling seems like "extending conversation"(lindley,1971). While usual statistical Model depicts the relationship between the observed variables and structural parameters, LVM introduces the latent variables like "extending conversation" and combines the observations and parameters. What is the utility of this "redundancy"?This approach makes it possible to handle the individual differences and various types of observed data by the unified treatments and we believe that the Bayesian hierarchical analysis is the most suitable methodology to derive the advantage of this model. We focus on the following two topics.
(1)Treatment of Pair Comparison Data: Inferences of latent variables and parameters based on pair comparison data can be done directly without introducing additional parameters.
(2)Direct evaluation of the parameters in the structural equation model:Probabilistic evaluation about parameters or their function is done by using Gibbs Sampling. Gibbs sampling among many MCMC techniques is easier to see the structure of the program and more flexible to handle the complex models but a drawback is that it is more time consuming and need more memory. To avoid it, we developed the Gibbs sampling techniques based on "sufficient statistics" of latent variables.

Haruo Yanai(National Center for University Entrance Examinations,Tokyo,Japan)

Matrix�@Methods�@and its Relationships with Factor Analysis

ABSTRACT: Since the advent of the Spearman�fs two factor model in 1904, a number of references on factor analysis theories have been published over the last 100 years. In accordance with the development of factor analysis theory, a num- ber of matrix methods in relation with factor analysis have been developed as well. In this paper, the author intends to relate the matrix methods de- veloped in the 20th century to some important topics of factor analysis such as identifiability condition of factor analysis, communality problem, image factor analysis,factor rotation and also estimation of factor scores ,and try to extend some of the earlier theories of factor analysis . In particular, the author emphasizes the uses of generalized inverse and projection matrices which never fail to enable us to cope with some complicate theories of factor analysis quite easily.

Ke-Hai Yuan (University of Notre Dame) and Peter M. Bentler (University of California, Los Angeles)

Mean Comparison: Manifest Variable versus Latent Variable

ABSTRACT: Mean comparisons are of great importance in the application of statistics. Procedures for mean comparison with manifest variables have been well studied. However, few rigorous studies have been conducted on mean comparisons with latent variables, although the methodology has been widely used and documented. This paper studies the commonly used statistics in latent variable mean modeling and compares them with parallel manifest variable statistics in terms of power, asymptotic distributions, and empirical distributions. The robustness property of each statistic is also explored when the model is misspecified or when data are nonnormally distributed. Our results indicate that, under certain conditions, the likelihood ratio and Wald statistics used for latent mean comparisons do not always have greater power than the Hotelling T² statistics used for manifest mean comparisons. The noncentrality parameter corresponding to the T² statistic can be much greater than those corresponding to the likelihood ratio and Wald statistics, which we find to be different from those provided in the literature. Our results also indicate that the likelihood ratio statistic can be stochastically much greater than the corresponding Wald statistic, and neither of their distributions can be described by a chi-square distribution when the null hypothesis is not trivially violated. Recommendations and advice are provided for the use of each statistic.