Analysis of structural equation models with Incomplete Polytomous Data

A two-stage estimation procedure is developed to analyze structural equation models of polytomous variables based on incomplete data. At the first stage, the partition maximum likelihood approach is used to obtain the estimates of the elements in the correlation matrix. It will be shown that the asymptotic distribution of these estimates is jointly multivariate normal. The second stage estimates the structural parameters in the correlation matrix by the generalized least squared approach with a correctly specified weight matrix. Asymptotic properties of the second stage estimates are also provided. Extension of the theory to multisample models, and some illustrative examples are also included.     

A mixture latent variable model for modeling mixed data in heterogeneous populations and its applications

Latent variable models are widely used for jointly modeling of mixed data including nominal, ordinal, count and continuous data. In this paper, we consider a latent variable model for jointly modeling relationships between mixed binary, count and continuous variables with some observed covariates. We assume that, given a latent variable, mixed variables of interest are independent and count and continuous variables have Poisson distribution and normal distribution, respectively. As such data may be extracted from different subpopulations, consideration of an unobserved heterogeneity has to be taken into account. A mixture distribution is considered (for the distribution of the latent variable) which accounts the heterogeneity. The generalized EM algorithm which uses the Newton–Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. The standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. Analysis of the primary biliary cirrhosis data is presented as an application of the proposed model.     

Robustness of partial least-squares method for estimating latent variable quality structures

Latent variable structural models and the partial least-squares (PLS) estimation procedure have found increased interest since being used in the context of customer satisfaction measurement. The well-known property that the estimates of the inner structure model are inconsistent implies biased estimates for finite sample sizes. A simplified version of the structural model that is used for the Swedish Customer Satisfaction Index (SCSI) system has been used to generate simulated data and to study the PLS algorithm in the presence of three inadequacies: (i) skew instead of symmetric distributions for manifest variables; (ii) multi-collinearity within blocks of manifest and between latent variables; and (iii) misspecification of the structural model (omission of regressors). The simulation results show that the PLS method is quite robust against these inadequacies. The bias that is caused by the inconsistency of PLS estimates is substantially increased only for extremely skewed distributions and for the erroneous omission of a highly relevant latent regressor variable. The estimated scores of the latent variables are always in very good agreement with the true values and seem to be unaffected by the inadequacies under investigation.     

The partial least squares-fix point method of estimating interdependent systems with latent variables

This paper describes a method for estimating the unknown parameters of an interdependent simultaneous equations model with latent variables. For each latent variable there may be single or multiple indicators. Estimation proceeds in three stages: first, estimates of the latent variables are constructed from the associated manifest indicators; second, treating the estimates as directly observed, fix-point estimates of the structural form parameters are obtained; third, the location parameters are estimated. The method involves only repeated application of ordinary least squares and no distributional assumptions are needed. The paper concludes with an empirical application of the method.     

Latent Variable Models for Mixed Discrete and Continuous Outcomes

We propose a latent variable model for mixed discrete and continuous outcomes. The model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects, as well as direct modelling of covariates on the latent variable. An EM algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by-product of the analysis. A generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes. This method is applied to birth defects data, where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies. Infants who were exposed in utero to anticonvulsant medications are compared with controls.     

Log-linear modelling of data from matched case-control studies

A log-linear modelling approach is proposed for dealing with polytomous, unordered exposure variables in case-control epidemiological studies with matched pairs. Hypotheses concerning epidemiological parameters are shown to be expressable in terms of log-linear models for the expected frequencies of the case-by-control square concordance table representation of the matched data; relevant maximum likelihood estimates and goodness-of-fit statistics are presented. Possible extensions to account for ordered categorical risk factors and multiple controls are illustrated, and comparisons with previous work are discussed. Finally, the possibility of implementing the proposed method with GLIM is illustrated within the context of a data set already analyzed by other authors.     

Estimation in Truncated GLG Model for Ordered Categorical Spatial Data Using the SAEM Algorithm

In this article, we utilize a scale mixture of Gaussian random field as a tool for modeling spatial ordered categorical data with non-Gaussian latent variables. In fact, we assume a categorical random field is created by truncating a Gaussian Log-Gaussian latent variable model to accommodate heavy tails. Since the traditional likelihood approach for the considered model involves high-dimensional integrations which are computationally intensive, the maximum likelihood estimates are obtained using a stochastic approximation expectation–maximization algorithm. For this purpose, Markov chain Monte Carlo methods are employed to draw from the posterior distribution of latent variables. A numerical example illustrates the methodology.     

BAYESIAN PREDICTION FOR SPATIAL GENERALISED LINEAR MIXED MODELS WITH CLOSED SKEW NORMAL LATENT VARIABLES

Spatial generalised linear mixed models are used commonly for modelling non‐Gaussian discrete spatial responses. In these models, the spatial correlation structure of data is modelled by spatial latent variables. Most users are satisfied with using a normal distribution for these variables, but in many applications it is unclear whether or not the normal assumption holds. This assumption is relaxed in the present work, using a closed skew normal distribution for the spatial latent variables, which is more flexible and includes normal and skew normal distributions. The parameter estimates and spatial predictions are calculated using the Markov Chain Monte Carlo method. Finally, the performance of the proposed model is analysed via two simulation studies, followed by a case study in which practical aspects are dealt with. The proposed model appears to give a smaller cross‐validation mean square error of the spatial prediction than the normal prior in modelling the temperature data set.     

A comparison of three methods of estimating the logistic regression coefficients

A Sampling experiment performed using data collected for a large clinical trial shows that the discriminant function estimates of the logistic regression coefficients for discrete variables may be severely biased. The simulations show that the mixed variable location model coefficient estimates have bias which is of the same magnitude as the bias in the coefficient estimates obtained using conditional maximum likelihood estimates but require about one-tenth of the computer time.     

A comparison of bootstrap approaches for estimating uncertainty of parameters in linear mixed‐effects models

A version of the nonparametric bootstrap, which resamples the entire subjects from original data, called the case bootstrap, has been increasingly used for estimating uncertainty of parameters in mixed‐effects models. It is usually applied to obtain more robust estimates of the parameters and more realistic confidence intervals (CIs). Alternative bootstrap methods, such as residual bootstrap and parametric bootstrap that resample both random effects and residuals, have been proposed to better take into account the hierarchical structure of multi‐level and longitudinal data. However, few studies have been performed to compare these different approaches. In this study, we used simulation to evaluate bootstrap methods proposed for linear mixed‐effect models. We also compared the results obtained by maximum likelihood (ML) and restricted maximum likelihood (REML). Our simulation studies evidenced the good performance of the case bootstrap as well as the bootstraps of both random effects and residuals. On the other hand, the bootstrap methods that resample only the residuals and the bootstraps combining case and residuals performed poorly. REML and ML provided similar bootstrap estimates of uncertainty, but there was slightly more bias and poorer coverage rate for variance parameters with ML in the sparse design. We applied the proposed methods to a real dataset from a study investigating the natural evolution of Parkinson's disease and were able to confirm that the methods provide plausible estimates of uncertainty. Given that most real‐life datasets tend to exhibit heterogeneity in sampling schedules, the residual bootstraps would be expected to perform better than the case bootstrap. Copyright © 2013 John Wiley & Sons, Ltd.     

Maximum likelihood estimation of polychoric correlations in r×s ×t contingency tables

This paper discusses the maximum likelihood estimation of the polychoric correlation coefficient based on observed frequencies of three polytomous ordinal variables. The underlying latent variables are assumed to have a standardized trivariate normal distribution. The thresholds and correlations are estimated simultaneously via the scoring algorithm. Some practical applications of the method are discussed. An example is reported to illustrate the theory and some technical details are presented in the Appendix.     

Nonlinear mixed‐effects models with misspecified random‐effects distribution

Nonlinear mixed‐effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random‐effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random‐effects distribution in nonlinear mixed‐effects models. Our study is focused on Gauss‐Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random‐effects distribution in nonlinear mixed‐effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed‐effects parameters in nonlinear mixed‐effects models are generally robust to deviations from normality of the random‐effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random‐effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.     

Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

Longitudinal models for polytomous responses

Recent work by Miller and Landis (1991) discusses generalized variance component models for polytomous responses. This work is adapted to longitudinal models for repeated measures of individuals having polytomous responses. In this setting, individuals are considered to be “clusters”. The resulting simplifications are discussed. First, each response has a multinomial distribution with N=l. Second, observed cluster proportions in the variance component estimates must be replaced by their expectations. This technique accommodates patients with missing data in a sequence of repeated observations.     

Generalized Linear Latent Variable Modeling for Multi-Group Studies

ABSTRACT

Latent variable modeling is commonly used in behavioral, social, and medical science research. The models used in such analysis relate all observed variables to latent common factors. In many applications, the observations are highly non normal or discrete, e.g., polytomous responses or counts. The existing approaches for non normal observations can be considered lacking in several aspects, especially for multi-group samples situations. We propose a generalized linear model approach for multi-sample latent variable analysis that can handle a broad class of non normal and discrete observations, and that furnishes meaningful interpretation and inference in multi-group studies through maximum likelihood analysis. A Monte Carlo EM algorithm is proposed for parameter estimation. The convergence assessment and standard error estimation is addressed. Simulation studies are reported to show the usefulness of the our approach. An example from a substance abuse prevention study is also presented.     

Nonlinear structural equation modeling: is partial least squares an alternative?

Nonlinear structural equation modeling provides many advantages over analyses based on manifest variables only. Several approaches for the analysis of latent interaction effects have been developed within the last 15 years, including the partial least squares product indicator approach (PLS-PI), the constrained product indicator approach using the LISREL software (LISREL-PI), and the distribution-analytic latent moderated structural equations approach (LMS) using the Mplus program. An assumed advantage of PLS-PI is that it is able to deal with very large numbers of indicators, while LISREL-PI and LMS have not been investigated under such conditions. In a Monte Carlo study, the performance of LISREL-PI and LMS was compared to PLS-PI results previously reported in Chin et al. (2003) and Goodhue et al. (2007) for identical conditions. The latent interaction model included six indicator variables for the measurement of each latent predictor variable and the latent criterion, and sample size was N=100. The results showed that PLS-PI’s linear and interaction parameter estimates were downward biased, while parameter estimates were unbiased for LISREL-PI and LMS. True standard errors were smallest for PLS-PI, while the power to detect the latent interaction effect was higher for LISREL-PI and LMS. Compared to the symmetric distributions of interaction parameter estimates for LISREL-PI and LMS, PLS-PI showed a distribution that was symmetric for positive values, but included outlying negative estimates. Possible explanations for these findings are discussed.     

Weighting for item non-response in attitude scales by using latent variable models with covariates

We discuss the use of latent variable models with observed covariates for computing response propensities for sample respondents. A response propensity score is often used to weight item and unit responders to account for item and unit non-response and to obtain adjusted means and proportions. In the context of attitude scaling, we discuss computing response propensity scores by using latent variable models for binary or nominal polytomous manifest items with covariates. Our models allow the response propensity scores to be found for several different items without refitting. They allow any pattern of missing responses for the items. If one prefers, it is possible to estimate population proportions directly from the latent variable models, so avoiding the use of propensity scores. Artificial data sets and a real data set extracted from the 1996 British Social Attitudes Survey are used to compare the various methods proposed.     

A Framework for Skew-Probit Links in Binary Regression

We review several asymmetrical links for binary regression models and present a unified approach for two skew-probit links proposed in the literature. Moreover, under skew-probit link, conditions for the existence of the ML estimators and the posterior distribution under improper priors are established. The framework proposed here considers two sets of latent variables which are helpful to implement the Bayesian MCMC approach. A simulation study to criteria for models comparison is conducted and two applications are made. Using different Bayesian criteria we show that, for these data sets, the skew-probit links are better than alternative links proposed in the literature.     

The Co-Integrated Vector Autoregression with Errors–in–Variables

The co-integrated vector autoregression is extended to allow variables to be observed with classical measurement errors (ME). For estimation, the model is parametrized as a time invariant state-space form, and an accelerated expectation-maximization algorithm is derived. A simulation study shows that (i) the finite-sample properties of the maximum likelihood (ML) estimates and reduced rank test statistics are excellent (ii) neglected measurement errors will generally distort unit root inference due to a moving average component in the residuals, and (iii) the moving average component may–in principle–be approximated by a long autoregression, but a pure autoregression cannot identify the autoregressive structure of the latent process, and the adjustment coefficients are estimated with a substantial asymptotic bias. An application to the zero-coupon yield-curve is given.     

Latent Variable Modelling: A Survey*

Abstract. Latent variable modelling has gradually become an integral part of mainstream statistics and is currently used for a multitude of applications in different subject areas. Examples of ‘traditional’ latent variable models include latent class models, item–response models, common factor models, structural equation models, mixed or random effects models and covariate measurement error models. Although latent variables have widely different interpretations in different settings, the models have a very similar mathematical structure. This has been the impetus for the formulation of general modelling frameworks which accommodate a wide range of models. Recent developments include multilevel structural equation models with both continuous and discrete latent variables, multiprocess models and nonlinear latent variable models.