Owing to the nature of the problems and the design of questionnaires, discrete polytomous data are very common in behavioural, medical and social research. Analysing the relationships between the manifest and the latent variables based on mixed polytomous and continuous data has proven to be dif®cult. A general structural equation model is investigated for these mixed outcomes. Maximum likelihood (ML) estimates of the unknown thresholds and the structural parameters in the covariance structure are obtained. A Monte Carlo±EM algorithm is implemented to produce the ML estimates. It is shown that closed form solutions can be obtained for the M-step, and estimates of the latent variables are produced as a by-product of the analysis. The method is illustrated with a real example.
Factor analysis is an important technique in behavioural science research in explaining the interdependence and assessing causations and correlations of the observed variables and the latent factors. Recently, generalization of the model to handle polytomous variables has received a lot of attention. In this paper, a Bayesian approach to analysing the model with continuous and polytomous variables is developed. In the posterior analysis, the observed continuous and polytomous data are augmented with the latent factor scores and the unobserved measurements underlying the polytomous variables. Random observations from the posterior distributions are simulated via the Gibbs sampler algorithm. It is shown that the conditional distributions involved in the implementation of the algorithm are the familiar distributions, hence the simulation is rather straightforward. Joint Bayesian estimates of the unknown thresholds, structural parameters and the factor scores are produced simultaneously. The efficiency and accuracy of our approach are demonstrated by a real‐life example and a simulation study.
Shi et al. (2006) proposed a Gaussian process functional regression (GPFR)model to model functional response curves with a set of functional covariates.Two main problems are addressed by this method: modelling nonlinear and nonparametric regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very good results for curve fitting and prediction but side-steps the problem of heterogeneity. In this paper we present a new method for modelling functional data with 'spatially' indexed data, i.e., the heterogeneity is dependent on factors such as region and individual patient's information. For data collected from different sources, we assume that the data corresponding to each curve (or batch) follows a Gaussian process functional regression model as a lower-level model, and introduce an allocation model * Address for correspondence: School of Mathematics and Statistics, University of Newcastle, NE1 7RU, UK. Email: j.q.shi@ncl.ac.uk 1 for the latent indicator variables as a higher-level model. This higher-level model is dependent on the information related to each batch. This method takes advantage of both GPFR and mixture models and therefore improves the accuracy of predictions. The mixture model has also been used for curve clustering, but focusing on the problem of clustering functional relationships between response curve and covariates. The model is examined on simulated data and real data.
Two-level data with hierarchical structure and mixed continuous and polytomous data are very common in biomedical research. In this article, we propose a maximum likelihood approach for analyzing a latent variable model with these data. The maximum likelihood estimates are obtained by a Monte Carlo EM algorithm that involves the Gibbs sampler for approximating the E-step and the M-step and the bridge sampling for monitoring the convergence. The approach is illustrated by a two-level data set concerning the development and preliminary findings from an AIDS preventative intervention for Filipina commercial sex workers where the relationship between some latent quantities is investigated.
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