Showing papers on "Latent variable model published in 2001"
TL;DR: The book’s appeal is that it illustrates a wide range of results for many kinds of models that appear in stochastic processes and time series literature and provides general insights on the nonlinear model classes that have been discussed in the modeling and control literature.
Abstract: (2001). Latent Variable Models and Factor Analysis. Technometrics: Vol. 43, No. 1, pp. 111-111.
714 citations
01 Mar 2001
TL;DR: The authors discusses models with latent variables that are continuous and/or categorical, and gives an overview of modeling issues related to crosssectional analysis using latent class models, modeling of longitudinal data using Latent Class Models, and modeling a combination of continuous and categorical latent variables (growth mixture models).
Abstract: This chapter discusses models with latent variables that are continuous and/
or categorical. It also gives an overview of modeling issues related to crosssectional analysis using latent class models, modeling of longitudinal data using
latent class models, and modeling of longitudinal data using a combination of
continuous and categorical latent variables (growth mixture models). A series
of examples are presented. The analyses are carried out within a general latent
variable modeling f\work shown in the appendix using the Mplus program
(Muthen & Muthen, 1998). Mplus input specifications for these analyses can be
obtained from www.statmodel.com. To introduce the analyses, a brief overview
of modeling ideas is presented in Figs. 1.1 to 1.3.
634 citations
TL;DR: The results confirm the 3-dimensional model for 12-month prevalence of mental disorders and underline the argument for focusing on core psychopathological processes rather than on their manifestation as distinguished disorders in future population studies on common mental disorders.
Abstract: Background We analyzed the underlying latent structure of 12-month DSM-III-R diagnoses of 9 common disorders for the general population in the Netherlands. In addition, we sought to establish (1) the stability of the latent structure underlying mental disorders across a 1-year period (structural stability) and (2) the stability of individual differences in mental disorders at the level of the latent dimensions (differential stability). Methods Data were obtained from the first and second measurement of the Netherlands Mental Health Survey and Incidence Study (NEMESIS) (response rate at baseline: 69.7%, n = 7076; 1 year later, 79.4%, n = 5618). Nine common DSM-III-R diagnoses were assessed twice with the Composite International Diagnostic Interview with a time lapse of 1 year. Using structural equation modeling, the number of latent dimensions underlying these diagnoses was determined, and the structural and differential stability were assessed. Results A 3-dimensional model was established as having the best fit: a first dimension underlying substance use disorders (alcohol dependence, drug dependence); a second dimension for mood disorders (major depression, dysthymia), including generalized anxiety disorder; and a third dimension underlying anxiety disorders (simple phobia, social phobia, agoraphobia, and panic disorder). The structural stability of this model during a 1-year period was substantial, and the differential stability of the 3 latent dimensions was considerable. Conclusions Our results confirm the 3-dimensional model for 12-month prevalence of mental disorders. Results underline the argument for focusing on core psychopathological processes rather than on their manifestation as distinguished disorders in future population studies on common mental disorders.
536 citations
01 Jan 2001
TL;DR: These latent difference score analyses were previously presented at the International Society for Behavioral Development, Bern, Switzerland, in July 1998, and at the American Psychological Association conference "New Methods for the of Change," Pennsylvania State University, in October 1998 as discussed by the authors.
Abstract: These latent difference score analyses were previously presented to the International Society for Behavioral Development, Bern, Switzerland, in July 1998, and at the Amer ican Psychological Association conference "New Methods for the of Change," Pennsylvania State University, in October 1998. This research was supported by Grants AG02695, AG04704, and AG07137 from the National Institute on Aging. These Na tional Longitudinal Survey of Youth (NLSY) data were selected by Patrick Curran of Duke University for a presentation on comparative longitudinal analyses at the for Research on Child Development in April 1997. All NLSY data used here are and the computer program sCripts used here are available from John ]. McArdle, so all analyses should be relatively easy to reproduce from the available files or from the original data. We thank our colleagues John R. Nesselroade, Paolo Ghisletta, and Patrick CUrran for their comments on drafts of this chapter.
490 citations
TL;DR: Joint Bayesian estimation of all latent variables, model parameters, and parameters that determine the probability law of the latent process is carried out by a new MCMC method called permutation sampling.
Abstract: Bayesian estimation of a very general model class, where the distribution of the observations depends on a latent process taking values in a discrete state space, is discussed in this article. This model class covers finite mixture modeling, Markov switching autoregressive modeling, and dynamic linear models with switching. The consequences the unidentifiability of this type of model has on Markov chain Monte Carlo (MCMC) estimation are explicitly dealt with. Joint Bayesian estimation of all latent variables, model parameters, and parameters that determine the probability law of the latent process is carried out by a new MCMC method called permutation sampling. The permutation sampler first samples from the unconstrained posterior–which often can be done in a convenient multimove manner–and then applies a permutation of the current labeling of the states of the latent process. In a first run, the random permutation sampler used selected the permutation randomly. The MCMC output of the random permutation s...
473 citations
01 Jan 2001
458 citations
TL;DR: In this article, the authors investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables and show that the conventional approach performs very badly.
Abstract: Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.
339 citations
TL;DR: It is found that the joint model-parameter space search methods perform adequately but can be difficult to program and tune, whereas the marginal likelihood methods often are less troublesome and require less additional coding.
Abstract: The problem of calculating posterior probabilities for a collection of competing models and associated Bayes factors continues to be a formidable challenge for applied Bayesian statisticians. Current approaches that take advantage of modern Markov chain Monte Carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model, others that sample over the model space alone, and still others that attempt to estimate the marginal likelihood of each model directly (because the collection of these is equivalent to the collection of model probabilities themselves). We review several methods and compare them in the context of three examples: a simple regression example, a more challenging hierarchical longitudinal model, and a binary data latent variable model. We find that the joint model-parameter space search methods perform adequately but can be difficult to program and tune, whereas the marginal likelihood methods o...
290 citations
TL;DR: Analyses over several data sets suggest that LC factor models typically fit data better and provide results that are easier to interpret than the corresponding LC cluster models.
Abstract: We propose an alternative method of conducting exploratory latent class analysis that utilizes latent class factor models, and compare it to the more traditional approach based on latent class cluster models. We show that when formulated in terms of R mutually independent, dichotomous latent factors, the LC factor model has the same number of distinct parameters as an LC cluster model with R+1 clusters. Analyses over several data sets suggest that LC factor models typically fit data better and provide results that are easier to interpret than the corresponding LC cluster models. We also introduce a new graphical bi-plot display for LC factor models and compare it to similar plots used in correspondence analysis and to a barycentric coordinate display for LC cluster models. New results on identification of LC models are also presented. We conclude by describing various model extensions and an approach for eliminating boundary solutions in identified and unidentified LC models, which we have implemented in a new computer program.
290 citations
TL;DR: In this paper, the authors present standardized effect size measures for latent mean differences inferred from both structured means modeling and MIMIC approaches to hypothesis testing about differences among means on a single latent construct, which are then related to post hoc power analysis, a priori sample size determination, and a relevant measure of construct reliability.
Abstract: While effect size estimates, post hoc power estimates, and a priori sample size determination are becoming a routine part of univariate analyses involving measured variables (e.g., ANOVA), such measures and methods have not been articulated for analyses involving latent means. The current article presents standardized effect size measures for latent mean differences inferred from both structured means modeling and MIMIC approaches to hypothesis testing about differences among means on a single latent construct. These measures are then related to post hoc power analysis, a priori sample size determination, and a relevant measure of construct reliability.
260 citations
TL;DR: In this article, an alternative procedure is presented that solves the convergence problem of the Joreskog-Yang procedure and provides consistent estimators of the parameters of the Kenny-Judd interaction model.
Abstract: Kenny and Judd (1984) developed a latent variable interaction model for observed variables centered around their population means. They estimated the model by using a covariance matrix calculated from sample-mean-centered variables and products of these variables. Subsequently,Joreskog and Yang (1996) identified the need to include intercepts for the measurement and structural equations and estimated the model by using a covariance matrix calculated from noncentered observed variables and products of these variables, and means of the observed variables and the products of noncentered variables. Evidence is presented that the Joreskog-Yang procedure for estimating the Kenny-Judd interaction model is subject to severe convergence problems when implemented in LISREL8.3 and means for the indicators of the latent exogenous variables are nonzero. An alternative procedure is presented that solves the convergence problem and provides consistent estimators of the parameters.
TL;DR: In this article, the authors use higher order factor models to model longitudinal change directly in a latent construct, where the construct of interest is assumed to be indicated by several measured variables, all of which are observed at the same multiple time points.
Abstract: Methods of latent curve analysis (latent growth modeling) have recently emerged as a versatile tool for investigating longitudinal change in measured variables. This article, using higher order factor models as suggested by McArdle (1988) and Tisak and Meredith (1990), illustrates latent curve analysis for the purpose of modeling longitudinal change directly in a latent construct. The construct of interest is assumed to be indicated by several measured variables, all of which are observed at the same multiple time points. Examples with simultaneous estimation of covariance and mean structures are provided for both a single group and a two-group scenario.
TL;DR: In this paper, a general class of ordinal logit models that specify equality and inequality constraints on sums of conditional response probabilities are presented, and models are obtained that are similar to parametric and nonparametric item response models.
Abstract: A general class of ordinal logit models is presented that specifies equality and inequality constraints on sums of conditional response probabilities. Using these constraints in latent class analysis, models are obtained that are similar to parametric and nonparametric item response models. Maximum likelihood is used to estimate these models, making their assumptions testable with likelihood-ratio statistics. Because of the intractability of the asymptotic distribution of the goodness-of-fit measure when imposing inequality constraints, parametric bootstrapping is used to obtain estimates of p values. The proposed restricted latent class models are illustrated by an example using reported adult crying behavior.
TL;DR: It is argued here that factor analysis does not need to be restricted to linearity and that nonlinear factor analysis can be formulated and carried out as a useful statistical method.
Abstract: Factor analysis and its extensions are widely used in the social and behavioral sciences, and can be considered useful tools for exploration and model fitting in multivariate analysis. Despite its popu- larity in applications, factor analysis has attracted rather limited atten- tion from statisticians. Three issues, identification ambiguity, heavy reliance on normality, and limitation to linearity, may have contributed to statisticians' lack of interest in factor analysis. In this paper, the sta- tistical contributions to the first two issues are reviewed, and the third issue is addressed in detail. Linear models can be unrealistic even as an approximation in many applications, and often do not fit the data well without increasing the number of factors beyond the level explainable by the subject-matter theory. As an exploratory model, the conventional factor analysis model fails to address nonlinear structure underlying multivariate data. It is argued here that factor analysis does not need to be restricted to linearity and that nonlinear factor analysis can be formulated and carried out as a useful statistical method. In particular, for a general parametric nonlinear factor analysis model, the errors- in-variables parameterization is suggested as a sensible way to formu- late the model, and two procedures for model fitting are introduced and described. Tests for the goodness-of-fit of the model are also proposed. The procedures are studied through a simulation study. An example from personality testing is used to illustrate the issues and the methods.
TL;DR: In this paper, the authors developed a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family, giving rise to a number of factor models not considered previously and enabling the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases.
Abstract: We develop a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family. This gives rise to a number of factor models not considered previously and enables the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases. The framework accommodates a great variety of different measurement scales and accommodates cases where different latent variables have different distributions. The models are estimated with the method of simulated likelihood, which allows for higher dimensional factor solutions to be estimated than heretofore. The models are illustrated on synthetic data. We investigate their performance when the distribution of the latent variables is mis-specified and when part of the observations are missing. We study the properties of the simulation estimators relative to maximum likelihood estimation with numerical integration. We provide an empirical application to the analysis of attitudes.
TL;DR: In this paper, a maximum likelihood approach for analyzing a latent variable model with two-level data is proposed, which involves the Gibbs sampler for approximating the E-step and the M-step, and the bridge sampling for monitoring the convergence.
Abstract: Summary. 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.
01 Jan 2001
TL;DR: Confirmatory Factor Analysis (CFA) is a mainly dis-confirmatory quantitative data analysis method that belongs to the family of structural equation modeling (SEM) techniques.
Abstract: Confirmatory factor analysis (CFA) is a mainly dis-confirmatory quantitative data analysis method that belongs to the family of structural equation modeling (SEM) techniques CFA allows for the assessment of fit between observed data and an a prioriconceptualized, theoretically grounded model that specifies the hypothesized causal relations between latent factors and their observed indicator variables In this article, typical steps in a CFA are introduced First, during model specification, a model is conceptualized by indicating how latent, unobserved factors relate to measurable variables Second, if each parameter can be expressed as a function of the variances and covariances of observed variables, model identification is assured and parameters can be estimated Third, iterative techniques such as the maximum likelihood, generalized least squares, or asymptotically distribution free estimation methods can be utilized to estimate the unknown model parameters Fourth, assessments of fit between observed data and the a priori specified model(s) can be made via a multitude of absolute, parsimonious, and incremental fit indices Fifth, if data-model inconsistencies are observed, model modifications might be appropriate, provided they are consistent with underlying substantive theories and the modified model is cross-validated using an independent sample The article closes with applied and methodological references appropriate for a more in-depth study of CFA and SEM in the social and behavioral sciences
TL;DR: A Bayesian framework for estimating finite mixtures of the LISREL model is proposed, to augment the observed data of the manifest variables with the latent variables and the allocation variables and to obtain the Bayesian solution.
Abstract: In this paper, we propose a Bayesian framework for estimating finite mixtures of the LISREL model. The basic idea in our analysis is to augment the observed data of the manifest variables with the latent variables and the allocation variables. The Gibbs sampler is implemented to obtain the Bayesian solution. Other associated statistical inferences, such as the direct estimation of the latent variables, establishment of a goodness-of-fit assessment for a posited model, Bayesian classification, residual and outlier analyses, are discussed. The methodology is illustrated with a simulation study and a real example.
TL;DR: This work used nine traits' FA as markers of a latent variable of men's developmental instability, which was associated with the number of sexual partners and indicated a sizeable correlation between developmental instability and men's sexual history.
Abstract: A single trait's fluctuating asymmetry (FA) is expected to be a poor measure of developmental instability. Hence, studies that examine associations between FA and outcomes expected to covary with developmental instability often have little power in detecting meaningful relationships. One way of increasing the power of detecting relationships between developmental instability and outcomes is through the use of multiple traits' FA. The way multiple traits have typically been used is in trait aggregates. Here, we illustrate another way of examining relationships with developmental instability using multiple traits' FA: through structural equation modelling. Covariances between measures of FA and an outcome variable are interpreted within the context of an explicit model of associations between variables, which is tested for fit and the parameters specified within the model are estimated. We used nine traits' FA as markers of a latent variable of men's developmental instability, which was associated with the number of sexual partners. The results indicate a sizeable correlation between developmental instability and men's sexual history, despite small correlations between individual traits' FA and sexual history.
TL;DR: In this paper, the impact of three method factors (positive affectivity, negative affectivity and impression management) on substantive relations among work attitudes were examined using latent variable models, and the substantive relations were constituted by direct and indirect effects (through organizational commitment) from job satisfaction and perceived organizational support to intent to quit.
Abstract: Positive affectivity (PA), negative affectivity (NA), and impression management (IM), which have been commonly asserted to be method factors that artifactually inflate relations among self-reports of work attitudes, were simultaneously examined using latent variable models. The substantive relations among work attitudes were constituted by direct and indirect effects (through organizational commitment) from job satisfaction and perceived organizational support to intent to quit. Results showed a strong and negative latent correlation between NA and IM but only a weak and positive latent correlation between NA and PA. PA and IM were not correlated. PA had significant and substantial method-effects loadings on measures of work attitudes, NA had no significant method-effects loadings, and IM had significant method-effects loadings only on intent to quit. Latent variable model comparisons that provide direct tests for the impact of these 3 method effects on estimation of substantive relations among work attit...
TL;DR: In this article, the assumption of measurement invariance across occasions yields three-mode models that are suited for the analysis of multivariate longitudinal data, including autoregressive models and latent curve models as special cases.
Abstract: Multivariate longitudinal data are characterized by three modes: variables, occasions and subjects. Three-mode models are described as special cases of a linear latent variable model. The assumption of measurement invariance across occasions yields three-mode models that are suited for the analysis of multivariate longitudinal data. These so-called longitudinal three-mode models include autoregressive models and latent curve models as special cases. Empirical data from the field of industrial psychology are used in an example of how to test substantive hypotheses with the longitudinal, autoregressive and latent curve three-mode models.
TL;DR: In this paper, it was shown that the relative size of the regression coefficients from OLS and PLS is a function not only of the loadings but also of the error variances for the predictor variables.
Abstract: In standard linear regression where the predictor matrix X is of full rank, the regression coefficients are clearly defined as the parameters B appearing in the linear regression model. In latent variable models there is no direct relationship between the predictor variables and response variables. Rather they are both related to an underlying reduced-rank set of latent variables. Recent papers have proposed different methods for obtaining approximate covariance matrices for the estimates of the regression coefficients from methods such as partial least squares (PLS) and for using them to determine ‘confidence intervals’, for variable selection and for judging variable importance. However, in the latent variable model a matrix of regression coefficients, B, does not even appear as a parameter matrix. In the situation where the data follow such a model, it is therefore uncertain how the regression coefficients and, by extension, any covariance matrices and ‘confidence intervals’ should be interpreted. In this paper we show that any inference is critically dependent upon how one defines these regression coefficients. Two definitions for the regression coefficients are given that are consistent with the latent variable model. Which of these definitions is more relevant is shown to be highly dependent on the goals of the analysis. Therefore one must be clear on the definition one is using for these coefficients when building predictive models, when screening variables based on them or when using them to make interpretations about the system. Under standard normality assumptions, different estimation methods such as ordinary least squares (OLS) and PLS are shown to provide very different distributions for the regression coefficient estimates when the data follow a latent variable model. This is shown to be not just a matter of the PLS coefficients being biased or the OLS estimates having large variance, but of more complex differences implied by the structure of the model parameters in the latent variable model. How the distributions for these estimates relate to the definitions given in this paper is explored here. It is shown for a simple case that the relative size of the PLS estimates, on average, tends to reflect the latent variable loadings, whereas the relative size of the OLS estimates, on average, is a function not only of the loadings but also of the error variances for the predictor variables. Thus in this particular case it appears that the relative size of the B parameters from PLS reflects the underlying latent structure, whereas those from OLS also reflect the error structure of the predictor variables. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: It is concluded that the slider-bar and radiobutton user interfaces both yield similar latent structures of survey item responses, which replicated across both taxometric procedures and across multiple survey items.
Abstract: Although Web-based surveys are increasing in popularity, very little research has been conducted on the psychometric implications of using different user interfaces for eliciting responses to survey items. The purpose of this study was to compare the latent structures of responses to two different user interface response formats in a Web-based survey. Two different coherent cut kinetics taxometric procedures—MAMBAC and L-mode factor analysis—were used to examine the latent structure of responses to a survey of library service quality using an unnumbered slider-bar user interface versus a radiobutton user interface. Strong evidence was found for a pure dimensional latent structure of responses from both user interfaces, which replicated across both taxometric procedures and across multiple survey items. It is concluded that the slider-bar and radiobutton user interfaces both yield similar latent structures of survey item responses. The implications of these findings for the construction of Web-based survey...
01 Jan 2001
TL;DR: A Bayesian learning scheme using the variational paradigm to learn the parameters of the model, estimate the source densities, and together with Automatic Relevance Determination (ARD) to infer the number of latent dimensions is proposed.
Abstract: Independent Component Analysis (ICA) is an important tool for extracting structure from data. ICA is traditionally performed under a maximum likelihood scheme in a latent variable model and in the absence of noise. Although extensively utilised, maximum likelihood estimation has well known drawbacks such as overfitting and sensitivity to local-maxima. In this paper, we propose a Bayesian learning scheme using the variational paradigm to learn the parameters of the model, estimate the source densities, and together with Automatic Relevance Determination (ARD) to infer the number of latent dimensions. We illustrate our method by separating a noisy mixture of images, estimating the noise and correctly inferring the true number of sources.
TL;DR: Pertaining to PER and PhA, the results are in accordance with prior research showing a latent class structure and a base rate of about 12% for schizotypy, however, for MI, there was no evidence of a taxonic structure.
Abstract: The schizotypy model proposed by Meehl (1990) assumes a discontinuous distribution of schizophrenia liability. The "schizogene" is thought to determine if one is at risk for psychosis (i.e., whether one is a member of the taxon or its complement, which are considered to be the two latent classes). Using a German non-student sample (n = 809) we wanted to (1) replicate the results of prior research pertaining to the latent structure and base rate of schizotypy assessed by the Perceptual Aberration Scale (PER; Chapman, Chapman, & Raulin, 1978), and (2) investigate whether the same holds true for two other prominent psychometric indices, the Magical Ideation Scale (MI; Eckblad & Chapman, 1983) and the Physical Anhedonia Scale (PhA; Chapman et al., 1976), if one uses the same kind of analysis--the MAXCOV-HITMAX analysis based on subsets of items (Meehl, 1973). Pertaining to PER and PhA, our results are in accordance with prior research showing a latent class structure and a base rate of about 12% for schizotypy. However, for MI, there was no evidence of a taxonic structure. Possible reasons for MI's negative results are discussed as well as the role of the concept "anhedonia."
TL;DR: Evaluated latent structure of DSM-IV schizotypal personality disorder diagnostic criteria found oddness, aloofness, and social withdrawal, rather than positive symptoms, best characterized SPD even in clinical samples.
Abstract: The aim of the study was to evaluate the latent structure of DSM-IV schizotypal personality disorder (SPD) diagnostic criteria. The sample consisted of 564 consecutively admitted inpatients and outpatients. Exploratory latent class analysis identified a four-class model as the best fitting model for DSM-IV SPD criteria. The first of the SPD latent classes was mainly characterized by odd thinking, inappropriate affect, and interpersonal features; the second class by cognitive/perceptual difficulties; the third class by paranoid features; and the fourth class by absence of SPD features. The conditional probability pattern of the four-class solution could be safely replicated across confounder strata. Unlike previous findings, oddness, aloofness, and social withdrawal, rather than positive symptoms, best characterized SPD even in clinical samples.
01 Jan 2001
TL;DR: This work proposes and test a structure with a hierachical nonlinear model for variances and means and proposes a cost function which can be used for updating the variables as well as optimising the model structure.
Abstract: We introduce building blocks from which a large variety of latent variable models can be built. The blocks include continuous and discrete variables, summation, addition, nonlinearity and switching. Ensemble learning provides a cost function which can be used for updating the variables as well as optimising the model structure. The blocks are designed to fit together and to yield efficient update rules. Emphasis is on local computation which results in linear computational complexity. We propose and test a structure with a hierachical nonlinear model for variances and means.
TL;DR: The partial least-squares algorithm is derived probabilistically in terms of stochastic variables where sample estimates calculated using data matrices are employed at the end.
Abstract: Traditionally the partial least-squares (PLS) algorithm, commonly used in chemistry for ill-conditioned multivariate linear regression, has been derived (motivated) and presented in terms of data matrices. In this work the PLS algorithm is derived probabilistically in terms of stochastic variables where sample estimates calculated using data matrices are employed at the end. The derivation, which offers a probabilistic motivation to each step of the PLS algorithm, is performed for the general multiresponse case and without reference to any latent variable model of the response variable and also without any so-called “inner relation”. On the basis of the derivation, some theoretical issues of the PLS algorithm are briefly considered: the complexity of the original motivation of PLS regression which involves an “inner relation”; the original motivation behind the prediction stage of the PLS algorithm; the relationship between uncorrelated and orthogonal latent variables; the limited possibilities to make n...
TL;DR: A latent class model is presented that accounts for the structure in a set of correlated, categorical variables measured at discrete time periods, drawing information from these variables to form a smaller number of latent classes.
Abstract: In longitudinal behavioural studies, it is common to have multiple categorical indicators for measuring a theoretical construct of interest. A latent class model is presented that accounts for the structure in a set of correlated, categorical variables measured at discrete time periods, drawing information from these variables to form a smaller number of latent classes. The dependence of the resulting latent class model parameters on suspected factors over time is simultaneously modelled using a baseline-category logistic regression model. Estimation of the model parameters is achieved using an estimating equations procedure. A motivating example is provided from a longitudinal study of suspected linkages between monitoring or supervision by parents and the occurrence of drug use behaviours in an epidemiologic sample of school-attending youths.