Mixture of latent trait analyzers for model-based clustering of categorical data

Flexible methods that relax restrictive conditional independence assumptions of latent classanalysis (LCA) are described. Dichotomous and ordered category manifest variables are viewed asdiscretized latent continuous variables. The latent continuous variables are assumed to have a mixtureofmultivariate-normals distribution. Within a latent class, conditional dependence is modeled as the mutual association of all or some latent continuous variables with a continuous latent trait (or in special cases, multiple latent traits). The relaxation of conditional independence assumptions allows LCA to better model natural taxa. Comparisons of specific restricted and unrestricted models permit statistical tests of specific aspects of latent taxonic structure. Latent class, latent trait, and latent distribution analysis can be viewed as special cases of the mixed latent trait model. The relationship between the multivariate probit mixture model proposed here and Rost’s mixed Rasch (1990, 1991) model is discussed. Two studies illustrate different uses of the proposed model.

The latent profile model is a latent variable model with a categorical latent variable and continuous manifest indicators. It was introduced in 1968 by Lazarsfeld and Henry. Although under different names, very similar models were proposed in the same period by Day and Wolfe. Over the past ten years, there has been renewed interest in this type of latent variable model, especially as a tool for cluster analysis. The latent profile model can be seen as a probabilistic or model-based variant of traditional non-hierarchical cluster analysis procedures such as the K-means method. It has been shown that such a model-based clustering procedure outperforms the more ad hoc traditional methods. It should be noted that only a few authors use the term latent profile model. More common names are mixture of normal components, mixture model clustering, model-based clustering, latent discriminant analysis, and latent class clustering. Possible social sciences applications of the latent profile model include building typologies and constructing diagnostic instruments. A sociologist might use it to build a typology of countries based on a set of socio-economic and political indicators. A psychologist may apply the method to combine various test scores into a single diagnostic instrument. As in latent class analysis, latent profile analysis assumes that the population consists of C unobserved subgroups that can be referred to as latent profiles, latent classes, or mixture components. Because the indicators are continuous variables, it is most natural to assume that their conditional distribution is normal. The most general model is obtained with unrestricted multivariate normal distributions; that is,

Often quantities of interest in psychology cannot be observed directly. These unobservable quantities, known as latent variables, tend to be complex, often multidimensional, constructs. In many cases these constructs are categorical, such that individuals belong to mutually exclusive and exhaustive unobservable subgroups. Latent class analysis (LCA) is a statistical approach to modeling a discrete latent variable using multiple, discrete observed variables as indicators. Examples of latent class variables that appear in the psychology literature include temperament type, substance use behavior, teaching style, stages of change in the transtheoretical model, and latent classes of risk. The first section of this chapter provides a conceptual introduction to the concept of a latent class followed by a technical introduction to the mathematical model, including multiple-groups LCA and LCA with covariates. This is followed by a discussion of parameter restrictions, model selection, and goodness-of-fit. The second section demonstrates LCA using the empirical example of depression subtypes in adolescence. Five latent classes were identified based on responses to eight questionnaire items assessing depression symptoms: Non-depressed (characterized by a low probability of reporting all eight depression symptoms), sad, disliked, sad + disliked, and depressed (characterized by a high probability of reporting all eight depression symptoms). The third section presents longitudinal extensions of the model, including repeated-measures LCA and latent transition analysis (LTA). The empirical example is extended to examine change in depression subtypes over time. The final sections describe recent extensions to the latent class model and areas that merit additional research in the future. Keywords: latent class analysis; latent transition analysis; latent variable model; categorical data

Chapter 33 - Graphical Display of Latent Budget Analysis and Latent Class Analysis, with Special Reference to Correspondence Analysis

Complications in the research into personality disorders may be rooted in the assumption within psychiatric diagnosis that underlying constructs are measured with equally valid observed items without rank or recognition of measurement error. The aim of this paper is to investigate the internal validity of DSM-IV (APA, 2000) paranoid personality disorder while accounting for measurement error and the continuous and categorical nature of the construct. General population data from the British Psychiatric Morbidity Survey (Singleton et al., 2001) was obtained from the Data Archives, University of Essex, England. Information from individuals with responses in the paranoid personality disorder section (n = 8393) of the Structured Clinical Interview for DSM-IV Axis II Disorders (SCID-II; First et al., 1997) screening questionnaire was analysed using confirmatory factor analysis (CFA), item response theory (IRT), latent class analysis (LCA) and latent class factor analysis (LCFA) mixture modelling. Results indicated that a one-factor model adequately represented the data, and that all items had reasonable factor loadings. However IRT analysis indicated that only four of the seven criteria discriminate well between individuals along different points of the underlying continuum. LCA and LCFA provided another perspective on the evaluation of paranoid personality disorder and indicated the presence of four underlying sub-populations. This is useful in terms of clinical and primary health settings as specific groups of interest can be investigated further in terms of characteristics, covariates and predictors.

It is known that how individuals perceive the situation result in choices. However, it is unknown, how drivers process the information of vehicle malfunctions based on the situation at hand. In this study, we only considered drivers' actions in cases where vehicles had some forms of malfunction at the time of the crash. Also due to endogeneity bias, or aggregation of the error term, we did not include the vehicle malfunction directly in our model. In addition, we used the latent variable (LV) for class allocation. Although the latent class model considering latent variable (LCLV) is not comparable with other considered models, by means of log likelihood, the results expanded our understanding regarding the indirect impact of an indicator of vehicle malfunction by means of latent variable on drivers' actions. Due to inclusion of latent variables in both, the indicator and the choice models, the relationship between those two models could be revealed based on the impacts of the latent variables. It was found that, for instance, the latent variable, which decreases the likelihood of driving vehicles with brake defect, at the same time increases taking driver action of speeding. Or female drivers are less likely to be involved in speeding driving actions before crashes due to various vehicles malfunctions. Driving under the influence of drug or alcohol or being under emotional conditions were some of the factors impacting various drivers actions. This is the earliest study implemented the technique in the context of traffic safety.

In many ways, Chap. 5 is an extension of the methods and techniques used to gather evidence based on the internal structure of the instrument and continues to describe the latent variable approach to instrument development. In this chapter, we introduce two common instrument development situations where discrete mathematical structures are encountered: latent class analysis (LCA) and item response theory (IRT). In LCA, both the latent variable and the indicator variable are best represented by a categorical structure. In IRT, the latent trait is continuous, but the items are the items are categorical. IRT has historically been used in educational achievement applications, however, it is gaining popularity in some affective characteristic measurement scenarios. The last section of the chapter is devoted to the topic of measurement invariance and the analytic techniques that allow instrument developers to explore whether the scale functions in the same manner (exhibits the same internal structure) across subgroups. If the items that reflect their latent constructs and the connections between the constructs operate in a fundamentally different way depending on group membership, cross group comparisons become difficult, if not impossible. This chapter discusses the most common statistical techniques (along with examples) for establishing invariance for both scenarios where the indicators are either continuous or categorical.

One of the challenges in clustering categorical data is the curse of dimensionality caused by the inherent sparsity of high-dimensional data, the records of which include a large number of attributes. The latent class model (LCM) assumes local independence between the variables in clusters, and is a parsimonious model-based clustering approach that has been used to circumvent the problem. The mixture of a log-linear model is more flexible but requires more parameters to be estimated. In this research, we recognize that each categorical observation can be conceived as a network with pairwise linked nodes, which are the response levels of the observation attributes. Therefore, the categorical data for clustering is considered a finite mixture of different component layer networks with distinct patterns. We apply a penalized composite likelihood approach to a finite mixture of networks for sparse multivariate categorical data to reduce the number of parameters, implement the EM algorithm to estimate the model parameters, and show that the estimates are consistent and satisfy asymptotic normality. The performance of the proposed approach is shown to be better in comparison with the conventional methods for both synthetic and real datasets.

Recurrent wheezing or asthma is a common problem in children that has increased considerably in prevalence in the past few decades. The causes and underlying mechanisms are poorly understood and it is thought that a numb er of distinct diseases causing similar symptoms are involved. Due to the lack of a biologically founded classification system, children are classified according to their observed disease related features (symptoms, signs, measurements) into phenotypes. The objectives of this PhD project were a) to develop tools for analysing phenotypic variation of a disease, and b) to examine phenotypic variability of wheezing among children by applying these tools to existing epidemiological data. A combination of graphical methods (multivariate co rrespondence analysis) and statistical models (latent variables models) was used. In a first phase, a model for discrete variability (latent class model) was applied to data on symptoms and measurements from an epidemiological study to identify distinct phenotypes of wheezing. In a second phase, the modelling framework was expanded to include continuous variability (e.g. along a severity gradient) and combinations of discrete and continuo us variability (factor models and factor mixture models). The third phase focused on validating the methods using simulation studies. The main body of this thesis consists of 5 articles (3 published, 1 submitted and 1 to be submitted) including applications, methodological contributions and a review. The main findings and contributions were: 1) The application of a latent class model to epidemiological data (symptoms and physiological measurements) yielded plausible pheno types of wheezing with distinguishing characteristics that have previously been used as phenotype defining characteristics. 2) A method was proposed for including responses to conditional questions (e.g. questions on severity or triggers of wheezing are asked only to children with wheeze) in multivariate modelling.ii 3) A panel of clinicians was set up to agree on a plausible model for wheezing diseases. The model can be used to generate datasets for testing the modelling approach. 4) A critical review of methods for defining and validating phenotypes of wheeze in children was conducted. 5) The simulation studies showed that a parsimonious parameterisation of the models is required to identify the true underlying structure of the data. The developed approach can deal with some challenges of real-life cohort data such as variables of mixed mode (continuous and categorical), missing data and conditional questions. If carefully applied, the approach can be used to identify whether the underlying phenotypic variation is discrete (classes), continuous (factors) or a combination of these. These methods could help improve precision of research into causes and mechanisms and contribute to the development of a new classification of wheezing disorders in children and other diseases which are difficult to classify.

Categorical data are quantified as either nominal variables--distinguishing different groups, for example, based on socio-economic status, education, and political persuasion--or ordinal variables--distinguishing levels of interest, such as the preferred politician or the preferred type of punishment for committing burglary. This new book is a collection of up-to-date studies on modern categorical data analysis methods, emphasizing their application to relevant and interesting data sets.This volume concentrates on latent class analysis and item response theory. These methods use latent variables to explain the relationships among observed categorical variables. Latent class analysis yields the classification of a group of respondents according to their pattern of scores on the categorical variables. This provides insight into the mechanisms producing the data and allows the estimation of factor structures and regression models conditional on the latent class structure. Item response theory leads to the identification of one or more ordinal or interval scales. In psychological and educational testing these scales are used for individual measurement of abilities and personality traits.The focus of this volume is applied. After a method is explained, the potential of the method for analyzing categorical data is illustrated by means of a real data example to show how it can be used effectively for solving a real data problem. These methods are accessible to researchers not trained explicitly in applied statistics. This volume appeals to researchers and advanced students in the social and behavioral sciences, including social, developmental, organizational, clinical and health psychologists, sociologists, educational and marketing researchers, and political scientists. In addition, it is of interest to those who collect data on categorical variables and are faced with the problem of how to analyze such variables--among themselves or in relation to metric variables.

Purpose Our objectives were to (a) identify oral and pharyngeal physiologic swallowing impairment severity classes based on latent class analyses (LCAs) of the Modified Barium Swallow Impairment Profile (MBSImP) swallow task scores and (b) quantify the probability of severity class membership given composite MBSImP oral total (OT) and pharyngeal total (PT) scores. Method MBSImP scores were collected from a patient database of 319 consecutive modified barium swallow studies. Because of missing swallow task scores, LCA was performed using 25 multiply imputed data sets. Results LCA revealed a three-class structure for both oral and pharyngeal models. We identified OT and PT score intervals to assign subjects to oral and pharyngeal impairment latent severity classes, respectively, with high probability (probability of class membership ≥ 0.9 given OT or PT scores within specified ranges) and high confidence (95% credible interval [CI] widths ≤ 0.24 for all total scores within specified ranges). OT scores ranging from 0 to 10 and from 14 to 18 yielded assignments in Oral Latent Classes 1 and 2, respectively, while OT = 22 was assigned to Oral Latent Class 3. PT scores ranging from 0 to 13 and from 18 to 24 yielded assignments in Pharyngeal Latent Classes 1 and 2, respectively, while PT = 26 was assigned to Pharyngeal Latent Class 3. Conclusions LCA of MBSImP task-level data revealed significant underlying oral and pharyngeal ordinal class structures representing increasingly severe gradations of physiologic swallow impairment. Clinically meaningful OT and PT score ranges were derived facilitating latent class assignment. Supplemental Material https://doi.org/10.23641/asha.12315677.

Missing data and especially dropouts frequently arise in longitudinal data. Maximum likelihood estimates are consistent when data are missing at random (MAR) but, as this assumption is not checkable, pattern mixture models (PMM) have been developed to deal with informative dropout. More recently, latent class models (LCM) have been proposed as a way to relax PMM assumptions. The aim of this paper is to compare PMM and LCM in order to tackle informative dropout in a longitudinal study of cognitive ageing measured by several psychometric tests. Using a multivariate longitudinal model with a latent process, a sensitivity analysis was performed to compare estimates under the MAR assumption, from a PMM and from two LCM. In the PMM, dropout patterns are included as covariates in the multivariate longitudinal model. In the simple LCM, they are predictors of the class membership probabilities while, in the joint LCM, the dropout time is jointly modeled using a proportional hazard model depending on latent classes. We show that parameter interpretation is different in the two kinds of models and thus can lead to different estimated values. PMM parameters are adjusted on the dropout patterns while LCM parameters are adjusted on the latent classes. This difference is highlighted in our data set because the latent classes exhibit much more heterogeneity than dropout patterns. We suggest several complementary analyses to investigate the characteristics of latent classes in order to understand the meaning of the parameters when using LCM to deal with informative dropout.

This article describes several latent variable approaches that can support rigorous quantitative inquiry in music education. I provide a definition of latent variables, list several advantages associated with their use for the measurement of constructs, and review three types of latent variables featuring utility for music education scholars: factor models, item response models, and latent class (mixture) models. Each type of model is illustrated with exemplar studies from music education literature. In addition, I report the results of a systematic analysis of latent variables’ use in published music education research. Over the past decade, latent variables were used in only 14.29% of the quantitative research reports in two prominent journals: the Bulletin of the Council for Research in Music Education and the Journal of Research in Music Education. The majority of these latent variable applications (80.56%) were factor models; however, there are myriad additional possibilities. While not appropriate for every study, latent variable models have as-yet unrealized potential to improve the richness and rigor in nearly all areas of music education research.

Relationships between aging, disease risks, and longevity are not yet well understood. For example, joint increases in cancer risk and total survival observed in many human populations and some experimental aging studies may be linked to a trade-off between cancer and aging as well as to the trade-off(s) between cancer and other diseases, and their relative impact is not clear. While the former trade-off (between cancer and aging) received broad attention in aging research, the latter one lacks respective studies, although its understanding is important for developing optimal strategies of increasing both longevity and healthy life span. In this paper, we explore the possibility of trade-offs between risks of cancer and selected major disorders. First, we review current literature suggesting that the trade-offs between cancer and other diseases may exist and be linked to the differential intensity of apoptosis. Then we select relevant disorders for the analysis (acute coronary heart disease [ACHD], stroke, asthma, and Alzheimer disease [AD]) and calculate the risk of cancer among individuals with each of these disorders, and vice versa, using the Framingham Study (5209 individuals) and the National Long Term Care Survey (NLTCS) (38,214 individuals) data. We found a reduction in cancer risk among old (80+) men with stroke and in risk of ACHD among men (50+) with cancer in the Framingham Study. We also found an increase in ACHD and stroke among individuals with cancer, and a reduction in cancer risk among women with AD in the NLTCS. The manifestation of trade-offs between risks of cancer and other diseases thus depended on sex, age, and study population. We discuss factors modulating the potential trade-offs between major disorders in populations, e.g., disease treatments. Further study is needed to clarify possible impact of such trade-offs on longevity.

Aging and Health in Old Age