A Unified Framework for the Comparison of Treatments with Ordinal Responses

Different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable. In this paper, we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses. The latent variable model is built upon the location-scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables, including the proportional odds model, the ordered probit-type model, and the proportional hazards model. A flexible estimation procedure is proposed for the identification and estimation of the general latent variable model, which allows for the location and scale parameters to be freely estimated. The framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data, thus allowing for important statistical inferences such as location and/or dispersion comparisons among treatments to be conveniently drawn. Analysis on real data sets is used to illustrate the proposed methods.     

A Comparison of Imputation Strategies for Ordinal Missing Data on Likert Scale Variables

This article compares a variety of imputation strategies for ordinal missing data on Likert scale variables (number of categories = 2, 3, 5, or 7) in recovering reliability coefficients, mean scale scores, and regression coefficients of predicting one scale score from another. The examined strategies include imputing using normal data models with naïve rounding/without rounding, using latent variable models, and using categorical data models such as discriminant analysis and binary logistic regression (for dichotomous data only), multinomial and proportional odds logistic regression (for polytomous data only). The result suggests that both the normal model approach without rounding and the latent variable model approach perform well for either dichotomous or polytomous data regardless of sample size, missing data proportion, and asymmetry of item distributions. The discriminant analysis approach also performs well for dichotomous data. Naïvely rounding normal imputations or using logistic regression models to impute ordinal data are not recommended as they can potentially lead to substantial bias in all or some of the parameters.     

定序变量回归模型在心理数据分析中的应用

定序变量在心理现象和心理数据中随处可见, 采用综合的定序变量回归分析模型可以对“镜像模式”和“漏斗模型”的心理现象做出合理的解释和预测。首先通过非参数检验对影响因素进行初步降维, 其次用Probit定序回归对降维后的影响因素贡献率进行判别, 从而进一步筛选具有显著性判断水平的有效指标, 最后用Logistic回归模型对某种特定的心理现象发生与否进行信息量足够大的解释和预测。大学毕业生工作生活质量满意度的预测对这种综合定序变量回归分析模型的实例拟合, 证实了综合定序变量回归分析模型在心理现象和心理数据分析中的应用价值。     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

Dynamic analysis of multivariate panel data with non-linear transformations

Many models for multivariate data analysis can be seen as special cases of the linear dynamic or state space model. Contrary to the classical approach to linear dynamic systems analysis, in which high-dimensional exact solutions are sought, the model presented here is developed from a social science framework where low-dimensional approximate solutions are preferred. Borrowing concepts from the theory on mixture distributions, the linear dynamic model can be viewed as a multi-layered regression model, in which the output variables are imprecise manifestations of an unobserved continuous process. An additional layer of mixing makes it possible to incorporate non-normal as well as ordinal variables.Using the EM-algorithm, we find estimates of the unknown model parameters, simultaneously providing stability estimates. The model is very general and cannot be well estimated by other estimation methods. We illustrate the applicability of the obtained procedure through an example with generated data.     

Regression among factor scores

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.     

A Bayesian Semiparametric Latent Variable Model for Mixed Responses

In this paper we introduce a latent variable model (LVM) for mixed ordinal and continuous responses, where covariate effects on the continuous latent variables are modelled through a flexible semiparametric Gaussian regression model. We extend existing LVMs with the usual linear covariate effects by including nonparametric components for nonlinear effects of continuous covariates and interactions with other covariates as well as spatial effects. Full Bayesian modelling is based on penalized spline and Markov random field priors and is performed by computationally efficient Markov chain Monte Carlo (MCMC) methods. We apply our approach to a German social science survey which motivated our methodological development. We thank the editor and the referees for their constructive and helpful comments, leading to substantial improvements of a first version, and Sven Steinert for computational assistance. Partial financial support from the SFB 386 “Statistical Analysis of Discrete Structures” is also acknowledged.     

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.     

Multilevel and Latent Variable Modeling with Composite Links and Exploded Likelihoods

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation. We wish to thank The Research Council of Norway for a grant supporting our collaboration.     

A general class of latent variable models for ordinal manifest variables with covariate effects on the manifest and latent variables

Previous work on a general class of multidimensional latent variable models for analysing ordinal manifest variables is extended here to allow for direct covariate effects on the manifest ordinal variables and covariate effects on the latent variables. A full maximum likelihood estimation method is used to estimate all the model parameters simultaneously. Goodness‐of‐fit statistics and standard errors are discussed. Two examples from the 1996 British Social Attitudes Survey are used to illustrate the methodology.     

Multilevel IRT using dichotomous and polytomous response data

A structural multilevel model is presented where some of the variables cannot be observed directly but are measured using tests or questionnaires. Observed dichotomous or ordinal polytomous response data serve to measure the latent variables using an item response theory model. The latent variables can be defined at any level of the multilevel model. A Bayesian procedure Markov chain Monte Carlo (MCMC), to estimate all parameters simultaneously is presented. It is shown that certain model checks and model comparisons can be done using the MCMC output. The techniques are illustrated using a simulation study and an application involving students' achievements on a mathematics test and test results regarding management characteristics of teachers and principles.     

Synergistic Effects of Expectancy and Value on Homework Engagement: The Case for a Within-Person Perspective

Historically, expectancy–value models of motivation assumed a synergistic relation between expectancy and value: motivation is high only when both expectancy and value are high. Motivational processes were studied from a within-person perspective, with expectancies and values being assessed or experimentally manipulated across multiple domains and the focus being placed on intraindividual differences. In contrast, contemporary expectancy–value models in educational psychology concentrate almost exclusively on linear effects of expectancy and value on motivational outcomes, with a focus on between-person differences. Recent advances in latent variable methodology allow both issues to be addressed in observational studies. Using the expectancy–value model of homework motivation as a theoretical framework, this study estimated multilevel structural equation models with latent interactions in a sample of 511 secondary school students and found synergistic effects between domain-specific homework expectancy and homework value in predicting homework engagement in 6 subjects. This approach not only brings the “×” back into expectancy–value theory but also reestablishes the within-person perspective as the appropriate level of analysis for latent expectancy–value models.     

Latent variable models for multivariate longitudinal ordinal responses

The paper proposes a full information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Two latent variable models are proposed that account for dependencies among items within time and between time. One model fits item‐specific random effects which account for the between time points correlations and the second model uses a common factor. The relationships between the time‐dependent latent variables are modelled with a non‐stationary autoregressive model. The proposed models are fitted to a real data set.     

Two-Step Estimation of Models Between Latent Classes and External Variables

We consider models which combine latent class measurement models for categorical latent variables with structural regression models for the relationships between the latent classes and observed explanatory and response variables. We propose a two-step method of estimating such models. In its first step, the measurement model is estimated alone, and in the second step the parameters of this measurement model are held fixed when the structural model is estimated. Simulation studies and applied examples suggest that the two-step method is an attractive alternative to existing one-step and three-step methods. We derive estimated standard errors for the two-step estimates of the structural model which account for the uncertainty from both steps of the estimation, and show how the method can be implemented in existing software for latent variable modelling.     

A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.     

Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations

The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients. Frank Rijmen was partly supported by the Fund for Scientific Research Flanders (FWO).     

A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.     

Generalized multilevel structural equation modeling

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.gllamm can be downloaded from http://www.gllamm.org. The paper was written while Sophia Rabe-Hesketh was employed at and Anders Skrondal was visiting the Department of Biostatistics and Computing, Institute of Psychiatry, King's College London.