Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed. Under this exogenous-x approach, missing covariates are handled primarily via listwise deletion. This sacrifices efficiency and does not allow missingness to depend on observed outcomes. Here we describe a modified joint likelihood approach that (a) allows inference about parameters of the exogenous-x conditional mixture even with nonnormal covariates, unlike a conventional multivariate mixture; (b) retains all cases under missing at random assumptions; (c) yields lower bias and higher efficiency than the exogenous-x approach under a variety of conditions with missing covariates; and (d) is straightforward to implement in available commercial software. The proposed approach is illustrated with an empirical analysis predicting membership in latent classes of conduct problems. Recommendations for practice are discussed.     

On the sampling interpretation of confidence intervals and hypothesis tests in the context of conditional maximum likelihood estimation

In the context ofconditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given theincidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that theconditional asymptotic confidence intervals are also valid under the other two sampling schemes. I am indebted to Theo Eggen, Norman Verhelst and one of Psychometrika's reviewers for their helpful comments.     

Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of and, or, not and conditioning are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.     

A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the metropolis-hastings algorithm

Nonlinear latent variable models are specified that include quadratic forms and interactions of latent regressor variables as special cases. To estimate the parameters, the models are put in a Bayesian framework with conjugate priors for the parameters. The posterior distributions of the parameters and the latent variables are estimated using Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis-Hastings algorithm. The proposed estimation methods are illustrated by two simulation studies and by the estimation of a non-linear model for the dependence of performance on task complexity and goal specificity using empirical data.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

The Gaussian Graphical Model in Cross-Sectional and Time-Series Data

We discuss the Gaussian graphical model (GGM; an undirected network of partial correlation coefficients) and detail its utility as an exploratory data analysis tool. The GGM shows which variables predict one-another, allows for sparse modeling of covariance structures, and may highlight potential causal relationships between observed variables. We describe the utility in three kinds of psychological data sets: data sets in which consecutive cases are assumed independent (e.g., cross-sectional data), temporally ordered data sets (e.g., n = 1 time series), and a mixture of the 2 (e.g., n > 1 time series). In time-series analysis, the GGM can be used to model the residual structure of a vector-autoregression analysis (VAR), also termed graphical VAR. Two network models can then be obtained: a temporal network and a contemporaneous network. When analyzing data from multiple subjects, a GGM can also be formed on the covariance structure of stationary means—the between-subjects network. We discuss the interpretation of these models and propose estimation methods to obtain these networks, which we implement in the R packages graphicalVAR and mlVAR. The methods are showcased in two empirical examples, and simulation studies on these methods are included in the supplementary materials.     

Inequality constrained analysis of variance: a Bayesian approach

Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means.     

Graphical regression models for polytomous variables

When modeling the relationship between two nominal categorical variables, it is often desirable to include covariates to understand how individuals differ in their response behavior. Typically, however, not all the relevant covariates are available, with the result that the measured variables cannot fully account for the associations between the nominal variables. Under the assumption that the observed and unobserved variables follow a homogeneous conditional Gaussian distribution, this paper proposesRC(M) regression models to decompose the residual associations between the polytomous variables. Based on Goodman's (1979, 1985)RC(M) association model, a distinctive feature ofRC(M) regression models is that they facilitate the joint estimation of effects due to manifest and omitted (continuous) variables without requiring numerical integration. TheRC(M) regression models are illustrated using data from the High School and Beyond study (Tatsuoka & Lohnes, 1988). This article was accepted for publication, when Willem J. Heiser was the Editor ofPsychometrika. This research was supported by grants from the National Science Foundation (#SBR96-17510 and #SBR94-09531) and the Bureau of Educational Research at the University of Illinois. We thank Jee-Seon Kim for comments and computational assistance.     

Fitting and Testing Conditional Multinormal Partial Credit Models

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in Psychometrika 55:5?C18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores, which are sufficient statistics. Attention is paid to maximum likelihood estimation of item parameters, multivariate moments of latent variables, and person parameters. It is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques. More general models are discussed which can be used for testing the model, and it is shown how models with different numbers of latent variables can be tested against each other. In addition, multi-group extensions are proposed, which can be used for testing both measurement invariance and latent population differences. Models and procedures discussed are demonstrated in an empirical data example.     

Parental conditional regard as a predictor of deficiencies in young children's capacities to respond to sad feelings

This study explored the relationship between parents' use of conditional regard (PCR, Assor, Roth, & Deci, 2004; Roth, 2008) to promote suppression of sad feelings and the following emotional skills in young children: (1) recognition of sadness in facial expressions, (2) awareness of sad feelings in oneself, and (3) empathic response to others' sad feelings. The study distinguished between two PCR components: conditional negative regard (similar to love withdrawal) and conditional positive regard. Emotion‐focused interviews were conducted with 102 children aged 5–6 years, and their parents completed questionnaires assessing parents' use of conditional regard. As expected, both PCR components correlated negatively with the emotional skills. Regression analyses showed that the seemingly benign practice of conditional positive regard had unique negative associations with the three emotional skills also when effects of negative PCR were controlled. Copyright © 2010 John Wiley & Sons, Ltd.     

Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models

Using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal. This asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix. Using quasi-maximum likelihood theory a Hausman misspecification test is developed. This test is sensitive to misspecification caused by errors that are correlated with the independent variables. This misspecification cannot be detected by the test statistics currently used in covariance structure analysis.For helpful comments on a previous draft of the paper we are indebted to Kenneth A. Bollen, Ulrich L. Küsters, Michael E. Sobel and the anonymous reviewers of Psychometrika. For partial research support, the first author wishes to thank the Department of Sociology at the University of Arizona, where he was a visiting professor during the fall semester 1987.     

Marginal likelihood inference for a model for item responses and response times

Marginal maximum‐likelihood procedures for parameter estimation and testing the fit of a hierarchical model for speed and accuracy on test items are presented. The model is a composition of two first‐level models for dichotomous responses and response times along with multivariate normal models for their item and person parameters. It is shown how the item parameters can easily be estimated using Fisher's identity. To test the fit of the model, Lagrange multiplier tests of the assumptions of subpopulation invariance of the item parameters (i.e., no differential item functioning), the shape of the response functions, and three different types of conditional independence were derived. Simulation studies were used to show the feasibility of the estimation and testing procedures and to estimate the power and Type I error rate of the latter. In addition, the procedures were applied to an empirical data set from a computerized adaptive test of language comprehension.     

Fitting multivariage normal finite mixtures subject to structural equation modeling

This paper is about fitting multivariate normal mixture distributions subject to structural equation modeling. The general model comprises common factor and structural regression models. The introduction of covariance and mean structure models reduces the number of parameters to be estimated in fitting the mixture and enables one to investigate a variety of substantive hypotheses concerning the differences between the components in the mixture. Within the general model, individual parameters can be subjected to equality, nonlinear and simple bounds constraints. Confidence intervals are based on the inverse of the Hessian and on the likelihood profile. Several illustrations are given and results of a simulation study concerning the confidence intervals are reported.     

Treatment effects on count outcomes with non-normal covariates

The effects of a treatment or an intervention on a count outcome are often of interest in applied research. When controlling for additional covariates, a negative binomial regression model is usually applied to estimate conditional expectations of the count outcome. The difference in conditional expectations under treatment and under control is then defined as the (conditional) treatment effect. While traditionally aggregates of these conditional treatment effects (e.g., average treatment effects) are computed by averaging over the empirical distribution, a recently proposed moment-based approach allows for computing aggregate effects as a function of distribution parameters. The moment-based approach makes it possible to control for (latent) multivariate normally distributed covariates and provides more reliable inferences under certain conditions. In this paper we propose three different ways to account for non-normally distributed continuous covariates in this approach: an alternative, known non-normal distribution; a plausible factorization of the joint distribution; and an approximation using finite Gaussian mixtures. A saturated model is used for categorical covariates, making a distributional assumption obsolete. We further extend the moment-based approach to allow for multiple treatment conditions and the computation of conditional effects for categorical covariates. An illustrative example highlighting the key features of our extension is provided.     

命题内容对青少年条件推理的影响

选取3个具体内容的条件命题作为实验材料,以小四、初一、高一、大三的学生为被试,探讨了命题内容对青少年条件推理的影响机制及其发展特点。结果表明:(1)对同一年级而言,不同内容的条件命题的相同推理(MP、MT、DA、AC)之间表现出显著的差异;对不同年级而言,相同内容的条件命题的四种推理之间也存在显著的差异。(2)青少年的条件推理过程似乎是一种基于对事件发生概率估计的直觉判断,这一判断过程主要取决于个体知识经验的增长和主体认知水平的提高,而用形式逻辑的标准来衡量个体条件推理能力的高低似乎并不妥当。     

Penalization approaches in the conditional maximum likelihood and Rasch modelling context

Recent detection methods for Differential Item Functioning (DIF) include approaches like Rasch Trees, DIFlasso, GPCMlasso and Item Focussed Trees, all of which - in contrast to well established methods - can handle metric covariates inducing DIF. A new estimation method shall address their downsides by mainly aiming at combining three central virtues: the use of conditional likelihood for estimation, the incorporation of linear influence of metric covariates on item difficulty and the possibility to detect different DIF types: certain items showing DIF, certain covariates inducing DIF, or certain covariates inducing DIF in certain items. Each of the approaches mentioned lacks in two of these aspects. We introduce a method for DIF detection, which firstly utilizes the conditional likelihood for estimation combined with group Lasso-penalization for item or variable selection and L1-penalization for interaction selection, secondly incorporates linear effects instead of approximation through step functions, and thirdly provides the possibility to investigate any of the three DIF types. The method is described theoretically, challenges in implementation are discussed. A dataset is analysed for all DIF types and shows comparable results between methods. Simulation studies per DIF type reveal competitive performance of cmlDIFlasso, particularly when selecting interactions in case of large sample sizes and numbers of parameters. Coupled with low computation times, cmlDIFlasso seems a worthwhile option for applied DIF detection.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

The EffectLiteR Approach for Analyzing Average and Conditional Effects

We present a framework for estimating average and conditional effects of a discrete treatment variable on a continuous outcome variable, conditioning on categorical and continuous covariates. Using the new approach, termed the EffectLiteR approach, researchers can consider conditional treatment effects given values of all covariates in the analysis and various aggregates of these conditional treatment effects such as average effects, effects on the treated, or aggregated conditional effects given values of a subset of covariates. Building on structural equation modeling, key advantages of the new approach are (1) It allows for latent covariates and outcome variables; (2) it permits (higher order) interactions between the treatment variable and categorical and (latent) continuous covariates; and (3) covariates can be treated as stochastic or fixed. The approach is illustrated by an example, and open source software EffectLiteR is provided, which makes a detailed analysis of effects conveniently accessible for applied researchers.     

Power and sample size calculations for multivariate linear models with random explanatory variables

This article considers the problem of power and sample size calculations for normal outcomes within the framework of multivariate linear models. The emphasis is placed on the practical situation that not only the values of response variables for each subject are just available after the observations are made, but also the levels of explanatory variables cannot be predetermined before data collection. Using analytic justification, it is shown that the proposed methods extend the existing approaches to accommodate the extra variability and arbitrary configurations of the explanatory variables. The major modification involves the noncentrality parameters associated with the F approximations to the transformations of Wilks likelihood ratio, Pillai trace and Hotelling-Lawley trace statistics. A treatment of multivariate analysis of covariance models is employed to demonstrate the distinct features of the proposed extension. Monte Carlo simulation studies are conducted to assess the accuracy using a child’s intellectual development model. The results update and expand upon current work in the literature.The author wishes to thank the associate editor and the referees for comments which improve the paper considerably. This research was partially supported by a grant from the Natural Science Council of Taiwan.