Proportion of third‐level variation in multi‐level studies: A note on an interval estimation procedure

An interval estimation procedure is outlined that can be used for evaluating the proportion of observed variance in a response variable, which is due to the third level of nesting in a hierarchical design. The approach is also useful when it is of concern to address the necessity of including a third level in analyses of data from a multi‐level study, relative to an alternative of proceeding with two‐level modelling. The proposed method is illustrated with an empirical example.     

Latent variable models for partially ordered responses and trajectory analysis of anger‐related feelings

A general framework is presented for the analysis of partially ordered set (poset) data. The work is motivated by the need to analyse poset data such as multi‐componential responses in psychological measurement and partially accomplished cognitive tasks in educational measurement. It is shown how the generalized loglinear model can be used to represent poset data that form a lattice and how latent‐variable models can be constructed by further specifying the canonical parameters of the loglinear representation. The approach generalizes a class of latent‐variable models for completely ordered data. We apply the methods to analyse data on the frequency and intensity of anger‐related feelings. Furthermore, we propose a trajectory analysis to gain insight into the response function of partially ordered emotional states.     

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.     

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.     

Generalized latent variable models with non‐linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two‐parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non‐linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non‐linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non‐linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration–maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non‐linear model.     

A latent trait model for response times on tests employing the proportional hazards model

For computer-administered tests, response times can be recorded conjointly with the corresponding responses. This broadens the scope of potential modelling approaches because response times can be analysed in addition to analysing the responses themselves. For this purpose, we present a new latent trait model for response times on tests. This model is based on the Cox proportional hazards model. According to this model, latent variables alter a baseline hazard function. Two different approaches to item parameter estimation are described: the first approach uses a variant of the Cox model for discrete time, whereas the second approach is based on a profile likelihood function. Properties of each estimator will be compared in a simulation study. Compared to the estimator for discrete time, the profile likelihood estimator is more efficient, that is, has smaller variance. Additionally, we show how the fit of the model can be evaluated and how the latent traits can be estimated. Finally, the applicability of the model to an empirical data set is demonstrated.     

Higher-order latent trait models for cognitive diagnosis

Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling theϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example. This research was funded by National Institute of Health grant R01 CA81068. We would like to thank William Stout and Sarah Hartz for many useful discussions, three anonymous reviewers for helpful comments and suggestions, and Kikumi Tatsuoka and Curtis Tatsuoka for generously sharing data.     

Response style analysis with threshold and multi‐process IRT models: A review and tutorial

Two different item response theory model frameworks have been proposed for the assessment and control of response styles in rating data. According to one framework, response styles can be assessed by analysing threshold parameters in Rasch models for ordinal data and in mixture‐distribution extensions of such models. A different framework is provided by multi‐process item response tree models, which can be used to disentangle response processes that are related to the substantive traits and response tendencies elicited by the response scale. In this tutorial, the two approaches are reviewed, illustrated with an empirical data set of the two‐dimensional ‘Personal Need for Structure’ construct, and compared in terms of multiple criteria. Mplus is used as a software framework for (mixed) polytomous Rasch models and item response tree models as well as for demonstrating how parsimonious model variants can be specified to test assumptions on the structure of response styles and attitude strength. Although both frameworks are shown to account for response styles, they differ on the quantitative criteria of model selection, practical aspects of model estimation, and conceptual issues of representing response styles as continuous and multidimensional sources of individual differences in psychological assessment.     

Testing multivariate mean collinearity via latent variable modelling

A procedure for testing mean collinearity in multidimensional spaces is outlined, which is applicable in settings with missing data and can be used when examining group mean differences. The approach is based on non‐linear parameter restrictions and is developed within the framework of latent variable modelling. The method provides useful information about the constellation of multiple response centroids in the populations studied, and is illustrated with an example.     

A method for testing group differences of scale validity in multiple population studies

A method for testing equality in validity of multi‐component measuring instruments across populations is outlined. The approach is developed within the framework of covariance structure modelling and complements earlier research on examining group differences in scale reliability. The procedure is particularly useful for purposes of ascertaining comparability of validity when constructing and developing measuring instruments. The method also provides ranges of plausible values for differences in composite validity across several populations and allows one to evaluate group discrepancies in validity of behavioural scales. The approach is illustrated using data from a cognitive intervention study.     

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 hierarchical latent response model for inferences about examinee engagement in terms of guessing and item-level non-response

In low-stakes assessments, test performance has few or no consequences for examinees themselves, so that examinees may not be fully engaged when answering the items. Instead of engaging in solution behaviour, disengaged examinees might randomly guess or generate no response at all. When ignored, examinee disengagement poses a severe threat to the validity of results obtained from low-stakes assessments. Statistical modelling approaches in educational measurement have been proposed that account for non-response or for guessing, but do not consider both types of disengaged behaviour simultaneously. We bring together research on modelling examinee engagement and research on missing values and present a hierarchical latent response model for identifying and modelling the processes associated with examinee disengagement jointly with the processes associated with engaged responses. To that end, we employ a mixture model that identifies disengagement at the item-by-examinee level by assuming different data-generating processes underlying item responses and omissions, respectively, as well as response times associated with engaged and disengaged behaviour. By modelling examinee engagement with a latent response framework, the model allows assessing how examinee engagement relates to ability and speed as well as to identify items that are likely to evoke disengaged test-taking behaviour. An illustration of the model by means of an application to real data is presented.     

Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data

Missing data are very common in behavioural and psychological research. In this paper, we develop a Bayesian approach in the context of a general nonlinear structural equation model with missing continuous and ordinal categorical data. In the development, the missing data are treated as latent quantities, and provision for the incompleteness of the data is made by a hybrid algorithm that combines the Gibbs sampler and the Metropolis‐Hastings algorithm. We show by means of a simulation study that the Bayesian estimates are accurate. A Bayesian model comparison procedure based on the Bayes factor and path sampling is proposed. The required observations from the posterior distribution for computing the Bayes factor are simulated by the hybrid algorithm in Bayesian estimation. Our simulation results indicate that the correct model is selected more frequently when the incomplete records are used in the analysis than when they are ignored. The methodology is further illustrated with a real data set from a study concerned with an AIDS preventative intervention for Filipina sex workers.     

Logistic response models with item interactions

Items that are clustered according to shared content may violate the principle of conditional independence commonly used in item response theory. This paper investigates the capabilities of a logistic item response model in relation to locally dependent item responses. The model includes main effect and interaction parameters that are computed as linear functions of the latent trait. The paper explains the interpretation of the parameters, the maximum likelihood estimation algorithm, the information matrix and some results concerning parameter identifiability. The problem of over-fitting the data is addressed in a simulation study, and two real data examples are described to illustrate the approach, one from the context of a sample survey and the other from ability testing using testlets.     

A comparison of methods for estimating quadratic effects in nonlinear structural equation models

Two Monte Carlo simulations were performed to compare methods for estimating and testing hypotheses of quadratic effects in latent variable regression models. The methods considered in the current study were (a) a 2-stage moderated regression approach using latent variable scores, (b) an unconstrained product indicator approach, (c) a latent moderated structural equation method, (d) a fully Bayesian approach, and (e) marginal maximum likelihood estimation. Of the 5 estimation methods, it was found that overall the methods based on maximum likelihood estimation and the Bayesian approach performed best in terms of bias, root-mean-square error, standard error ratios, power, and Type I error control, although key differences were observed. Similarities as well as disparities among methods are highlight and general recommendations articulated. As a point of comparison, all 5 approaches were fit to a reparameterized version of the latent quadratic model to educational reading data.     

Generalized linear models with ordinally‐observed covariates

An ordinally‐observed variable is a variable that is only partially observed through an ordinal surrogate. Although statistical models for ordinally‐observed response variables are well known, relatively little attention has been given to the problem of ordinally‐observed regressors. In this paper I show that if surrogates to ordinally‐observed covariates are used as regressors in a generalized linear model then the resulting measurement error in the covariates can compromise the consistency of point estimators and standard errors for the effects of fully‐observed regressors. To properly account for this measurement error when making inferences concerning the fully‐observed regressors, I propose a general modelling framework for generalized linear models with ordinally‐observed covariates. I discuss issues of model specification, identification, and estimation, and illustrate these with examples.     

Non‐ignorable missingness item response theory models for choice effects in examinee‐selected items

Examinee‐selected item (ESI) design, in which examinees are required to respond to a fixed number of items in a given set, always yields incomplete data (i.e., when only the selected items are answered, data are missing for the others) that are likely non‐ignorable in likelihood inference. Standard item response theory (IRT) models become infeasible when ESI data are missing not at random (MNAR). To solve this problem, the authors propose a two‐dimensional IRT model that posits one unidimensional IRT model for observed data and another for nominal selection patterns. The two latent variables are assumed to follow a bivariate normal distribution. In this study, the mirt freeware package was adopted to estimate parameters. The authors conduct an experiment to demonstrate that ESI data are often non‐ignorable and to determine how to apply the new model to the data collected. Two follow‐up simulation studies are conducted to assess the parameter recovery of the new model and the consequences for parameter estimation of ignoring MNAR data. The results of the two simulation studies indicate good parameter recovery of the new model and poor parameter recovery when non‐ignorable missing data were mistakenly treated as ignorable.     

A Bayesian Model For The Estimation Of Latent Interaction And Quadratic Effects When Latent Variables Are Non-Normally Distributed

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we present a Bayesian model for the estimation of latent nonlinear effects when the latent predictor variables are nonnormally distributed. The nonnormal predictor distribution is approximated by a finite mixture distribution. We conduct a simulation study that demonstrates the advantages of the proposed Bayesian model over contemporary approaches (Latent Moderated Structural Equations [LMS], Quasi-Maximum-Likelihood [QML], and the extended unconstrained approach) when the latent predictor variables follow a nonnormal distribution. The conventional approaches show biased estimates of the nonlinear effects; the proposed Bayesian model provides unbiased estimates. We present an empirical example from work and stress research and provide syntax for substantive researchers. Advantages and limitations of the new model are discussed.     

Evaluation of latent construct correlations in the presence of missing data: a note on a latent variable modelling approach

A latent variable modelling approach is discussed, which can be used to evaluate indices of linear relationship between latent constructs in incomplete data sets. The method is based on an application of maximum-likelihood estimation and inclusion of covariates predictive of missing values. The approach can be employed for point and interval estimation of latent correlations in the presence of missing data, and capitalizes on enhanced plausibility of the assumption of data missing at random through introduction of informative covariates. The method is illustrated on empirical data.