Information-theoretic latent distribution modeling: distinguishing discrete and continuous latent variable models

Distinguishing between discrete and continuous latent variable distributions has become increasingly important in numerous domains of behavioral science. Here, the authors explore an information-theoretic approach to latent distribution modeling, in which the ability of latent distribution models to represent statistical information in observed data is emphasized. The authors conclude that loss of statistical information with a decrease in the number of latent values provides an attractive basis for comparing discrete and continuous latent variable models. Theoretical considerations as well as the results of 2 Monte Carlo simulations indicate that information theory provides a sound basis for modeling latent distributions and distinguishing between discrete and continuous latent variable models in particular.     

Continuous and discrete latent structure models for item response data

Relations are examined between latent trait and latent class models for item response data. Conditions are given for the two-latent class and two-parameter normal ogive models to agree, and relations between their item parameters are presented. Generalizationss are then made to continuous models with more than one latent trait and discrete models with more than two latent classes, and methods are presented for relating latent class models to factor models for dichotomized variables. Results are illustrated using data from the Law School Admission Test, previously analyzed by several authors.     

The trade-off between accuracy and precision in latent variable models of mediation processes

Social psychologists place high importance on understanding mechanisms and frequently employ mediation analyses to shed light on the process underlying an effect. Such analyses can be conducted with observed variables (e.g., a typical regression approach) or latent variables (e.g., a structural equation modeling approach), and choosing between these methods can be a more complex and consequential decision than researchers often realize. The present article adds to the literature on mediation by examining the relative trade-off between accuracy and precision in latent versus observed variable modeling. Whereas past work has shown that latent variable models tend to produce more accurate estimates, we demonstrate that this increase in accuracy comes at the cost of increased standard errors and reduced power, and examine this relative trade-off both theoretically and empirically in a typical 3-variable mediation model across varying levels of effect size and reliability. We discuss implications for social psychologists seeking to uncover mediating variables and provide 3 practical recommendations for maximizing both accuracy and precision in mediation analyses.     

The chronometry of single neuron activity: testing discrete and continuous models of information processing

The authors propose to study information transmission by comparing the effects of experimental factors on reaction time (RT) with the latency of the changes in activity of single-neurons. An experiment was conducted in which a monkey (Macaca mulatta) performed a tactilo-manual 2-choice RT task and the compatibility of the stimulus-response mapping was manipulated. Task-related neurons were recorded in the monkey's primary somesthetic and motor cortices. The changes in activity of 105 of these neurons were classified either as sensory-like or as motor-like. The sensory-like changes occurred before the motor-like ones. The stimulus-response mapping exerted its entire effect on the RT after the sensory-like changes and before the motor-like ones. These findings suggest that the information was transmitted discretely from the processes affected by the mapping to the processes implemented by the motor-like changes.     

Bayesian latent variable models for the analysis of experimental psychology data

In this paper, we address the use of Bayesian factor analysis and structural equation models to draw inferences from experimental psychology data. While such application is non-standard, the models are generally useful for the unified analysis of multivariate data that stem from, e.g., subjects’ responses to multiple experimental stimuli. We first review the models and the parameter identification issues inherent in the models. We then provide details on model estimation via JAGS and on Bayes factor estimation. Finally, we use the models to re-analyze experimental data on risky choice, comparing the approach to simpler, alternative methods.     

Temporal properties of human information processing: Tests of discrete versus continuous models

Cognitive psychologists have characterized the temporal properties of human information processing in terms of discrete and continuous models. Discrete models postulate that component mental processes transmit a finite number of intermittent outputs (quanta) of information over time, whereas continuous models postulate that information is transmitted in a gradual fashion. These postulates may be tested by using an adaptive response-priming procedure and analysis of reaction-time mixture distributions. Three experiments based on this procedure and analysis are reported. The experiments involved varying the temporal interval between the onsets of a prime stimulus and a subsequent test stimulus to which a response had to be made. Reaction time was measured as a function of the duration of the priming interval and the type of prime stimulus. Discrete models predict that manipulations of the priming interval should yield a family of reaction-time mixture distributions formed from a finite number of underlying basis distributions, corresponding to distinct preparatory states. Continuous models make a different prediction. Goodness-of-fit tests between these predictions and the data supported either the discrete or the continuous models, depending on the nature of the stimuli and responses being used. When there were only two alternative responses and the stimulus-response mapping was a compatible one, discrete models with two or three states of preparation fit the results best. For larger response sets with an incompatible stimulus-response mapping, a continuous model fit some of the data better. These results are relevant to the interpretation of reaction-time data in a variety of contexts and to the analysis of speed-accuracy trade-offs in mental processes.     

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.     

The phenomenology of spatial integration: data and models

A briefly presented visual stimulus followed by darkness seems to persist beyond its physical offset. We are concerned here with the relation between two characteristics of this visible persistence: first, its phenomenological resemblance to the stimulus that spawned it and second, its usefulness as a basis for integrating visual stimuli that are separated in time. We describe two experiments using a task in which two halves of a visual stimulus were presented successively and observers reported how complete the stimulus appeared to be. Stimuli appeared less complete with increases in both the duration of the interval intervening between presentation of the two halves and the duration of the initially presented stimulus half. This data pattern is similar to that obtained in tasks in which spatial integration of two temporally disparate stimuli is necessary for correct responding. On the basis of this similarity, we argue that phenomenological appearance and ability to integrate stimuli over time are two facets of the same perceptual events. We describe a formal model to account for these and other data.     

Separable effects of factors on activation functions in discrete and continuous models: d' and evoked potentials

This article derives tests for partial-output models of information processing. A process in a partial-output model sends output, while executing, to its successors. In a stage model a process sends output to its successors only when it stops. Both models can be analyzed by manipulating factors, each affecting a different process. In discrete serial models, such factors have additive effects on reaction time (Sternberg, 1969) and sometimes on log percent correct (Schweickert, 1985). In partial-output models, such factors produce a simple pattern, the rectangle condition, in performance curves. Conditions for a partial-output representation are based on decomposable structures (Krantz et al., 1971). Tests of discrete and partial-output models are simultaneously satisfied only if performance is a linear function of time. Evoked potentials (Coles et al., 1985) and d' (Dosher, 1981) illustrate results.     

Cognitive psychology meets psychometric theory: on the relation between process models for decision making and latent variable models for individual differences

This article analyzes latent variable models from a cognitive psychology perspective. We start by discussing work by Tuerlinckx and De Boeck (2005), who proved that a diffusion model for 2-choice response processes entails a 2-parameter logistic item response theory (IRT) model for individual differences in the response data. Following this line of reasoning, we discuss the appropriateness of IRT for measuring abilities and bipolar traits, such as pro versus contra attitudes. Surprisingly, if a diffusion model underlies the response processes, IRT models are appropriate for bipolar traits but not for ability tests. A reconsideration of the concept of ability that is appropriate for such situations leads to a new item response model for accuracy and speed based on the idea that ability has a natural zero point. The model implies fundamentally new ways to think about guessing, response speed, and person fit in IRT. We discuss the relation between this model and existing models as well as implications for psychology and psychometrics.     

Non‐linear structural equation models with correlated continuous and discrete data

Structural equation models (SEMs) have been widely applied to examine interrelationships among latent and observed variables in social and psychological research. Motivated by the fact that correlated discrete variables are frequently encountered in practical applications, a non‐linear SEM that accommodates covariates, and mixed continuous, ordered, and unordered categorical variables is proposed. Maximum likelihood methods for estimation and model comparison are discussed. One real‐life data set about cardiovascular disease is used to illustrate the methodologies.     

A general structural equation model with dichotomous,ordered categorical,and continuous latent variable indicators

A structural equation model is proposed with a generalized measurement part, allowing for dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A computationally feasible three-stage estimator is proposed for any combination of observed variable types. This approach provides large-sample chi-square tests of fit and standard errors of estimates for situations not previously covered. Two multiple-indicator modeling examples are given. One is a simultaneous analysis of two groups with a structural equation model underlying skewed Likert variables. The second is a longitudinal model with a structural model for multivariate probit regressions.This research was supported by Grant No. 81-IJ-CX-0015 from the National Institute of Justice, by Grant No. DA 01070 from the U.S. Public Health Service, and by Grant No. SES-8312583 from the National Science Foundation. I thank Julie Honig for drawing the figures. Requests for reprints should be sent to Bengt Muthén, Graduate School of Education, University of California, Los Angeles, California 90024.     

The development of verbal and visual working memory processes: a latent variable approach

Working memory (WM) processing in children has been studied with different approaches, focusing on either the organizational structure of WM processing during development (factor analytic) or the influence of different task conditions on WM processing (experimental). The current study combined both approaches, aiming to distinguish verbal and visual processing in order to investigate WM development. We investigated recall performance under different task conditions in a sample of 5- to 13-year-olds, applying latent class regression analysis. In this analysis, we examined latent classes (subgroups) within the sample that differed in terms of processing type. The interpretations of the latent classes were validated internally using characteristics of the latent classes and externally using recall performance of words and figures. The results showed that children of different developmental stages used the same type of processing under the same conditions. However, due to developmental differences, their overall performances differed, showing groups of children who were successful in verbal or visual processing and groups of children who were not. This study shows and discusses the importance of disentangling the influence of task conditions from the influence of WM development when interpreting recall performance in children.     

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.     

The continuous end-state comfort effect: weighted integration of multiple biases

The grasp orientation when grasping an object is frequently aligned in anticipation of the intended rotation of the object (end-state comfort effect). We analyzed grasp orientation selection in a continuous task to determine the mechanisms underlying the end-state comfort effect. Participants had to grasp a box by a circular handle—which allowed for arbitrary grasp orientations—and then had to rotate the box by various angles. Experiments 1 and 2 revealed both that the rotation’s direction considerably determined grasp orientations and that end-postures varied considerably. Experiments 3 and 4 further showed that visual stimuli and initial arm postures biased grasp orientations if the intended rotation could be easily achieved. The data show that end-state comfort but also other factors determine grasp orientation selection. A simple mechanism that integrates multiple weighted biases can account for the data.     

The reliability and validity of discrete and continuous measures of psychopathology: a quantitative review

In 2 meta-analyses involving 58 studies and 59,575 participants, we quantitatively summarized the relative reliability and validity of continuous (i.e., dimensional) and discrete (i.e., categorical) measures of psychopathology. Overall, results suggest an expected 15% increase in reliability and 37% increase in validity through adoption of a continuous over discrete measure of psychopathology alone. This increase occurs across all types of samples and forms of psychopathology, with little evidence for exceptions. For typical observed effect sizes, the increase in validity is sufficient to almost halve sample sizes necessary to achieve standard power levels. With important caveats, the current results, considered with previous research, provide sufficient empirical and theoretical basis to assume a priori that continuous measurement of psychopathology is more reliable and valid. Use of continuous measures in psychopathology assessment has widespread theoretical and practical benefits in research and clinical settings.     

The use of latent trajectory models in psychopathology research

Despite the recent surge in the development of powerful modeling strategies to test questions about individual differences in stability and change over time, these methods are not currently widely used in psychopathology research. In an attempt to further the dissemination of these new methods, the authors present a pedagogical introduction to the structural equation modeling based latent trajectory model, or LTM. They review several different types of LTMs, discuss matching an optimal LTM to a given question of interest, and highlight several issues that might be particularly salient for research in psychopathology. The authors augment each section with a review of published applications of these methods in psychopathology-related research to demonstrate the implementation and interpretation of LTMs in practice.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.