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.     

Modeling and Testing Differential Item Functioning in Unidimensional Binary Item Response Models with a Single Continuous Covariate: A Functional Data Analysis Approach

Differential item functioning (DIF), referring to between-group variation in item characteristics above and beyond the group-level disparity in the latent variable of interest, has long been regarded as an important item-level diagnostic. The presence of DIF impairs the fit of the single-group item response model being used, and calls for either model modification or item deletion in practice, depending on the mode of analysis. Methods for testing DIF with continuous covariates, rather than categorical grouping variables, have been developed; however, they are restrictive in parametric forms, and thus are not sufficiently flexible to describe complex interaction among latent variables and covariates. In the current study, we formulate the probability of endorsing each test item as a general bivariate function of a unidimensional latent trait and a single covariate, which is then approximated by a two-dimensional smoothing spline. The accuracy and precision of the proposed procedure is evaluated via Monte Carlo simulations. If anchor items are available, we proposed an extended model that simultaneously estimates item characteristic functions (ICFs) for anchor items, ICFs conditional on the covariate for non-anchor items, and the latent variable density conditional on the covariate—all using regression splines. A permutation DIF test is developed, and its performance is compared to the conventional parametric approach in a simulation study. We also illustrate the proposed semiparametric DIF testing procedure with an empirical example.     

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.     

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.     

Conditional Direction Dependence Analysis: Evaluating the Causal Direction of Effects in Linear Models with Interaction Terms

Abstract

Direction dependence analysis (DDA) makes use of higher than second moment information of variables (x and y) to detect potential confounding and to probe the causal direction of linear variable relations (i.e., whether x?→?y or y?→?x better approximates the underlying causal mechanism). The “true” predictor is assumed to be a continuous nonnormal exogenous variable. Existing methods compatible with DDA, however, are of limited use when the relation of a focal predictor and an outcome is affected by a moderator. This study presents a conditional direction dependence analysis (CDDA) framework which enables researchers to evaluate the causal direction of conditional regression effects. Monte–Carlo simulations were used to evaluate two different moderation scenarios: Study 1 evaluates the performance of CDDA tests when a moderator affects the strength of the causal effect x?→?y. Study 2 evaluates cases in which the causal direction itself (x?→?y vs y?→?x) depends on moderator values. Study 3 evaluates the robustness of DDA tests in the presence of functional model misspecifications. Results suggest that significance tests compatible with CDDA are suitable in both moderation scenarios, i.e., CDDA allows one to discern regions of a moderator in which the causal direction is uniquely identifiable. An empirical example is provided to illustrate the approach.     

Nonlinear Indicator-Level Moderation in Latent Variable Models

Linear, nonlinear, and nonparametric moderated latent variable models have been developed to investigate possible interaction effects between a latent variable and an external continuous moderator on the observed indicators in the latent variable model. Most moderation models have focused on moderators that vary across persons but not across the indicators (e.g., moderators like age and socioeconomic status). However, in many applications, the values of the moderator may vary both across persons and across indicators (e.g., moderators like response times and confidence ratings). Indicator-level moderation models are available for categorical moderators and linear interaction effects. However, these approaches require respectively categorization of the continuous moderator and the assumption of linearity of the interaction effect. In this article, parametric nonlinear and nonparametric indicator-level moderation methods are developed. In a simulation study, we demonstrate the viability of these methods. In addition, the methods are applied to a real data set pertaining to arithmetic ability.     

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.     

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 Propensity Score Adjustment for Multiple Group Structural Equation Modeling

In the behavioral and social sciences, quasi-experimental and observational studies are used due to the difficulty achieving a random assignment. However, the estimation of differences between groups in observational studies frequently suffers from bias due to differences in the distributions of covariates. To estimate average treatment effects when the treatment variable is binary, Rosenbaum and Rubin (1983a) proposed adjustment methods for pretreatment variables using the propensity score. However, these studies were interested only in estimating the average causal effect and/or marginal means. In the behavioral and social sciences, a general estimation method is required to estimate parameters in multiple group structural equation modeling where the differences of covariates are adjusted. We show that a Horvitz–Thompson-type estimator, propensity score weighted M estimator (PWME) is consistent, even when we use estimated propensity scores, and the asymptotic variance of the PWME is shown to be less than that with true propensity scores. Furthermore, we show that the asymptotic distribution of the propensity score weighted statistic under a null hypothesis is a weighted sum of independent χ² ₁ variables. We show the method can compare latent variable means with covariates adjusted using propensity scores, which was not feasible by previous methods. We also apply the proposed method for correlated longitudinal binary responses with informative dropout using data from the Longitudinal Study of Aging (LSOA). The results of a simulation study indicate that the proposed estimation method is more robust than the maximum likelihood (ML) estimation method, in that PWME does not require the knowledge of the relationships among dependent variables and covariates.     

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.     

Experimental Personality Designs: Analyzing Categorical by Continuous Variable Interactions

Theories hypothesizing interactions between a categorical and one or more continuous variables are common in personality research. Traditionally, such hypotheses have been tested using nonoptimal adaptations of analysis of variance (ANOVA). This article describes an alternative multiple regression-based approach that has greater power and protects against spurious conclusions concerning the impact of individual predictors on the outcome in the presence of interactions. We discuss the structuring of the regression equation, the selection of a coding system for the categorical variable, and the importance of centering the continuous variable. We present in detail the interpretation of the effects of both individual predictors and their interactions as a function of the coding system selected for the categorical variable. We illustrate two- and three-dimensional graphical displays of the results and present methods for conducting post hoc tests following a significant interaction. The application of multiple regression techniques is illustrated through the analysis of two data sets. We show how multiple regression can produce all of the information provided by traditional but less optimal ANOVA procedures.     

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.     

Model Fit and Item Factor Analysis: Overfactoring,Underfactoring, and a Program to Guide Interpretation

In exploratory item factor analysis (IFA), researchers may use model fit statistics and commonly invoked fit thresholds to help determine the dimensionality of an assessment. However, these indices and thresholds may mislead as they were developed in a confirmatory framework for models with continuous, not categorical, indicators. The present study used Monte Carlo simulation methods to investigate the ability of popular model fit statistics (chi-square, root mean square error of approximation, the comparative fit index, and the Tucker–Lewis index) and their standard cutoff values to detect the optimal number of latent dimensions underlying sets of dichotomous items. Models were fit to data generated from three-factor population structures that varied in factor loading magnitude, factor intercorrelation magnitude, number of indicators, and whether cross loadings or minor factors were included. The effectiveness of the thresholds varied across fit statistics, and was conditional on many features of the underlying model. Together, results suggest that conventional fit thresholds offer questionable utility in the context of IFA.     

When does measurement error in covariates impact causal effect estimates? Analytic derivations of different scenarios and an empirical illustration

The average causal treatment effect (ATE) can be estimated from observational data based on covariate adjustment. Even if all confounding covariates are observed, they might not necessarily be reliably measured and may fail to obtain an unbiased ATE estimate. Instead of fallible covariates, the respective latent covariates can be used for covariate adjustment. But is it always necessary to use latent covariates? How well do analysis of covariance (ANCOVA) or propensity score (PS) methods estimate the ATE when latent covariates are used? We first analytically delineate the conditions under which latent instead of fallible covariates are necessary to obtain the ATE. Then we empirically examine the difference between ATE estimates when adjusting for fallible or latent covariates in an applied example. We discuss the issue of fallible covariates within a stochastic theory of causal effects and analyse data of a within-study comparison with recently developed ANCOVA and PS procedures that allow for latent covariates. We show that fallible covariates do not necessarily bias ATE estimates, but point out different scenarios in which adjusting for latent covariates is required. In our empirical application, we demonstrate how latent covariates can be incorporated for ATE estimation in ANCOVA and in PS analysis.     

Buy three but get only two: The smallest effect in a 2 × 2 ANOVA is always uninterpretable

Loftus (Memory & Cognition 6:312–319, 1978) distinguished between interpretable and uninterpretable interactions. Uninterpretable interactions are ambiguous, because they may be due to two additive main effects (no interaction) and a nonlinear relationship between the (latent) outcome variable and its indicator. Interpretable interactions can only be due to the presence of a true interactive effect in the outcome variable, regardless of the relationship that it establishes with its indicator. In the present article, we first show that same problem can arise when an unmeasured mediator has a nonlinear effect on the measured outcome variable. Then we integrate Loftus’s arguments with a seemingly contradictory approach to interactions suggested by Rosnow and Rosenthal (Psychological Bulletin 105:143–146, 1989). We show that entire data patterns, not just interaction effects alone, produce interpretable or noninterpretable interactions. Next, we show that the same problem of interpretability can apply to main effects. Lastly, we give concrete advice on what researchers can do to generate data patterns that provide unambiguous evidence for hypothesized interactions.     

Single and Multiple Ability Estimation in the SEM Framework: A Noninformative Bayesian Estimation Approach

Latent variable models with many categorical items and multiple latent constructs result in many dimensions of numerical integration, and the traditional frequentist estimation approach, such as maximum likelihood (ML), tends to fail due to model complexity. In such cases, Bayesian estimation with diffuse priors can be used as a viable alternative to ML estimation. This study compares the performance of Bayesian estimation with ML estimation in estimating single or multiple ability factors across 2 types of measurement models in the structural equation modeling framework: a multidimensional item response theory (MIRT) model and a multiple-indicator multiple-cause (MIMIC) model. A Monte Carlo simulation study demonstrates that Bayesian estimation with diffuse priors, under various conditions, produces results quite comparable with ML estimation in the single- and multilevel MIRT and MIMIC models. Additionally, an empirical example utilizing the Multistate Bar Examination is provided to compare the practical utility of the MIRT and MIMIC models. Structural relationships among the ability factors, covariates, and a binary outcome variable are investigated through the single- and multilevel measurement models. The article concludes with a summary of the relative advantages of Bayesian estimation over ML estimation in MIRT and MIMIC models and suggests strategies for implementing these methods.     

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.     

Building an identifiable latent class model with covariate effects on underlying and measured variables

In recent years, latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest. Such models summarize shared features of the multiple indicators as an underlying categorical variable, and the indicators' substantive associations with predictors are built directly and indirectly in unique model parameters. In this paper, we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership, but that also allow direct effects of secondary covariates on the indicators themselves. Theory for model identification is developed. We detail an Expectation-Maximization algorithm for parameter estimation, standard error calculation, and convergent properties. Comparison of the proposed model with models underlying existing latent class modeling software is provided. A detailed analysis of how visual impairments affect older persons' functioning requiring distance vision is used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03 and National Institutes of Mental Health grant R01-MH-56639-01A1. Dr. Bandeen-Roche is a Brookdale National Fellow. The authors wish to thank Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available. We also thank the Editor, the Associate Editor, and three referees for their valuable comments.     

Testing the degree of cross-sectional and longitudinal dependence between two discrete dynamic processes

Developmental research often involves studying change across 2 or more processes or constructs simultaneously. A natural question in this work is whether change in these 2 processes is related or independent. Associative latent transition analysis (ALTA) was designed to test hypotheses about the degree to which change in 2 discrete latent variables is related. The ALTA model is a type of latent class model, which is a categorical latent variable model based on categorical indicators. In the ALTA approach, level and change on 1 variable is predicted by level and change in another. Two types of hypotheses are discussed: (a) broad hypotheses of dependence between the 2 discrete latent variables and (b) targeted hypotheses comparing specific patterns of change between levels of the discrete variables. Both types of hypotheses are tested via nested model comparisons. Analyses of relations between psychological state and substance use illustrate the model. Recent psychological state and recent substance use were found to be associated cross-sectionally and longitudinally, implying that change in recent substance use was related to change in recent psychological state.     

A Latent Growth Curve Modeling Approach to Pooled Interrupted Time Series Analyses

This paper presents a latent variable approach for the estimation of treatment effects within a pooled interrupted time series (ITS) design. Although considered quasi-experimental, the ITS design has been noted as representing one of the strongest alternatives to the randomized experiment, making it highly appropriate for use in documenting the presence of effects that might warrant further evaluation in a large-scale randomized study. Results suggest that the latent variable growth modeling (LGM) is capable of detecting simultaneous differences in both level and slope, and provides tests of significance for these two necessary indicators of an ITS intervention effect. As shown in the analyses, the LGM framework provides a comprehensive and flexible approach to research design and data analysis, making available to a wide audience of researchers an analytical framework for a variety of analyses of growth and developmental processes.