Daniel J. Bauer

Computational Tools for Probing Interactions in Multiple Linear Regression, Multilevel Modeling, and Latent Curve Analysis

Simple slopes, regions of significance, and confidence bands are commonly used to evaluate interactions in multiple linear regression (MLR) models, and the use of these techniques has recently been extended to multilevel or hierarchical linear modeling (HLM) and latent curve analysis (LCA). However, conducting these tests and plotting the conditional relations is often a tedious and error-prone task. This article provides an overview of methods used to probe interaction effects and describes a unified collection of freely available online resources that researchers can use to obtain significance tests for simple slopes, compute regions of significance, and obtain confidence bands for simple slopes across the range of the moderator in the MLR, HLM, and LCA contexts. Plotting capabilities are also provided.

Conceptualizing and testing random indirect effects and moderated mediation in multilevel models: New procedures and recommendations.

The authors propose new procedures for evaluating direct, indirect, and total effects in multilevel models when all relevant variables are measured at Level 1 and all effects are random. Formulas are provided for the mean and variance of the indirect and total effects and for the sampling variances of the average indirect and total effects. Simulations show that the estimates are unbiased under most conditions. Confidence intervals based on a normal approximation or a simulated sampling distribution perform well when the random effects are normally distributed but less so when they are nonnormally distributed. These methods are further developed to address hypotheses of moderated mediation in the multilevel context. An example demonstrates the feasibility and usefulness of the proposed methods.

Probing Interactions in Fixed and Multilevel Regression: Inferential and Graphical Techniques.

Many important research hypotheses concern conditional relations in which the effect of one predictor varies with the value of another. Such relations are commonly evaluated as multiplicative interactions and can be tested in both fixed- and random-effects regression. Often, these interactive effects must be further probed to fully explicate the nature of the conditional relation. The most common method for probing interactions is to test simple slopes at specific levels of the predictors. A more general method is the Johnson-Neyman (J-N) technique. This technique is not widely used, however, because it is currently limited to categorical by continuous interactions in fixed-effects regression and has yet to be extended to the broader class of random-effects regression models. The goal of our article is to generalize the J-N technique to allow for tests of a variety of interactions that arise in both fixed- and random-effects regression. We review existing methods for probing interactions, explicate the analytic expressions needed to expand these tests to a wider set of conditions, and demonstrate the advantages of the J-N technique relative to simple slopes with three empirical examples.

Distributional assumptions of growth mixture models: implications for overextraction of latent trajectory classes.

Growth mixture models are often used to determine if subgroups exist within the population that follow qualitatively distinct developmental trajectories. However, statistical theory developed for finite normal mixture models suggests that latent trajectory classes can be estimated even in the absence of population heterogeneity if the distribution of the repeated measures is nonnormal. By drawing on this theory, this article demonstrates that multiple trajectory classes can be estimated and appear optimal for nonnormal data even when only 1 group exists in the population. Further, the within-class parameter estimates obtained from these models are largely uninterpretable. Significant predictive relationships may be obscured or spurious relationships identified. The implications of these results for applied research are highlighted, and future directions for quantitative developments are suggested.

The Disaggregation of Within-Person and Between-Person Effects in Longitudinal Models of Change

Longitudinal models are becoming increasingly prevalent in the behavioral sciences, with key advantages including increased power, more comprehensive measurement, and establishment of temporal precedence. One particularly salient strength offered by longitudinal data is the ability to disaggregate between-person and within-person effects in the regression of an outcome on a time-varying covariate. However, the ability to disaggregate these effects has not been fully capitalized upon in many social science research applications. Two likely reasons for this omission are the general lack of discussion of disaggregating effects in the substantive literature and the need to overcome several remaining analytic challenges that limit existing quantitative methods used to isolate these effects in practice. This review explores both substantive and quantitative issues related to the disaggregation of effects over time, with a particular emphasis placed on the multilevel model. Existing analytic methods are reviewed, a general approach to the problem is proposed, and both the existing and proposed methods are demonstrated using several artificial data sets. Potential limitations and directions for future research are discussed, and recommendations for the disaggregation of effects in practice are offered.

The Integration of Continuous and Discrete Latent Variable Models: Potential Problems and Promising Opportunities

Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study.

Local Solutions in the Estimation of Growth Mixture Models.

Finite mixture models are well known to have poorly behaved likelihood functions featuring singularities and multiple optima. Growth mixture models may suffer from fewer of these problems, potentially benefiting from the structure imposed on the estimated class means and covariances by the specified growth model. As demonstrated here, however, local solutions may still be problematic. Results from an empirical case study and a small Monte Carlo simulation show that failure to thoroughly consider the possible presence of local optima in the estimation of a growth mixture model can sometimes have serious consequences, possibly leading to adoption of an inferior solution that differs in substantively important ways from the actual maximum likelihood solution. Often, the defaults of current software need to be overridden to thoroughly evaluate the parameter space and obtain confidence that the maximum likelihood solution has in fact been obtained.

Observations on the Use of Growth Mixture Models in Psychological Research.

Psychologists are applying growth mixture models at an increasing rate. This article argues that most of these applications are unlikely to reproduce the underlying taxonic structure of the population. At a more fundamental level, in many cases there is probably no taxonic structure to be found. Latent growth classes then categorically approximate the true continuum of individual differences in change. This approximation, although in some cases potentially useful, can also be problematic. The utility of growth mixture models for psychological science thus remains in doubt. Some ways in which these models might be more profitably used are suggested.

Estimating Multilevel Linear Models as Structural Equation Models

Multilevel linear models (MLMs) provide a powerful framework for analyzing data collected at nested or non-nested levels, such as students within classrooms. The current article draws on recent analytical and software advances to demonstrate that a broad class of MLMs may be estimated as structural equation models (SEMs). Moreover, within the SEM approach it is possible to include measurement models for predictors or outcomes, and to estimate the mediational pathways among predictors explicitly, tasks which are currently difficult with the conventional approach to multilevel modeling. The equivalency of the SEM approach with conventional methods for estimating MLMs is illustrated using empirical examples, including an example involving both multiple indicator latent factors for the outcomes and a causal chain for the predictors. The limitations of this approach for estimating MLMs are discussed and alternative approaches are considered.

Psychometric approaches for developing commensurate measures across independent studies: traditional and new models.

When conducting an integrative analysis of data obtained from multiple independent studies, a fundamental problem is to establish commensurate measures for the constructs of interest. Fortunately, procedures for evaluating and establishing measurement equivalence across samples are well developed for the linear factor model and commonly used item response theory models. A newly proposed moderated nonlinear factor analysis model generalizes these models and procedures, allowing for items of different scale types (continuous or discrete) and differential item functioning across levels of categorical and/or continuous variables. The potential of this new model to resolve the problem of measurement in integrative data analysis is shown via an empirical example examining changes in alcohol involvement from ages 10 to 22 years across 2 longitudinal studies.