Longitudinal Analysis

Modeling Within-Person Fluctuation and Change

Lesa Hoffman

• Publisher: Routledge
• Year: 2015
• ISBN:
  9781317591085
• Edition: 1
• Topic: Psicologia
• Subtopic: Ricerche e metodologie nella psicologia

Longitudinal Analysis provides an accessible, application-oriented treatment of introductory and advanced linear models for within-person fluctuation and change. Organized by research design and data type, the text uses in-depth examples to provide a complete description of the model-building process. The core longitudinal models and their extensions are presented within a multilevel modeling framework, paying careful attention to the modeling concerns that are unique to longitudinal data. Written in a conversational style, the text provides verbal and visual interpretation of model equations to aid in their translation to empirical research results. Overviews and summaries, boldfaced key terms, and review questions will help readers synthesize the key concepts in each chapter.

Written for non-mathematically-oriented readers, this text features:

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• A description of the data manipulation steps required prior to model estimation so readers can more easily apply the steps to their own data

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• An emphasis on how the terminology, interpretation, and estimation of familiar general linear models relates to those of more complex models for longitudinal data

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• Integrated model comparisons, effect sizes, and statistical inference in each example to strengthen readers' understanding of the overall model-building process

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• Sample results sections for each example to provide useful templates for published reports

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• Examples using both real and simulated data in the text, along with syntax and output for SPSS, SAS, STATA, and M plus at www.PilesOfVariance.com to help readers apply the models to their own data

The book opens with the building blocks of longitudinal analysis—general ideas, the general linear model for between-person analysis, and between- and within-person models for the variance and the options within repeated measures analysis of variance. Section 2 introduces unconditional longitudinal models including alternative covariance structure models to describe within-person fluctuation over time and random effects models for within-person change. Conditional longitudinal models are presented in section 3, including both time-invariant and time-varying predictors. Section 4 reviews advanced applications, including alternative metrics of time in accelerated longitudinal designs, three-level models for multiple dimensions of within-person time, the analysis of individuals in groups over time, and repeated measures designs not involving time. The book concludes with additional considerations and future directions, including an overview of sample size planning and other model extensions for non-normal outcomes and intensive longitudinal data.

Class-tested at the University of Nebraska-Lincoln and in intensive summer workshops, this is an ideal text for graduate-level courses on longitudinal analysis or general multilevel modeling taught in psychology, human development and family studies, education, business, and other behavioral, social, and health sciences. The book's accessible approach will also help those trying to learn on their own. Only familiarity with general linear models (regression, analysis of variance) is needed for this text.

Section II
Modeling the Effects of Time

Chapter 4
Describing Within-Person Fluctuation Over Time

The first three chapters in this text were meant to provide the necessary building blocks for conducting longitudinal analysis. These initial building blocks included new concepts, new vocabulary, and an overview of how the longitudinal models to be presented will differ from traditional approaches (chapter 1), followed by a review of how to interpret main effects and interactions among predictors (chapter 2). Recall that while the purpose of the model for the means is to predict the outcome values, the model for the variance tries to predict the pattern of variance and covariance of the outcome residuals across observations instead. In chapter 3 we examined how and why the model for the variance needs to differ from those of traditional general linear models in order to be suitable for longitudinal data: to separate between-person from within-person variance and to allow residuals from the same person to covary over time, so that the effects of within-person predictors can be tested accurately. Chapter 3 also introduced how to compare the fit of alternative models, which we’ll be doing frequently.

The next three chapters will use these building blocks in order to model the effects of time in longitudinal data. All models from this point forward will be estimated using likelihood-based approaches (maximum likelihood or restricted maximum likelihood, as introduced at the end of chapter 3) and thus can include persons with incomplete outcomes. Recall from chapter 1 that different types of within-person variation can be located on a data continuum ranging from within-person fluctuation over time (i.e., as is typical for short-term studies) to within-person change over time (i.e., as is typical for longer-term studies in which time is meaningfully sampled with the goal of observing systematic change). The models in chapter 4 focus on describing within-person fluctuation over time, whereas the models in chapters 5 and 6 will focus on describing within-person change instead.

Accordingly, this chapter introduces alternative covariance structure (ACS) models for describing the pattern (or structure) of the variance and covariance over time in a longitudinal outcome that shows within-person fluctuation over time. We will see that the compound symmetry and unstructured models (found in the repeated measures ANOVA models from chapter 3) will reappear as special cases of the ACS models, and other new variants will be introduced as well. Only the ACS models most commonly used for dependency related to time will be presented in this chapter, although there are similar models for other types of covariance as well (e.g., for spatial dependency in SAS MIXED; see Littell, Milliken, Stroup, Wolfinger, & Schabenberger, 2006, chapter 11). Furthermore, the ACS models presented here will be distinguished into models without a random intercept (which are relatively well known) and models with a random intercept (which are less well known). An excellent additional treatment of both types of ACS models can also be found in Hedeker and Gibbons (2006, chapters 6–7).

It is important to note before beginning that most of the ACS models to be presented in this chapter require equal-interval time observations that are balanced across persons—that is, although the data from persons with missing outcomes can still be included, everyone must have the same set of equidistant time observations (the few exceptions to this requirement will be noted). Furthermore, ACS models simply describe the pattern of outcome variance and covariance over time as observed in the data—ACS models say nothing about the potential reasons for any non-constant variance and covariance over time. In contrast, we will see in chapter 5 how models that include random slopes for time do not require equal-interval and balanced time observations and explicitly postulate individual differences in change as the source behind differing variance and covariance over time instead. But for outcomes in which people show within-person fluctuation over time rather than systematic change over time, random time slopes for individual differences in change won’t be relevant for describing patterns of outcome variance and covariance over time—this is where ACS models are more useful.

1. Unconditional Models for Describing Within-Person Fluctuation Over Time

Before diving into the specifics of alternative covariance structure (ACS) models, it is useful to examine the bigger picture—which data characteristics will be important to consider, and how they can be used to construct an organizational schema by which specific variants of ACS models can then be distinguished. This section covers each of these topics in turn.

1.A. Concerns in Evaluating Alternative Models for the Variance

When systematic change over time is not expected, the ACS models to be presented for such outcomes showing only within-person fluctuation will generally not need any fixed effects of time in their model for the means. However, fixed effects of time could be added and tested as needed to account for any unexpected mean differences across occasions (i.e., as in the saturated means model in chapter 3). Everything that follows in this chapter will focus exclusively on the model for the variance—what parameters are needed to describe the pattern or structure of the outcome variance and covariance over time as parsimoniously but accurately as possible (i.e., using as few parameters as needed to achieve good fit to the data).

In selecting an ACS model for the variance for a longitudinal outcome, we will need to consider two main issues. First, is the variance across persons the same or different across occasions (i.e., homogeneous variance or heterogeneous variance, respectively)? Second, although we expect the model residuals from the same person to be related across occasions (i.e., to covary or show covariance), are these covariances the same or different over time? So far we have seen two limited choices through repeated measures ANOVA models—either the outcome variance and covariance is the same over time (univariate or compound symmetry model), or the outcome variances and covariances all differ over time (multivariate or unstructured model). Although the ACS models presented in this chapter only permit homogeneous or heterogeneous variance over time (i.e., all the same or all different), they do offer more flexibility in the patterns of covariance they predict over time. The random slopes models presented in chapters 5 and 6 will be even more flexible in the patterns of non-constant variance and covariance they can predict.

But why should we care? Even if the model for the variance is not of theoretical interest, it is nevertheless an important concern when modeling longitudinal data for two reasons. First, as was shown in chapter 3, if the variances and covariances predicted by the model over time do not adequately reflect those in the actual data, then the standard errors and p-values for the fixed effects of the model predictors may be incorrect—they can be too conservative or too liberal, depending on how the model for the variance is misspecified. This is why we will evaluate both the model for the means and the model for the variance with respect to time before testing the effects of other predictors—so that they will be tested as accurately as possible. However, the model for the variance should be re-evaluated after examining all predictors to ensure it is still adequate (and perhaps it can then be simplified without compromising the accuracy of the SEs and p-values for the fixed effects of predictors that were retained).

Second, understanding the structure of outcome variance and covariance over time can be informative for subsequent model building. For instance, consider an intraclass correlation (from the compound symmetry model in chapter 3) that reflects how much outcome variance is due to between-person mean (intercept) differences rather than due to within-person variation over time. If most of an outcome’s variance is between persons, then stable person characteristics will be more useful for explaining its variance than will person characteristics that change over time. Furthermore, if a compound sym...