This volume began as a nightmare.
Once upon a time, life for social and behavioral scientists was (relatively) simple. When a research design called for repeated measures data, the data were analyzed with repeated measures analysis of variance. The BMDP 2V module was frequently the package of choice for the calculations.
Life today is more complicated. There are many more choices. Does the researcher need to model behavior at the level of the individual as well as at the level of the group? Should the researcher use the familiar and well-understood least-squares criterion? Should the researcher turn to the maximum likelihood criterion for assessing the overall fit of a model? Is it possible and is it desirable to represent the repeated measures data within structural equation modeling?
So the nightmare began as (shall we be dishonest and say) one night of deliberations among these choices. The thought then arose that it would be useful to have the statistical experts writing in the same volume about the possibilities and some of the dimensions that are pertinent to making these choices. Hence the origin of the present volume.
The issue of the analysis of repeated measures data has commonly been examined within the context of the study of change, particularly with respect to longitudinal data (cf., Collins & Horn, 1991; Gottman, 1995). This volume contains three chapters whose primary focus is on the study of growth over several years time (Raudenbush, chapter 2; Curran & Hussong, chapter 3; Duncan, Duncan, Li, & Strycker, chapter 7). Studies of change typically imply the expectation that variation, movement in scores, is generally unidirectional—generally up or generally down. Not all repeated measures data are concerned with change, and change is only one aspect of the variability that occurs within individuals. To illustrate, consider an example from the study of social behavior.
Personality, social, and organizational psychologists are often interested in the effects of situations on behavior: to what extent are individuals’ behaviors consistent across sets of situations and to what extent does the behavior of individuals change as a function of the situation. For example, the focus might be on how people’s dominant and submissive behaviors change as a function of being in a subordinate, co-equal, or supervisory work role. There might also be interest in whether people’s responses to these situations vary as a function of their level on personality characteristics. Some people, let’s say extraverts, may change more in their behavior than other individuals in responding to these different situations.
This could be studied in the laboratory in which individuals participate in situations in which they are placed in a subordinate role, a co-equal role, and a supervisory role and their responses are recorded. This would be an example of a balanced design. All participants would participate in three situations. These data can be analyzed in the familiar technique of repeated measures analysis of variance. We might introduce the personality variable of extraversion to examine the interaction between individual differences and situation.
However, there is considerable error variance in a measure based on a one-occasion assessment (Epstein, 1979; Moskowitz & Schwarz, 1982). Measurements of the individual in each situation on several occasions would improve the quality of measurement. This is possible but difficult in the laboratory, so sometimes researchers make use of naturalistic techniques for collecting this kind of data (see Kenny, Bolger, & Kashy, chapter 1; also see Moskowitz, Suh, & Desaulniers, 1994).
Despite whether the researcher remains in the laboratory or whether the researcher uses a naturalistic methodology, the researcher is confronted with decisions about how to handle the data. The multiple measures for each situation could be aggregated (averaged) to provide a single measurement in each situation for each individual. If this is done within the context of the laboratory, this provides a balanced design with better measures. Unfortunately, this strategy throws away information. Some people would have less variability in their measures than other people. It may be of interest to know who has more variability in their responses to such situations as being the boss or being the supervisee.
As an alternative to the laboratory context, the researcher might use a naturalistic data collection method such as event contingent recording (Moskowitz, 1986; Wheeler & Reis, 1991). In event contingent recording, participants are given standardized forms and asked to record their behavior after being in certain kinds of events, such as all interpersonal events at work. The form could request information about characteristics of the situation as well as the person’s behavior so interpersonal events can be categorized into situations with a boss, situations with a co-equal, and situations with a supervisee. This method has appeal because it provides records of behavior in real life rather than responses to possibly artificial situations in the laboratory.
However, the structure of such data presents data analytic decisions. Individuals will report differing numbers of events. Individuals will report differing numbers of events in different kinds of situations. Individuals may report events corresponding to some of the targeted situations (e.g., in the subordinate and co-equal situations) but not to other targeted situations such as having the supervisory responsibilities of the boss situation.
The data structure could be simplified by aggregating across events referring to the same kind of situation to obtain one measure per situation and only including people in the sample who reported events in all three kinds of situations. The simplification of the data structure would provide a balanced design, and consequently the familiar data analytic techniques of repeated measures analysis of variance and repeated measures analysis of covariance could be used.
However, such simplification would also eliminate information. The simplification would (1) not take into account variability in people’s responses across events of the same type of situation; (2) throw away that portion of a sample that has “missing data”; that is, individuals whose data do not include the representation of all kinds of specified events, and (3) disregard the time ordering of events.
Once one becomes involved with recording multiple assessments of individuals behavior and affect responses, the variability of people’s responses across events becomes salient and compels modeling. For example, diurnal and weekly rhythms have been demonstrated for affect and behavior (Brown & Moskowitz, 1998; Larsen, 1987; Watson, Wiese, Vaidya, & Tellegen, 1999). Behavior and affect co-occur over time in ways that cannot be identified from static assessments of these variables (Moskowitz & Cote, 1995). Similarity and dissimilarity among measures or items from occasion to occasion may be of interest (see Nesselroade, McArdle, Aggen, & Meyers, chapter 9). The shape of variation can be of considerable importance, such as the shape of change in response to stress or psychotherapy or recovery from illness (e.g., Bolger & Zuckerman, 1995; also see Ramsay, chapter 4). The time ordering of events can be used to make inferences about antecedent-consequent relations (see Hillmer, chapter 8).
So the focus of this volume is the examination of how individuals behave across time and to what degree this behavior changes, fluctuates, is stable, or is not stable. We call this change in individual behavior “intraindividual variability.” Intraindividual variability can be contrasted with “interindividual variability.” The latter describes individual differences among different people; the former describes differences within a single person. Although most behavioral and social scientists believe that behavior does differ from one occasion to the next, sophisticated techniques for exploring intraindividual variability have been underutilized. Several factors have contributed to the reluctance of analysts to utilize these techniques. One factor is their newness, many of them having only been developed within the last few years. A second factor is the perceived difficulty of implementing these techniques; descriptions tend to be highly technical and inaccessible to nonmathematically trained researchers. A third factor is the unavailability of computer programs to do the analyses, a situation that has recently been much improved with the release of new computer programs.
The primary goal of this volume is to make accessible to a wide audience of researchers and scholars the latest techniques used to assess intraindividual variability. The chapters of this volume represent a group of distinguished experts who have written on a range of available techniques. The emphasis is generally at an introductory level; the experts have minimized mathematical detail and provided concrete empirical examples applying the techniques.
The volume opens with a chapter by David Kenny, Niall Bolger, and Deborah Kashy, who contrast several procedures for the analysis of repeated measures data. They note two problems with using traditional analysis of variance (ANOVA) procedures for analyzing many contemporary designs using repeated measures data. The first is that research participants often will not have the same number of data points. The second is that the predictor variable generally does not have the same distribution across measurement points for all research participants. They approach the analysis of intraindividual variability within the context of multilevel analyses in which research participant are independent units and the repeated observations for each individual are not assumed to be independent. They illustrate that a strength of alternative procedures to ANOVA is that they more readily permit the evaluation of random effects that reflect the extent of variability among individuals to fixed effects. They compare features of three alternative procedures for modeling the group of research participants and the variability within the group of research participants: a two-step ordinary least-squares regression procedure, a weighted least-squares variation of multiple regression, and a procedure based on a maximum likelihood criterion.
Stephen Raudenbush compares advantages of the hierarchical linear model (a multilevel model), structural equation modeling, and the generalized multivariate linear model in the analysis of repeated measures data. He argues for the flexibility of the hierarchical linear model (HLM). HLM permits the inclusion of all available data, allows unequal spacing of time points across participants, can incorporate a variety of ways of characterizing change in the data such as rate of change and rate of acceleration, and can provide for the clustering of individuals within groups such as schools or organizations. He then combines ideas from the standard hierarchical linear model and the multivariate model to produce a hierarchical multivariate model that allows for different distributions within persons of randomly missing data and time-varying covariates, permits the testing of a variety of covariance structures, and examines the sensitivity of inferences about change to alternative specification of the covariance structure. The procedure discussed permits the examination of whether alternative models are consistent with key inferences about the shape of change.
Patrick Curran and Andrea Hussong describe how repeated measures data can be represented in structural equation models. They discuss the advantages and disadvantages of two kinds of structural equation models for representing longitudinal change: the autoregressive crosslagged panel model and the latent curve analysis model. They emphasize the latent curve approach, an approach that first estimates growth factors underlying observed measures and then uses the growth factors in subsequent analyses. Latent curve analysis provides two key advances over autoregressive crosslagged panel models. The first is the capability to model data sets with more than two time points. The second is the capability to provide estimates of the extent of variability among individuals, both the extent of variability in starting points and in rates of change. An applied example concerning the development of antisocial behavior and reading proficiency is used to illustrate the latent curve analysis model. The example illustrates that predictors of behavior at single time points (e.g., initial status) are different from the predictors of the shape of change over time. They also use the example to illustrate several options for incorporating nonlinear forms of growth in structural equation models.
James Ramsay provides a commentary on issues connecting the chapters by Raudenbush, Curran and Hussong, and Kenny, Bolger, and Kashy. He makes several points relative to the study of longitudinal data, considering the implications of missing data, the number of points necessary to define characteristics of growth curves such as level, slopes, and bumps, and the possibility that the curves for individuals are not registered such that the curves for individuals may show a similar shape but reflect different timings of events. His chapter further extends the discussion of repeated measurements to the case where there are many measurements and makes the point that such data can be represented by a sample of curves using a set of techniques referred to as functional data analysis. His chapter ends on a note of caution, reminding the reader that moving to the more complex models that are sometimes presented in this book has costs that need to be considered. For example, the maximum-likelihood procedures are sensitive to the mis-specification of the variance–covariance structure. Moreover, adding random coefficient parameters uses up degrees of freedom leading to a loss of power and potentially unstable estimates of fixed effects. Thus, the cautious researcher who has a moderate sample size may prefer to keep the model simple such as by remaining with a least-squares–based regression procedure (cf. Kenny, Bolger, & Kashy, chapter 1).
There is considerable complexity in the analysis of the models that make use of random as well as fixed effects (see chapters 1 and 2). The chapters by Judith Singer and by Dennis Wallace and Samuel Green present detailed description of how to analyze and interpret such models using a commonly available package, the PROC MIXED procedure from SAS.
Dennis Wallace and Samuel Green’s chapter provides extensive information about how to estimate fixed and random effects. They provide detailed explanations of the meaning of the underlying statistics, such as maximum-likelihood and restricted maximum-likelihood methods, and an introduction to some of the structures that may be found in the variance–covariance matrices. They provide an outline of recommended steps for estimating models incorporating fixed and random effects. These steps are illustrated using an example from a longitudinal study of the effect of two treatment interventions for reducing substance abuse among homeless individuals; the illustration includes an examination of whether the effectiveness of the treatment programs vary as a function of changing levels of depression.
Judith Singer’s discussion provides practical advice for all stages of the analysis including data preparation and writing computer code. She illustrates a process that is sometimes mysterious for the novice researcher in this area. Models for the representation of individuals’ variability across time are sometimes presented as single equations at multiple levels (Bryk & Raudenbush, 1992) and sometimes by single equations that specify multiple sources of variation (cf. Goldstein, 1995). She demonstrates how separate equations can be written at multiple levels and then elements can be substituted in to arrive at a representation in a single equation. The presentation is situated in the context of individual growth models; the presentation can also be extended and applied to cases with repeated measures data that are unidirectional as described in the extended example presented earlier and in the chapter by Kenny, Kashy, and Bolger.
Stephen Raudenbush comments that the use of structural equation modeling has not typically been extended to the case where individuals are clustered. Terry Duncan, Susan Duncan, Fuzhong Li, and Lisa Strycker take the step of providing such an extension. They provide an introduction to representing multilevel models in structural equation models using an example from an analysis of change in adolescents’ use of alcohol. They compare the strengths and weaknesses of three approaches for modeling longitudinal data that are clustered and unbalanced. One method, a full information hierarchical linear model (HLM), is familiar from the chapter by Raudenbush. A second method, a limited information multilevel latent growth model (MLGM), is an extension of latent growth modeling that was presented in the chapter by Curran and Hussong. The third approach is based on a full information maximum likelihood (FIML) latent growth modeling using an extension of a factor of curves model which has not previously been discussed in the book. They provide examples of programming in both HLM (Bryk, Raudenbush, & Congdon, 1996) and Mplus (Muthén & Muthén, 1998).
Steven Hillmer provides a basic introduction to using time series models to predict intraindividual change. In a time series model, data points for the same variable are arranged sequentially in time, and a basic goal is to identify a model that best represents the sequencing of these data. Hillmer reviews the differences between the main kinds of models that might be used. He contrasts two classes of models: stationary models in which the joint probability of any set of observations is unaffected by a shift backward or forward in the time series and nonstationary models in which parts of the series behave similarly although not identically to other parts of the series. He reviews the steps of building a time series model, providing extensive graphical material for understanding the issues that might arise. The chapter includes an example of an interrupted time series data in which an intervention occurs during the course of a time series and the effect of the intervention is estimated. The extended example provided is drawn from the business literature on sales. Time series analyses can also be applied to the modeling of variability within a person when sufficient data points have been collected.
John Nesselroade, John McArdle, Steven Aggen, and Jonathan Meyers provide an introduction to dynamic factor analysis. Dynamic factor analysis permits the examination of the similarity and dissimilarity of data from occasion to occasion. They introduce the topic by describing the P-technique factor analysis, which uses the common factor model to model the covariation of multiple variables measured across time for a single individual. They note problems with this model in the representation of process changes over time, such as the representation of effects that dissipate or strengthen over time. They present two models that allow for time-related dependencies and illustrate the application of these two dynamic factor analysis methods using reports of daily moods. The necessary LISREL code for conducting these analyses is included.
The initial organization for this volume was done within the context of two symposia presented at the 1997 meeting of the American Psychological Association. We thank Lisa Harlow, the editor of the Erlbaum Multivariate Applications Series for suggesting that we prepare a volume based on these symposia. We also thank James Ramsay, Yoshio Takane, and David Zuroff for comments on drafts of these chapters. We are also grateful to Chantale Bousquet and Serge Arsenault for their preparation of the text in 
. Preparation of this volume was partially supported by funds from the Social Sciences and Humanities Research Council of Canada.
We hope that the volume provides readers with a sense of the range of reasonable options for analyzing repeated measures data and stimulates new questions and more interest in repeated measures designs that extend beyond the context of longitudinal data.
Pleasant dreams…
Debbie S. Moskowitz
Scott L. Hershberger
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