The Analysis of Change

John Mordechai Gottman

• Publisher: Psychology Press
• Year: 2013
• Print ISBN: 9780805813562
• eBook ISBN: 9781134778058
• Topic: Psychology
• Subtopic: History & Theory in Psychology
• Index: Psychology

Continuity and change have been major concerns of the social and behavioral sciences -- in the study of human development and in the study of processes that unfold in various ways across time. There has been a veritable explosion of techniques for studying change over time which have fundamentally changed how we need to think of and study change. Unfortunately, many of the old precepts and beliefs are still among us. The field of methodology for the study of change is itself ready to change. Recently, there have been many analytic and conceptual developments questioning our cherished beliefs about the study of change. As such, how are individuals to think about issues and correctly analyze change? The chapters in this volume address these issues. Divided into two sections, this book deals with designs that analyze change in multiple subjects, and with change in single subjects and an interacting system. Papers presented in this volume are accessible to scientists who are not methodologists. The character of the papers are more like primers than basic treatises on methodology, written for other methodologists. It is time that people stop thinking in rigid ways about how to study change and be introduced to a range of many possibilities. Change, stability, order and chaos are elusive concepts. The pursuit of the laws of change must be approached in as flexible and creative a fashion as possible. This book should help to lead the way.

I

ANALYZING CHANGE IN MULTIPLE SUBJECTS

1

Myths and Methods: “Myths About Longitudinal Research” plus Supplemental Questions

David Rogosa

Stanford University

Rogosa’s 1988 chapter “Myths about longitudinal research” challenged many classic assumptions about the analysis of change. Unfortunately, Rogosa’s work is not widely enough followed even today. Many scientists and reviewers just assume that change scores are unreliable, that the analysis of covariance with time-1 scores as the covariate is an appropriate way for studying change, that regression toward the mean is almost a law of nature, and so on. Rogosa’s work shows that many of these assumptions are wrong. He also offers a viable alternative. In this chapter, the original “Myths about longitudinal research” chapter is reprinted, augmented by new material on data analysis and the measurement of change in the form of supplemental questions and answers.

—Editor

PREAMBLE

A little more than 10 years ago I prepared the first version of a colloquium presentation “Myths About Longitudinal Research” (given at University of California, Berkeley, May 1983). The written version of that presentation appeared as Rogosa (1988), and is reprinted as part of this chapter. The purpose of that presentation was to summarize my technical publications on longitudinal research and the measurement of change, the theme of the work being that statistical models for collections of individual trajectories are the proper way to approach research questions about growth and change. The underlying two-part approach is to construct models for the individual time trajectory and then to represent individual differences by differences over individuals in the values of the model parameters (or even the form of the individual time trajectory). And thus the individual history, rather than between-variable relations such as time-1, time-2 scatterplots, provides the basic information for addressing longitudinal research questions.

In the 10 years since I constructed this Myths talk, I have found that the message to focus on the individual unit model carries over to many other settings outside the nonexperimental longitudinal research setting of these Myths. Much of my own work uses individual unit models as the fundamental starting point in various areas of behavioral science research. A critical distinction is between models which start with the individual process versus models for relations among variables, of which path analysis, covariance structure analysis, and other causal modeling strategies are prominent examples. My work on statistical models and methods for behavioral observations (Rogosa, Floden, & Willett, 1984; Rogosa and Ghandour, 1991) is based on this approach, using a renewal process model for the behavioral observations on an individual unit and allowing the parameters of the model to vary over individuals (in contrast to the analysis of variance or generalizability theory models that have mainly been applied in this context). Also, in an examination of methods for aptitude-treatment interaction research in Rogosa (1991), the basic modeling approach is through a combination of models for individual maturation in the outcome variable and models for individual differences in response to an intervention. From this formulation, the performance of standard statistical approaches to ATI research (e.g., time-2, time-1 regression analysis) for assessing the effects of the treatment versus control comparisons is studied. A more important example is the analysis of the estimation of causal effects from an intervention in Holland (1988). In this examination of “encouragement designs,” simple representations of the individual processes define causal effects of interest at the level of the individual unit, and effects averaged over individuals yield analytic results to show the failure of path analysis methods to obtain the relevant effects (even though the example fits perfectly the “direct and indirect effects” template for path analysis).

To provide a partial update to the Myths, this chapter adds a set of supplemental questions and answers to the original chapter. The topics for this additional material are failures of structural equation models, descriptive and inferential data analysis based on individual growth curve models, limitations of time-1, time-2 data (including residual change scores), regression toward the mean, and methods for constructing longitudinal data examples. These supplemental items follow the same notation as in the Myths chapter, and numbering of figures, tables, and equations continues the sequence of the Myths chapter (e.g., the first figure is Figure 1.8). Original references are left as part of the reprinted chapter; additional references from this preamble and from the supplemental material are listed at the end of this chapter.

The data analysis discussion is perhaps the most important addition. The original Myths presentation can be regarded as providing the first half of the story for longitudinal research, by explaining productive ways to think about longitudinal panel data and also to demonstrate approaches that should be avoided. A full treatment of methods for longitudinal data analysis would complete that exposition (and a small part is given here). The overall message is that the basis for analyzing longitudinal data is the individual history, and an individual unit model, such as an individual growth curve, simply serves to smooth and summarize the individual history data.

MYTHS ABOUT LONGITUDINAL RESEARCH

This chapter is concerned with methods for the analysis of longitudinal data. Longitudinal research in the behavioral and social sciences has been dominated, for the past 50 years or more, by a collection of damaging myths and misunderstandings. The development and application of useful methods for the analysis of longitudinal data have been impeded by these myths. In debunking these myths the chapter seeks to convey “right thinking” about longitudinal research; in particular, productive statistical analyses require the identification of sensible research questions, appropriate statistical models, and unambiguous quantities to be estimated. The heroes of this chapter are statistical models for collections of individual growth (learning) curves. The myths to be discussed are:

1.Two observations a longitudinal study make.

2.The difference score is intrinsically unreliable and unfair.

3.You can determine from the correlation matrix for the longitudinal data whether or not you are measuring the same thing over time.

4.The correlation between change and initial status is

(a) negative

(b) zero

(c) positive

(d) all of the above

5.You can’t avoid regression toward the mean.

6.Residual change cures what ails the difference score.

7.Analyses of covariance matrices inform about change.

8.Stability coefficients estimate

(a) the consistency over time of an individual

(b) the consistency over time of an average individual

(c) the consistency over time of individual differences

(d) none of the above

(e) some of the above

9.Casual analyses support causal inferences about reciprocal effects.

The most prevalent type of longitudinal data in the behavioral and social sciences is longitudinal panel data. Longitudinal panel data consist of observations on many individual cases (persons) on relatively few (two or more) occasions (waves) of observation. An observation on a variable X at time t_i, for individual p is written as X_ip where i = 1, . . . , T, and p = 1, . . . , n. (For statistical methods based on individual growth curves, observations need not be made at the same times for all individuals. But as this is necessary for the standard methods that predominate in the behavioral and social sciences, in my examples all individuals have the same values of t_i, which means everyone is measured at the same times.)

The X_ip are presumed to be composed of a true score ξ_p(t_i) and an error of measurement ɛ_ip according to the classical test theory model: X_ip = ξ_p(t_i) + ɛ_ip. Many of the examples are in terms of the ξ_p(t_i) and thus assume good measurement. The justification is that perfect measurement serves as a baseline for the examination of analysis methods. A statistical procedure that works poorly even with perfect measurement is clearly not attractive. Estimation of individual growth curves is not jeopardized by the presence of measurement error within reasonable bounds, but measurement errors cause more severe problems for methods based on the covariance matrix of the X_i (e.g., regression-based procedures).

The individual growth curves are functions of true score over time, ξ_p(t). Research questions about growth, development, learning, and the like center on the systematic change in an attribute over time, and thus the individual growth curves are...

Table of contents

• Front Cover
• Half Title
• Title Page
• Copyright
• Contents
• Preface
• PART I: ANALYZING CHANGE IN MULTIPLE SUBJECTS
• PART II: ANALYZING CHANGE IN A SINGLE SUBJECT OR AN INTERACTING SYSTEM
• Author Index
• Subject Index