In longitudinal research investigators often measure multiple variables at multiple points in time and are interested in investigating individual differences in patterns of change on those variables. In the vast majority of applications, researchers focus on studying change in one variable at a time. In this article we consider methods for studying relations1.1ips between patterns of change on different variables. We show how the multilevel modeling framework, which is often used to study univariate change, can be extended to the multivariate case to yield estimates of covariances of parameters representing aspects of change on different variables. We illustrate this approach using data from a study of physiological response to marital conflict in older married couples, showing a substantial correlation between rate of linear change on different stress-related hormones during conflict. We also consider how similar issues can be studied using extensions of latent curve models to the multivariate case, and we show how such models are related to multivariate multilevel models.

A central problem in the application of exploratory factor analysis is deciding how many factors to retain (m). Although this is inherently a model selection problem, a model selection perspective is rarely adopted for this task. We suggest that Cudeck and Henly's (1991) framework can be applied to guide the selection process. Researchers must first identify the analytic goal: identifying the (approximately) correct m or identifying the most replicable m. Second, researchers must choose fit indices that are most congruent with their goal. Consistent with theory, a simulation study showed that different fit indices are best suited to different goals. Moreover, model selection with one goal in mind (e.g., identifying the approximately correct m) will not necessarily lead to the same number of factors as model selection with the other goal in mind (e.g., identifying the most replicable m). We recommend that researchers more thoroughly consider what they mean by "the right number of factors" before they choose fit indices.

Confirmatory factor analysis revealed that the Life Orientation Test (LOT) consisted of separate Optimism and Pessimism factors among middle-aged and older adults. Although the two factors were significantly negatively correlated among individuals facing a profound life challenge (i.e., caregiving), they were only weakly correlated among noncaregivers. Caregivers also expressed less optimism than noncaregivers and showed a trend toward greater pessimism, suggesting that life stress may affect these dispositions. Pessimism, not optimism, uniquely predicted subsequent psychological and physical health; however, optimism and pessimism were equally predictive for stressed and nonstressed samples. By exploring optimism and pessimism separately, researchers may better determine whether the beneficial effects of optimism result from thinking optimistically, avoiding pessimistic thinking, or a combination of the two. Research on dispositional optimism has increased rapidly in the past decade because of evidence linking an optimistic outlook to psychological and physical well-being (see review in Scheier & Carver, 1992). Dispositional optimism has been related to such diverse outcomes as success in an aftercare alcohol treatment program (Strack, Carver, & Blaney, 1987), adjustment to college (Aspinwall & Taylor, 1992), resistance to postpartum depression (Carver & Gaines, 1987), protection from distress following a failed attempt at in vitro fertilization (Litt, Tennen, Affleck, & Klock, 1992), and adjustment following surgery for breast cancer (Carver et al., 1993). In a striking demonstration of the benefits of optimism on health, Scheier et al. (1989) found that optimists recovered more quickly following coronary artery bypass surgery and returned to a normal life more rapidly than pessimists. All of the studies cited used the Life Orientation Test (LOT; Scheier & Carver, 1985), the most widely used measure of dispositional optimism, as a unidimensional measure of optimism. Scheier and Carver initially identified two factors on the LOT (one containing positively worded items; the other containing negatively worded items). Despite the two-factor structure of the measure, Scheier and Carver have preferred to treat the LOT as a unidimensional measure, suggesting that the

Standard chi-square-based fit indices for factor analysis and related models have a little known property: They are more sensitive to misfit when unique variances are small than when they are large. Consequently, very small correlation residuals indicating excellent fit can be accompanied by indications of bad fit by the fit indices when unique variances are small. An empirical example of this incompatibility between residuals and fit indices is provided. For illustrative purposes, an artificial example is provided that yields exactly the same correlation residuals as the empirical example but has larger unique variances. For this example, the fit indices indicate excellent fit. A theoretical explanation for this phenomenon is provided using relationships between unique variances and eigenvalues of the fitted correlation matrix.

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The power law (y = ax-b) has been shown to provide a good description of data collected in a wide range of fields in psychology. R, B. Anderson and Tweney (1997) suggested that the model's data-fitting success may in part be artifactual, caused by a number of factors, one of which is the use of improper data averaging methods. The present paper follows up on their work and explains causes of the power law artifact. A method for studying the geometric relations among responses generated by mathematical models is introduced that shows the artifact is a result of the combined contributions of three factors: arithmetic averaging of data that are generated from a nonlinear model in the presence of individual differences. Schooler, 1991), or could it be due to some artifact (Estes, 1956)? Concerns of the latter type were recently rekindled in a study on the mathematical form of the forgetting function (R. B. Anderson & Tweney, 1997). These authors were concerned about the superior data-fitting ability of the power function relative to the exponential function. Simulations showed that occurrence of the artifact depended on data averaging, whether arithmetic, Specifically, they showed that data generated by either function were fit best by a power model when the data of simulated subjects were arithmetically averaged, a phenomenon we refer to throughout the paper as the power law artifact. Geometrically averaging the data sometimes eliminated the artifact, improving the fit of the exponential model relative to the power model.However, Wixted and Ebbesen (1997) and R. B. Anderson and Tweney (1997) showed that the artifact cannot be due solely to the use of an improper averaging method, because if it were, geometric averaging should always improve the fit of the exponential model. Wixted and Ebbesen (1997) reanalyzed two data sets from a previous study on the form ofthe forgetting function (Wixted & Ebbesen, 1991). The type of averaging made very little difference. Not only did the power function fit the arithmetically and geometrically averaged data better than the exponential function, but it did so by virtually the same A measure of scientific advancement in psychology is the discovery of lawful, predictable behavior. Such discoveries are particularly satisfying when the operation of the underlying mental process that mediates between stimulus and response can be described using mathematical functions. One of the functions that has enjoyed considerable popularity, especially among cognitive psychologists, is the power function (y = ax-b). The power function has been shown to provide an excellent description of human behavior in a variety offields, such as psychophysics (Stevens, 1971), memory (Wixted & Ebbesen, 1991), skill learning (Newell & Rosenbloom, 1981), and judgment and decision making (Stevenson, 1993). For example, research on human memory has explored the rate at which information is forgotten over time (Ebbinghaus, 1964;Wickens, 1998; see Rubin & Wenzel, 1996, for a review). In a typical experimental set...

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Abrasive marital interactions may have physiological consequences even among older adults in long-term marriages.