In Equation (1), xtj is the time-related variable such as age, measurement wave, or the elapsed time following the occurrence of an event (e.g., surgery). Note that x(i has two subscripts, t and i, indicating it varies both over measurement occasions and across individuals. The intercept a. represents the predicted level of Individual i on the measure when x(i= 0. When time is scaled so that the first measurement occasion equals 0, af may be interpreted as the individual "initial status" or level on Y at the beginning of the study. The slope represents the individual growth rate, the change in Yper unit of time. The individual intercept a. and the individual slope form a pair of growth parameters that characterize the individual trajectory. Figure 21.3 shows hypothetical linear growth curves of three individuals on a variable Y over time. Note that the individuals start at different levels (different as) and grow at different rates (different J3s). Other time-varying covariates may be added as predictors to the Level 1 equation.

For example, suppose we collected daily measures of stressful events wti and well-being Y(. in each patient for 10 days immediately following minor surgery. We can add the time-varying covariate w(i to Equation (1). For patient i, we now have

a. is patient i's predicted well-being (initial status) at the completion of surgery; /3i is the rate of increase in well-being (slope). These parameters characterize each individual's growth function over and above the temporal disturbances accounted for by the time-varying covariate w r 7T. is the individually varying partial regression coefficient relating stress to well-being for Individual i, and £(. is the residual. Thus, Level 1 describes the change within individuals.

In the simplest Level 2 model, we assume that the set of as and the set of /3;s are normally distributed. The means and variances of these growth parameters are estimated at Level 2. The means of the growth parameters allow us to obtain a mean trajectory for the whole group. To the extent that the variances of the growth parameters are greater than 0, there are differences between individuals in

the growth patterns over time. With variation across individuals, the two individual growth parameters, ai and /3(, can become outcome variables to be regressed on time-invariant individual background covariate variables. These background variables can be experimental treatment conditions (e.g., presurgical psychological intervention versus no intervention) or stable individual difference variables (e.g., neuroticism). The Level 2 equations for the intercepts and the slopes may be expressed as a. = an + y Z + 8

i 0 ' a i ai where <x0 is the grand intercept (mean intercept across N individuals), ¡3() is the grand slope (mean slope across N individuals), and Zi is the timeinvariant covariate (e.g., neuroticism) and 8ai and d are the residuals associated with a. and respectively; and ya and y^ are the regression coefficients. Besides the linear growth parameters, additional Level 2 equations may be written to account for variation in the Level 1 regression coefficients for the time-varying variables (e.g., daily stress) if these are included in the model. Thus, at Level 2, we model between individual differences in the values of the growth parameters (intercept and slope) and the regression coefficients for the time-varying variables.

Although we have focused on linear growth, more complex patterns including quadratic growth, growth to an asymptote, and other nonlinear forms of growth may be modeled as the number of measurement waves increases (Cudeck, 1996; Singer & Willett, 2002). In addition, different time-related metrics may be of focal interest such as age or elapsed time since an event (e.g., surgery) or the beginning of a developmental period (see Biesanz et al„ 2003).

Standard growth curve models can also be estimated using structural equation modeling (Muthen & Khoo, 1998; Willett & Sayer, 1994). Mehta and West (2000) noted that the two approaches can both typically be used and produce the same results, but that some applications may be more amenable to one of the approaches. The hierarchical modeling approach discussed in this section may be more flexible in representing some nonlinear forms of growth. In contrast, the structural equation modeling approach often has more flexibility in modeling the measurement structure using multiple indicators of a construct at each time point and in modeling complex relationships between multiple series. Within the structural equation approach, features of autoregressive models (Curran & Bollen, 2001; McArdle, 2001) and features of latent trait-state models (Tisak & Tisak, 2002) can be combined with growth models.

The modeling of change using growth curve modeling described earlier calls for several very strong assumptions regarding the measurement scale. First, the repeated measurements must be made on at least an interval-level scale. Otherwise, the form of growth will be confounded by changes in the size of the measurement unit at each point in the scale. Second, there must be measurement invariance over time—the relationship between the observed measures and the underlying construct must remain constant with the passage of time. For example, items such as pushing and biting might measure physical aggression at age 4. However, at age 16 these items will no longer adequately reflect aggression, precluding meaningful study of change over time. On the other hand, if we measure aggression at age 16 with items like "threaten with gun or knife" and "hit with objects," then the meaning of the construct has changed. (See Patterson, 1995, on developmental change in constructs.) In such cases in which the items on instruments do change over the course of the study (e.g., different items on a measure of math ability in first and fourth grades), there is a need to ensure that the meaning of the construct remains the same. Educational researchers have been successful to some extent in the area of assessing skills and knowledge using vertical equating of overlapping test forms of increasing difficulty levels (see section on vertical equating). Similar techniques are not as well developed for longitudinal studies of psychological and affective constructs.

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