Growth Curve

Growth Curves in Longitudinal Research

One of the primary interests in longitudinal research is to describe patterns of change over time. For example, researchers might be interested in investigating depressive symptoms. Possible questions include the following: Do all people display similar initial levels of depressive symptoms (similar intercepts)? Do some people tend to have a greater increase or decrease in depressive symptoms than others (different slopes)? Separate growth curves can be estimated for each individual, using the following equation:

That is, the outcome variable Y for individual i is predicted by an intercept of β0i and a slope of β1i. The error term at each point in time, ∊it, represents the within-subject error. Each individual will have different growth parameters (i.e., different intercept and slope), and these individual growth curves are used to estimate an aggregate mean and variance for the group intercept and the group slope (see Figure 1). The intercept, also called the initial level or constant, represents the value of the outcome variable when the growth curve or change is first measured (when time = 0). The aggregate intercept determines the average outcome variable for all samples, whereas the aggregate slope indicates the average rate of change for the outcome variable for each incremental time point (e.g., year, month, or day).

The growth curve can be positive (an incline) or negative (a decline), linear (representing straight line), or nonlinear. Three or more repeated observations are generally recommended for growth curve analysis. Two waves of data offer very limited information about change and the shape of the growth curves. With three or more waves of data, a linear growth curve can be tested. With four or more waves of data, higher order polynomial alternatives (e.g., quadratic, cubic, logarithmic, or exponential) can be tested. A higher order polynomial growth curve is useful to describe patterns of change that are not the same over time. For example, a rapid increase in weight, height, and muscle mass tends to occur during the first 3 years of childhood; becomes less rapid as children reach their third birthday; and increases rapidly again as they reach puberty. This pattern illustrates the nonlinear trajectories of physical growth that can be captured with additional data points.

In order to measure quantitative changes over time, the study outcome variable must also change continuously and systematically over time. In addition, for each data point, the same instrument must be used to measure the outcome. Consistent measurements help ensure that the changes over time reflect growth and are not due to changes in measurement.

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