Latent Growth Modeling

Fixed versus Random Effects

To understand growth modeling, one needs to understand the concepts of fixed effects and random effects. In ordinary least-squares regression, a fixed intercept and a slope for each predictor are estimated. In growth modeling, it is often the case that each person has a different intercept and slope, which are called random effects. Consider a growth model of marital conflict reported by the mother across the first 12 months after the birth of a couple's first child. Conflict might be measured on a 0 to 10 scale right after the birth and then every 2 months for the first year. There are 7 time points (0, 2, …, 12) and a regression of the conflict scores on time might be done. Hypothetical results appear in Figure 1 in the graph labeled Both Intercept and Slope Are Fixed. The results reflect an intercept α0 = 2.5 and a slope α1 = 0.2. Thus, the conflict starts with an initial level of 2.5 and increases by 0.2 every 2 months. By the 12th month, the conflict would be moderate, 2.5 + 0.2×12=4.9. These results are fixed effects. However, women might vary in both their intercept and their slope.

In contrast, the graph in Figure 1 labeled Random Intercept and Fixed Slope allows for differences in the initial level and results in parallel lines. Mother A has the same intercept and slope as the fixed-effects model, 2.5 and 0.2, respectively. All three have a slope of 0.2, but they vary in their intercept (starting point). This random intercept model, by providing for individual differences in the intercept, should fit the data for all the mothers better than the fixed model, but the requirement that all lines are parallel might be unreasonable. An alternative approach is illustrated in the graph labeled Fixed Intercept and Random Slope. Here, all the mothers have a fixed initial level of conflict, but they are allowed to have different slopes (growth rates).

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