27 Predictors in Longitudinal Growth Models
27.1 Tips for growth models
Start with an unconditional growth model, i.e., don’t include any level-1 or level-2 predictors. This model provides useful empirical evidence for determining a proper specification of the individual growth equation and baseline statistics for evaluating more complicated level-2 models.
The nature of the predictor in longitudinal analysis determines where it gets added to the model: Time-invariant predictors always go in level-2 (subject level) model Time-varying predictors can go in level-1 and/or level-2. The level of the predictor dictates which variance component it seeks to describe: Level-2 describes level-2 variances and Level-1 describes level-1 variances. Although the order in which you add these predictors (in a series of successive models) may not ultimately matter, general practice is to add level-2 (time-invariant) predictors first.
How to decide where to add predictors? One strategy:
First fit an unconditional (i.e. no predictors) random intercept model. This isn’t really predictive, but we can use it as a baseline model that partitions variance into between and within-person variances. Singer & Willett (2003) call this the “unconditional means model”.
Calculate the ICC
If most of the variance is between-persons in the random intercept (level-2), then you’ll use person-level predictors to reduce that variance (i.e., account for inter-person differences)
If most of the variance is within-person (level-1 residual variance), you’ll need time-level predictors to reduce that variance (i.e. account for intra-person differences)
Because the time-specific subscript t can only appear in the level-1 model, all time-varying predictors must appear in the level-1 individual growth model. That is, person-specific predictors that vary over time appear at level-1, not level-2. Time-invariant predictors go in level-2. Furthermore, because they are time-invariant, this means they have no within-person variation to allow for a level-2 residual; thus, the level-2 growth rate parameter corresponding to this time-invariant predictor will not have an error term (i.e. it’s assumed to be zero). Interpretation wise, this assumes the effect of a person-specific effect is constant across population members. For a time-varying predictor, however, the associated level-2 growth parameter equation would have a residual term. This allows the effect of the time-varying predictor to vary randomly across the individuals in the population.
With only a few measurement points per person, we often lack sufficient data to estimate many variance components. Thus, it’s suggested that we resist the temptation to automatically allow the effects of time-varying predictors to vary at level-2 unless you have a good reason, and enough data, to do so.
So far in class, we’ve seen person-specific variables appear in level-2 submodels as predictors for level-1 growth parameters. You might therefore think that substantive predictors must always appear at level-2, but this isn’t true!
How inclusion of predictors affect variance components: Generally, when we include time-invariant predictors:
the level-1 variance component, \(\sigma^2_e\), remains pretty stable because time-invariant predictors can’t explain any within-person variation
the level-2 variance components, \(\tau_{00}\) and \(\tau_{01}\), will decrease if the time-invariant predictors explain some of the between-person variation in initial status or rates of change, respectively.
When we include time-varying predictors:
both level-1 and level-2 variance components might be affected because time-varying predictors vary both within a person and between people
we can interpret the resulting decrease in the level-1 variance component as amount of variation in the outcome explained by the time-varying predictors; however, it isn’t meaningful to interpret subsequent changes in level-2 variance components because adding the time-varying predictor changes the meaning of the individual growth parameters, which consequently alters the meaning of the level-2 variances, so it doesn’t make sense to compare the magnitude of these level-2 variances across successive models.