Evaluating Latent Growth Curve Models
I am using a latent growth curve approach to model change in problem behavior across four time points. Although my exploratory analyses suggested that a linear growth function would describe individual trajectories well for nearly all of the adolescents in my sample, the overall model fit (in terms of RMSEA and CFI) is poor. Is my model really that bad? — Signed, Fit to be Tied
The overall fit statistics for latent growth curve models assess the fit of the covariance structure as they do for any structural equation model. Although they are informative, they do not provide information about the fit of the functional form, the fit of individual trajectories, or the fit of the mean structure. Generally, one is interested in either the difference between the group mean trajectory and an individual’s observed trajectory or the difference between an individual’s observed trajectory and the same individual’s model implied (or predicted) trajectory. The former provides a method for detecting outliers whereas the latter provides a method for assessing the fit of individual trajectories.
The former is quantified in the individual loglikelihoods which, when summed over the entire sample, add up to the overall log-likelihood. Further details about individual log-likelihoods in the context latent growth curve models may be found in Coffman and Millsap (2006). For cases with large individual log-likelihood values, the lack of fit may be due to either incorrect functional form or the case may lie far from the group mean trajectory. Examining a plot of the individual’s observed trajectory provides an indication as to the reason for the lack of fit.
The difference between an individual’s observed trajectory and the same individual’s predicted trajectory is more difficult to quantify. This is because in the framework of factor analysis, the individual’s predicted trajectory is based on factor scores. These are not estimated as part of fitting the model and within the factor analysis tradition are considered indeterminate. Thus, there are several methods for obtaining these predicted trajectories, each leading to different predictions. Once a method for computing the predicted trajectories has been chosen, then one could summarize the difference between the predicted trajectory and the observed trajectory as a root mean square residual between the predicted and observed values for an individual.
It is important to note that examining the fit using individual log-likelihoods may be done only when there are no data missing. This is because individuals with more missing data will contribute less to the overall log-likelihood (their individual log-likelihood will be smaller) than those with less missing data. This may lead to the conclusion that a particular individual does not fit as well when in fact the individual’s log-likelihood is larger due to the absence of missing data.
Coffman, D. L., & Millsap, R. E. (2006). Evaluating latent growth curve models using individual fit statistics. Structural Equation Modeling, 13, 1-27.