Analyzing longitudinal data with latent growth modeling

Estimating growth trajectories from repeated measures

Longitudinal data reflect changes over time, differences between individuals, the context at each measurement occasion, and measurement error. Latent growth modeling estimates initial levels and rates of change from repeated measures and examines how they differ across individuals. Explaining the average growth trajectory alongside differences in individual trajectories is central to growth modeling.

Analytical choices connecting growth trajectories and covariates

A methodological paper organized around 12 questions links longitudinal research questions to model components (Curran et al., 2010). In a linear model, fixed effects represent the mean intercept and mean slope, and random effects represent the variance of individual intercepts and slopes. The primary estimation frameworks are multilevel models and structural equation modeling (SEM). In multilevel models, repeated measures are nested within individuals. SEM uses repeated measures as indicators of latent growth factors. Both approaches yield numerically equivalent results under various conditions. Multilevel models accommodate additional nested levels, such as repeated measures–children–classrooms. SEM allows measurement error to be separated through latent variable measurement models. Sample size requirements vary with model complexity and explained variance. The paper presents an applied example with n = 22, but a sample size of at least 100 is generally preferred. Typically, a minimum of 3 measurements per individual is required. Partial missingness that leaves some individuals with only 1~2 observations can also be accommodated.

Analysis begins by determining the functional form of change. Options range from a flat trajectory with only a random intercept to linear, quadratic, and cubic curves. Two or more linear segments can be connected through piecewise linear modeling. Slope factor loadings can be freely estimated in a fully latent curve model. In SEM, model fit is assessed using chi-square tests, RMSEA, CFI, and TLI. Nested models are compared using likelihood ratio tests, and non-nested models using AIC and BIC. Growth curve modeling handles unequally spaced measurements, partial missingness, and non-normal or discrete data. The analytical studies and simulations reviewed in the paper generally found that growth curve modeling had greater power than traditional repeated measures analysis. For partial missingness, the paper introduces direct maximum likelihood and multiple imputation. The number of imputations typically used at the time was 5~10. Both methods are interpreted under MCAR or MAR assumptions. Separate procedures are required for MNAR, where missingness depends on the unobserved values themselves.

Time-invariant covariates (TIC) predict individual differences in intercepts and slopes, and time-varying covariates (TVC) predict repeated measures at each time point while controlling for growth factors. Anxiety measured only at baseline can also serve as a TIC, so a variable’s role also depends on the measurement design. An interaction between a TVC and time or a TIC indicates whether the covariate’s effect varies across time points or individual characteristics. If the predictor variable itself changes systematically, the intercepts and slopes of both constructs can be estimated together using a multivariate growth model. The latent difference score model and autoregressive latent trajectory model address relationships between change and levels across repeated measurement occasions. Multiple-group growth models allow comparisons of means, variances, and other parameters across observed groups. Growth mixture models and latent class growth models explore unobserved trajectory groups under different assumptions and are selected according to the research hypothesis.

Reliability changes when separating time-specific variance

A study comparing first- and second-order latent growth curve models examines how measurement affects estimates of growth trajectories (Geiser et al., 2013). First-order models (FGMs) use a single observed variable, such as a total score, at each time point. Second-order models (SGMs) construct a common latent factor from at least two indicators at each time point and then estimate that factor’s growth trajectory. In Latent State-Trait Theory (LST theory), observed scores are decomposed into a trait component, a state residual reflecting situations and person-situation interactions, and measurement error. In an FGM with a single indicator, the state residual and random measurement error are not separated. Interpreting the entire residual as measurement error assumes that time-specific variance is zero. An SGM represents trait changes alongside time-specific variance, assuming that the indicators at each time point are unidimensional. Measurement invariance and indicator method effects can also be examined. In LST theory, reliability is the proportion of observed variance explained by the latent state and therefore includes time-specific variance. In a basic SGM, reliability is calculated as the sum of CC, the proportion explained by the trait and trait change, and OS, the proportion attributable to time-specificity.

In the simulation, data were generated using an SGM with two tau-parallel indicators at each of four time points, with the intercept-slope covariance set to 0. Monte Carlo samples of N = 1,000 were generated 1,000 times for each condition, with conditions ranging from zero to high time-specific variance. Both models were estimated using maximum likelihood in Mplus 6.1. The arithmetic mean of the two indicators was entered into the FGM. The true reliability of the composite score was calculated using the Spearman-Brown formula. The SGM reproduced the true reliability almost exactly in all conditions. The FGM also yielded unbiased estimates when time-specific variance was zero. In condition 4, where time-specific variance was high, the relative bias in FGM reliability reached −64.91 %. While the true reliability increased from .7805 to .9412 in condition 5, the FGM estimate remained at approximately .68, and the SGM estimate increased from .7795 to .9412. In conditions 2–4, the true reliability remained constant at .8889, but the FGM estimate increased over time. This reliability comparison is best interpreted in light of the conditions under which the data were generated: a linear growth model with two tau-parallel indicators across four time points and a zero intercept-slope covariance.

The empirical application used data from Proyecto: La Familia, which surveyed the health and adjustment of Mexican American children every 2 years beginning in the 5th grade. The analysis included 591 mothers and 323 fathers with data at all three time points. It excluded 4 positive items from the 20-item CES-D, a 4-point scale. The average of the 16 negative items was entered into the FGM, and two test halves consisting of 8 items each were entered into the SGM. A method factor for the second test half was added to the mothers’ SGM. The fathers’ model allowed loadings and intercepts to differ between indicators while constraining them to be equal over time. The SGM fit indices were CFI = .997, RMSEA = .03 for mothers and CFI = .992, RMSEA = .05 for fathers. The mean slope was 0.004 (p = .75) for mothers and .03 (p = .05) for fathers. The slope variance was 0.01 for both groups and was not significant in either group. Growth parameters were identical or similar in the FGM. FGM reliability was .46–.56 for mothers and .49–.63 for fathers, whereas indicator-specific reliability in the SGM was .86–.91 and .76–.86, respectively. Time-specific effects accounted for 40–52 % of latent state variance for mothers and 30–45 % for fathers.

Exploring subgroups with different parameters in growth models

SEM Trees explores groups whose parameters vary across covariate values while retaining a theoretically specified template model (Brandmaier et al., 2013). An SEM fitted to the full dataset is compared with models fitted to groups defined by a candidate covariate. The procedure is repeated within the selected groups. With m free parameters and k groups formed by a split, the likelihood ratio test has (k − 1)m degrees of freedom. The paper presents Bonferroni correction and cross-validation (CV) as approaches to multiple testing. A split is accepted when the average improvement in fit for the hold-out data in 5-fold or 10-fold CV is greater than zero. A two-step procedure addresses selection bias favoring variables with many split points. It first identifies the optimal cutpoint for each variable and then selects among the variables. In a simulation with uninformative variables having 2, 4, and 10 categories, bias under this procedure was negligible. The OpenMx-based R package semtree implements tree generation, evaluation, pruning, parameter constraints, and tests of measurement invariance. Missingness in observed variables is handled using FIML. Cases with missing classification covariates remain at the corresponding internal node.

The longitudinal application used WISC data from 204 children assessed at approximately ages 6, 7, 9, and 11. Raw scores from 8 subtests in the verbal and performance domains were converted to a 0–100 percent correct scale and averaged. The slope loadings of a linear latent growth curve model were set to 0, 1, 3, and 5 to match the measurement intervals. The model had 6 free parameters: the means and variances of the intercept and slope, the measurement error, and the intercept-slope correlation. A total of 28 candidate binary splits were constructed from sex, age at first measurement, and parental education level. The significance level for Bonferroni correction was set at .05, and the minimum sample size for terminal nodes was set at 20. Both methods identified a split based on whether the parents had graduated from high school, and CV did not select sex. In all 100 bootstrap runs, the likelihood of the SEM Tree for cases excluded from the resampling was higher than that of the single growth curve model. The first split was based on mothers’ education 33 times and fathers’ education 67 times. The paper interpreted this in relation to the high correlation between the parental education variables.

Another application analyzed 1,680 individuals aged 18 and older from a total of 1,880 in WAIS–R data collected in 1976 and 1980. A single-factor model of verbal intelligence was constructed using Information, Comprehension, Similarities, and Vocabulary. The analysis examined 70 candidate binary splits derived from education, occupation, and age groups. In the analysis testing weak measurement invariance, groups with 0–7 years and 16 or more years of education were split off first. Further splits by occupation and years of education emerged among the remaining 1,335 individuals. An analysis without invariance constraints explored intelligence profiles with different factor loadings across groups. The interpretation of latent means depends on strong invariance, which constrains loadings and indicator intercepts to equality. The parameter differences reflected in the splits depend on global and local constraints. Assessing tree stability requires considering small groups, imbalanced splits, and a greedy algorithm that selects the best split at each step. Bootstrap evaluation and evaluation with independent data help assess tree stability.

Reading notes

When examining relationships between feedback experiences and performance in HRD and medical education, measurement timing and indicator composition depend on whether the analysis uses the initial experience to explain the subsequent rate of change or estimates changes in both processes simultaneously. Repeatedly measuring performance during a clerkship and collecting multiple indicators at each time point allows researchers to distinguish long-term change from variance shared at a specific time point. Connecting estimated time-specific variance to particular feedback or clinical experiences requires both measuring those experiences and specifying their relationships within the model. The interpretation of group differences in SEM Trees depends on which differences in slope, initial level, and variance are allowed across groups.

References

• Brandmaier, A. M., von Oertzen, T., McArdle, J. J., & Lindenberger, U. (2013). Structural equation model trees. Psychological Methods, 18(1), 71–86. https://doi.org/10.1037/a0030001
• Curran, P. J., Obeidat, K., & Losardo, D. (2010). Twelve frequently asked questions about growth curve modeling. Journal of Cognition and Development, 11(2), 121–136. https://doi.org/10.1080/15248371003699969
• Geiser, C., Keller, B. T., & Lockhart, G. (2013). First- versus second-order latent growth curve models: Some insights from latent state-trait theory. Structural Equation Modeling: A Multidisciplinary Journal, 20(3), 479–503. https://doi.org/10.1080/10705511.2013.797832