Growth curve modeling (also called latent growth modeling) provides a powerful framework for studying change over time. Rather than treating change as a nuisance to be averaged away, these models treat individual trajectories as the primary objects of analysis. Each person's trajectory is characterized by latent variables — an intercept (starting point) and one or more slope factors (rate of change) — and the model estimates both the average trajectory and the extent to which individuals differ from that average.
The Linear Growth Model
η₀_i = α₀ + ζ₀_i (intercept = mean + individual deviation)
η₁_i = α₁ + ζ₁_i (slope = mean + individual deviation)
λ_t = time scores: e.g., 0, 1, 2, 3 for four occasions
Var(ζ₀) = ψ₀₀, Var(ζ₁) = ψ₁₁, Cov(ζ₀,ζ₁) = ψ₀₁
In the SEM framework, the observed scores at each time point serve as indicators of the latent growth factors. Factor loadings on the intercept factor are all fixed to 1.0 (every observation includes the initial level). Factor loadings on the slope factor are fixed to values reflecting the time metric — typically 0, 1, 2, 3 for equally spaced assessments. The latent means α₀ and α₁ capture the average intercept and slope. The latent variances ψ₀₀ and ψ₁₁ capture individual differences in initial level and rate of change. The covariance ψ₀₁ reveals whether initial status predicts the rate of change.
Nonlinear and Conditional Models
The framework extends naturally to nonlinear growth by modifying the time scores. A quadratic model adds a third latent variable with loadings 0, 1, 4, 9 (squared time). A freed-loading model estimates some time scores rather than fixing them, allowing the data to reveal the functional form of change. Piecewise models use different slopes for different phases (e.g., a treatment phase and a follow-up phase), connected at a knot point.
η₁_i = α₁ + γ₁₁X_i + ζ₁_i
γ₀₁ = effect of predictor X on intercept
γ₁₁ = effect of predictor X on slope (predicting rate of change)
Growth mixture models (GMM) combine latent growth curves with latent class analysis, allowing for the possibility that the population contains distinct subgroups following qualitatively different trajectories. For instance, in studying recovery from depression, some individuals may show rapid improvement, others gradual improvement, and others chronic symptoms. GMM identifies these latent classes and estimates the trajectory parameters within each. Class membership can be predicted from covariates and can predict distal outcomes. The number of classes is selected using BIC, entropy, and the Lo-Mendell-Rubin test.
Advantages and Applications
Compared to traditional repeated-measures ANOVA, growth curve modeling offers several advantages: it models individual differences in change (not just average change), handles missing data naturally through full-information maximum likelihood, does not require equal spacing of assessments, and allows time-varying and time-invariant covariates. Compared to multilevel modeling — which can estimate equivalent models — SEM-based growth models offer the advantage of incorporating latent variables and multiple outcomes in a single model.
Growth curve models are widely applied in developmental psychology (tracking cognitive development, language acquisition, behavior problems), clinical psychology (modeling treatment response), educational research (learning trajectories), and aging research (cognitive decline). They provide a mathematically rigorous framework for the fundamental psychological question of how and why people change over time, transforming longitudinal data analysis from a description of average trends to an examination of the individual differences that drive those trends.