Growth Curve Model

Growth Curve Model: Concept Architecture

Follow an outcome across repeated observations

A growth curve model describes how an outcome changes over time and how trajectories vary between people or groups. This page explains the time origin, average and individual change, a simple fitted example, and the checks needed before using a trajectory in health economic analysis.

Define the clock and repeated outcome

Specify whether time means age, months since treatment, or another meaningful scale, and identify the outcome and its units. Repeated observations from the same person are correlated, so their timing and within-person pattern matter. A model may estimate an average trajectory and allow each person's starting level or rate of change to differ.

A common linear mixed-effects representation for person $i$ at time $t$ is:

$$Y_{it}=(\beta_0+b_{0i})+(\beta_1+b_{1i})t+\varepsilon_{it}.$$

The fixed coefficients $\beta_0$ and $\beta_1$ describe the population's mean intercept and slope; $b_{0i}$ and $b_{1i}$ represent person-specific deviations, and $\varepsilon_{it}$ is observation-level error. Their distributions and correlations require assumptions. A methodological account of repeated-measures analysis explains random intercepts and slopes, while research on individual quality-of-life growth demonstrates their use in health data.

Interpret a simple fitted trajectory

Suppose an illustrative health-related quality-of-life instrument runs from 0 to 100, and its fitted mean score is 70 points at treatment start and rises by 3 points per month over the observed first three months. The model's linear mean predictions are 70 at month 0 and $70+3(3)=79$ at month 3. A particular patient could start below 70 and improve more slowly because their intercept and slope differ from the mean.

The points are illustrative, and the actual instrument and direction must be stated for a real study. Extending the same line to month 24 would predict 142 points, which is impossible on the illustrative 0-to-100 scale. A good fit over three months is therefore not evidence that linear improvement continues for two years.

Allow clinically plausible shapes

Recovery, deterioration, and treatment response often change pace. Quadratic terms, splines, piecewise slopes, or other appropriate time functions can represent curvature or a change around a clinically meaningful point. More flexible forms need enough data and should respect the outcome's bounds and the decision's intended prediction period.

Growth curve modelling includes related implementations such as mixed-effects and latent growth curve models, which express individual trajectories with different parameterisations and estimation assumptions. They should not be equated with a cross-sectional regression of different people at different ages without considering cohort and selection effects. A longitudinal health-related quality-of-life cohort study used growth curves to study age-related trajectories and addressed death and institutionalization.

Compare groups at the same times

In a randomized study, a group-by-time term can estimate whether mean trajectories differ between treatment groups over the observed period. The contrast must be defined at a clinically meaningful time or as a justified summary over time; a difference in slopes alone can be misleading when curves are nonlinear or cross. Baseline adjustment, treatment switching, adherence, and post-randomization missingness affect the causal interpretation.

For observational groups, an apparent trajectory difference may also reflect confounding, selection, or different visit patterns. A fitted model handles correlation among repeated measures but does not create exchangeability between treatments. Report group-specific predicted trajectories and their uncertainty rather than only a coefficient whose units and time origin are obscure.

Address dropout and changing patient mix

Irregular visit times can be accommodated in a suitable longitudinal model, but missing outcomes are not automatically harmless. A conventional likelihood-based mixed model commonly relies on a missing-at-random assumption conditional on variables included in the analysis; if people with worsening health die, withdraw, or stop responding for reasons not captured by those variables, the observed or modelled trajectory may misrepresent the target population. State the estimand and missing-data assumptions, and consider joint, pattern-mixture, or other sensitivity analyses when dropout is informative.

Distinguish within-person change from differences between people who remain under observation. An oncology quality-of-life growth model considers irregular visits and missing measurements, while later work on informative dropout shows why a simple linear trajectory can fail in palliative settings. The model and estimand must match the health-economic question.

Connect a trajectory to model outcomes

A health economic model may use predicted utility, symptoms, resource use, or another repeated outcome at successive times. Check whether the fitted scale can be translated legitimately into health-state utilities or costs, and integrate over compatible intervals when estimating quality-adjusted life years. Propagate uncertainty in fixed effects, individual variation when relevant, and structural choices about the curve.

Report the time origin, observed range, measurement schedule, population, fixed and random terms, residual assumptions, fit checks, dropout handling, and extrapolation limits. A growth curve estimates change under the study and model assumptions; its slope alone does not establish a causal treatment effect or justify a lifetime projection.