Concept Architecture
Concept
Theoretically, Growth Curve Model is a longitudinal statistical model used to estimate individual patterns of change over time while simultaneously describing variation in trajectories across a population. It is founded on multilevel modelling and structural equation modelling, allowing repeated observations to be represented as functions of latent growth parameters. In health economics, growth curve models are used to evaluate disease progression, health-related quality of life, healthcare utilisation and treatment response over time.
Mathematically, growth curve models represent repeated measurements as functions of latent intercepts, slopes and, where appropriate, higher-order growth parameters. Fixed effects estimate the average population trajectory, while random effects quantify individual deviations from that average. Parameters are typically estimated using maximum likelihood or restricted maximum likelihood methods within a mixed-effects modelling framework.
In practice, growth curve models are implemented using repeated measurements collected from individuals over multiple time points. Linear, quadratic or more complex functional forms may be specified depending on the expected pattern of change. In health economics, these models are applied to estimate long-term treatment effects, model patient recovery trajectories and analyse longitudinal health outcomes while accounting for within-patient correlation.
Purpose
Used to estimate individual and population-level trajectories over time, quantify longitudinal change and evaluate treatment effects using repeated measurements in health economic studies.
Mathematical Formulae
Primary Formula
y?? = ?? + ??t + u?? + u??t + �??
where:
- y?? = outcome for individual i at time t
- ?? = population intercept
- ?? = population growth rate
- u?? = individual-specific random intercept
- u?? = individual-specific random slope
- �?? = residual error
Supporting Formulae
Quadratic growth model:
y?? = ?? + ??t + ??t� + u?? + u??t + �??
Variance components:
Var(y??) = ZGZ? + R
where:
- Z = random-effects design matrix
- G = covariance matrix of random effects
- R = residual covariance matrix
Related Mathematical Methods
- Linear Mixed Models
- Multilevel Models
- Random Effects Models
- Maximum Likelihood Estimation
- Restricted Maximum Likelihood
- Latent Growth Curve Models
- Hierarchical Linear Models
Example
A health economist evaluates EQ-5D utility scores measured every six months for five years following cardiac surgery. A growth curve model estimates an average annual improvement of 0.035 utility units while allowing each patient to have their own baseline utility and rate of recovery. Random slopes demonstrate substantial heterogeneity in long-term health improvement between patients.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B101,A2:A101,TRUE,TRUE) | Estimate simple linear growth trends |
| TREND | =TREND(B2:B101,A2:A101,A102:A110) | Predict longitudinal outcomes |
| SLOPE | =SLOPE(B2:B101,A2:A101) | Estimate average rate of change |
| INTERCEPT | =INTERCEPT(B2:B101,A2:A101) | Estimate baseline outcome |
| Solver | Minimise residual sum of squares | Estimate growth model parameters in simplified implementations |
VBA (Optional)
Automate longitudinal growth analyses by estimating repeated-measures trajectories, generating individual growth plots and exporting model summaries.
Sources
- Singer JD, Willett JB. Applied Longitudinal Data Analysis.
- Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis.
- Verbeke G, Molenberghs G. Linear Mixed Models for Longitudinal Data.
- Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a growth curve model?
A modelling approach describing how an outcome changes over time within individuals, estimating both an average trajectory and individual variation around it.
Source: Bollen & Curran 2006
What does a growth curve model describe about change within individuals?
A growth curve model describes how an outcome changes over time within individuals, fitting a trajectory to each person's repeated measurements. It estimates an average curve for the group, the typical course of change, together with how much individuals depart from it in their starting point and their rate of change. This captures both the shared pattern and the personal variation around it, which suits data tracking recovery, decline, or development over repeated visits. Modelling individual paths around an average is its purpose. Kirkwood and Sterne (2003) describe such models.
Source: Kirkwood & Sterne 2003
How does a growth curve model work?
A growth curve model works by modelling each individual's outcome over time with a trajectory, such as a straight line or curve, whose parameters, like the intercept and slope, vary between individuals around group averages. It estimates the average trajectory and the variability of the individual trajectories, often within a mixed model or structural equation framework. So a growth curve model works by fitting individual trajectories that share a common form but differ in their parameters, estimating both the mean trajectory and the between-individual variation, which allows the average course of change and the extent and predictors of individual differences in that course to be examined.
Source: Bollen & Curran 2006
When is a growth curve model used?
A growth curve model is used to study change over time in longitudinal data, when the interest is in how an outcome develops within individuals, the average pattern of change, and how and why individuals differ in their trajectories. It suits repeated measurements on the same individuals. So a growth curve model is used to analyse individual development or change, such as the course of a symptom, ability, or measure over time, which is valuable when both the typical trajectory and the individual variation around it, and its predictors, are of interest, making it a standard tool for longitudinal analysis.
Source: Bollen & Curran 2006
What does a growth curve model estimate?
A growth curve model estimates the average trajectory of the outcome over time, described by parameters such as an average intercept and slope, and the variation of individuals around this average, capturing how much people differ in their starting levels and rates of change. It can also relate these individual differences to predictors. So a growth curve model estimates both the mean pattern of change and the between-individual variability in that pattern, and often the factors that explain the variability, providing a description of how an outcome develops on average and how and why individuals diverge from that average over time.
Source: Bollen & Curran 2006
How does a growth curve model handle individual variation?
A growth curve model handles individual variation by allowing each individual's trajectory parameters, such as intercept and slope, to differ from the group averages, modelling these differences as random variation around the mean trajectory. This is typically done through random effects in a mixed model framework. So a growth curve model handles individual variation by treating individuals' trajectories as varying around a common average, estimating the extent of this variation and, where predictors are included, what accounts for it, which distinguishes it from a model assuming a single trajectory for everyone and allows individual differences in change to be quantified and explained.
Source: Bollen & Curran 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/growth-curve-model
- Term code
- HE-ES-SA-073
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