Concept Architecture
Concept
Theoretically, Group-Based Trajectory Model (GBTM) is a finite mixture modelling approach used to identify distinct latent groups of individuals who follow similar developmental or longitudinal trajectories over time. It is founded on finite mixture theory and latent class modelling, assuming that the population comprises a finite number of unobserved trajectory groups with different underlying patterns of change. In health economics, GBTMs are used to characterise heterogeneous disease progression, healthcare utilisation, medication adherence and long-term treatment response.
Mathematically, GBTMs model repeated observations using group-specific polynomial functions of time combined with latent class membership probabilities. The likelihood is expressed as a weighted mixture of trajectory-specific likelihoods, with parameters estimated by maximum likelihood using the Expectation-Maximisation algorithm or related optimisation methods. Individuals are assigned posterior probabilities of membership for each trajectory group.
In practice, GBTMs are implemented by selecting the number of trajectory groups, specifying the polynomial order for each trajectory and estimating model parameters using longitudinal data. Model selection commonly relies on the Bayesian Information Criterion (BIC), posterior classification probabilities and clinical interpretability. In health economics, GBTMs are applied to identify patient subgroups with distinct healthcare costs, disease progression or treatment adherence patterns.
Purpose
Used to identify latent longitudinal subgroups, model heterogeneous patterns of change over time and support subgroup-specific analyses in health economic research.
Mathematical Formulae
Primary Formula
L = ?? ????? �? f(y? | ??)
where:
- L = likelihood
- �? = probability of membership in trajectory group k
- f(y? | ??) = trajectory-specific probability distribution
- ?? = parameters for trajectory group k
- K = number of trajectory groups
Supporting Formulae
Trajectory model:
??(t) = ??? + ???t + ???t� + ?
Posterior probability of group membership:
P(k | y?) = [�?f(y? | ??)] � [????? �?f(y? | ??)]
Bayesian Information Criterion:
BIC = ?2ln(L) + p ln(n)
where:
- ??(t) = expected trajectory for group k
- ? = trajectory coefficients
- p = number of estimated parameters
- n = sample size
Related Mathematical Methods
- Finite Mixture Models
- Latent Class Analysis
- Maximum Likelihood Estimation
- Expectation-Maximisation Algorithm
- Bayesian Information Criterion
- Longitudinal Data Analysis
Example
A health economist analyses annual healthcare expenditure for 6,000 patients over eight years. A four-group trajectory model identifies stable low-cost, increasing-cost, decreasing-cost and persistently high-cost patient groups. Posterior probabilities exceed 0.85 for most individuals, indicating good classification accuracy and supporting subgroup-specific resource planning.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Calculate exponential likelihood components |
| LN | =LN(A2) | Compute log-likelihood values |
| SUMPRODUCT | =SUMPRODUCT(B2:B5,C2:C5) | Calculate weighted trajectory probabilities |
| Solver | Maximise log-likelihood | Estimate trajectory model parameters |
| LINEST | =LINEST(B2:B101,A2:A101^{1,2},TRUE,TRUE) | Estimate polynomial trajectory coefficients for exploratory analyses |
VBA (Optional)
Automate trajectory modelling by estimating latent group parameters, calculating posterior membership probabilities and generating trajectory plots for longitudinal health economic data.
Sources
- Nagin DS. Group-Based Modeling of Development. Harvard University Press.
- Nagin DS. Group-Based Trajectory Modeling: An Overview. Annals of Nutrition and Metabolism. 2014.
- Jones BL, Nagin DS, Roeder K. A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories. Sociological Methods & Research. 2001.
- McLachlan G, Peel D. Finite Mixture Models.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a group-based trajectory model?
A technique identifying distinct subgroups following similar patterns of change over time, without assuming everyone follows the same single trajectory.
Source: Nagin 1999
What does a group-based trajectory model identify over time?
A group-based trajectory model identifies distinct subgroups within a population that follow similar patterns of change over time, rather than assuming everyone moves along one common path. It sorts individuals into a small number of typical trajectories, such as those whose symptoms steadily improve, stay flat, or worsen, estimating both the shape of each path and how many people follow it. This reveals heterogeneity in how a condition unfolds that a single average would hide. Finding typical paths of change is its purpose. Nagin (2005) describes this technique.
Source: Nagin 2005
How does a group-based trajectory model work?
A group-based trajectory model works by fitting a finite mixture of trajectories to longitudinal data, estimating a set number of latent groups, each with its own average trajectory over time, and the probability that each individual belongs to each group. The number of groups and the shape of their trajectories are estimated from the data. So a group-based trajectory model works by identifying distinct groups whose members follow similar patterns of change, assigning individuals to groups probabilistically, which allows the different developmental or temporal patterns present in a heterogeneous population to be described and the proportion following each pattern to be estimated.
Source: Nagin 1999
When is a group-based trajectory model used?
A group-based trajectory model is used when a population is thought to contain distinct subgroups that follow qualitatively different patterns of change over time, such as different courses of a symptom or behaviour, and the aim is to identify and characterise these patterns. So a group-based trajectory model is used to reveal and describe heterogeneous developmental or temporal patterns, which is valuable when a single average trajectory would obscure meaningful differences between subgroups, allowing distinct courses, such as improving, stable, or worsening groups, to be identified and their sizes and characteristics estimated from longitudinal data.
Source: Nagin 1999
How does a group-based trajectory model differ from a growth curve model?
A group-based trajectory model represents change as a set of distinct latent groups each with its own average trajectory, while a standard growth curve model assumes a single average trajectory with continuous individual variation around it. The trajectory model captures qualitatively different subgroup patterns, whereas the growth curve model captures graded variation about one path. So the two differ in how they represent heterogeneity in change, with the group-based trajectory model using discrete groups and the growth curve model a continuous distribution of individual trajectories, and the choice depends on whether change is better described by distinct subgroups or by variation around a common trajectory.
Source: Nagin 1999
What are the limitations of a group-based trajectory model?
The limitations of a group-based trajectory model include the difficulty of choosing the number of groups, which affects the results; the risk of identifying groups that are statistical artefacts rather than real subpopulations; sensitivity to modelling assumptions; and the tendency to impose discrete groups on what may be continuous variation. So a group-based trajectory model is applied and interpreted with caution, since the trajectory groups it identifies require validation and may not correspond to genuinely distinct subpopulations, and the number and shape of the groups depend on modelling choices, which is why the plausibility and replicability of the identified trajectories are assessed rather than assumed.
Source: Nagin 1999
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-072
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