Concept Architecture
Concept
Theoretically, Growth Mixture Model (GMM) is a latent variable modelling approach that extends growth curve models by allowing the population to comprise multiple unobserved subgroups, each following a distinct longitudinal trajectory. It is founded on finite mixture theory, latent class modelling and structural equation modelling, recognising that individuals may belong to different latent populations with unique patterns of change over time. In health economics, growth mixture models are used to identify heterogeneous patterns of disease progression, treatment response, healthcare utilisation and quality-of-life trajectories.
Mathematically, a growth mixture model combines latent growth curve models with finite mixture distributions. Each latent class has its own growth parameters, while class membership probabilities determine the likelihood that an individual belongs to each subgroup. Model parameters are estimated using maximum likelihood estimation, typically through the Expectation-Maximisation algorithm or related numerical optimisation procedures.
In practice, growth mixture models are implemented using repeated longitudinal measurements, with the number of latent classes selected using statistical criteria such as the Bayesian Information Criterion together with clinical interpretability. Individuals are assigned posterior probabilities of class membership, allowing subgroup-specific estimates of growth trajectories. In health economics, these models are applied to identify patient populations with differing long-term healthcare costs, treatment adherence or health outcome trajectories.
Purpose
Used to identify latent subpopulations with distinct longitudinal trajectories, quantify 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 latent class k
- f(y? | ??) = class-specific likelihood
- ?? = class-specific growth parameters
- K = number of latent classes
Supporting Formulae
Class-specific growth model:
y?? = ??? + ???t + u?? + u??t + �??
Posterior probability of class membership:
P(k | y?) = [�?f(y? | ??)] � [????? �?f(y? | ??)]
Bayesian Information Criterion:
BIC = ?2ln(L) + p ln(n)
where:
- ??? = class-specific intercept
- ??? = class-specific slope
- p = number of estimated parameters
- n = sample size
Related Mathematical Methods
- Growth Curve Models
- Finite Mixture Models
- Latent Class Analysis
- Latent Growth Curve Models
- Maximum Likelihood Estimation
- Expectation-Maximisation Algorithm
- Bayesian Information Criterion
Example
A health economist analyses annual EQ-5D utility scores for 4,500 patients following stroke over six years. A three-class growth mixture model identifies patients with stable recovery, gradual decline and rapid deterioration. Posterior class probabilities exceed 0.90 for most individuals, enabling subgroup-specific estimates of future healthcare costs and resource requirements.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Calculate likelihood components |
| LN | =LN(A2) | Calculate log-likelihood values |
| SUMPRODUCT | =SUMPRODUCT(B2:B5,C2:C5) | Calculate weighted class likelihoods |
| Solver | Maximise log-likelihood | Estimate mixture model parameters |
| LINEST | =LINEST(B2:B101,A2:A101^{1,2},TRUE,TRUE) | Explore polynomial growth trajectories |
VBA (Optional)
Automate growth mixture modelling by estimating latent trajectory classes, calculating posterior membership probabilities and generating subgroup trajectory reports.
Sources
- Muth�n B, Shedden K. Finite Mixture Modeling with Mixture Outcomes Using the EM Algorithm. Biometrics. 1999.
- Muth�n BO. Latent Variable Analysis: Growth Mixture Modeling and Related Techniques for Longitudinal Data.
- Jung T, Wickrama KAS. An Introduction to Latent Class Growth Analysis and Growth Mixture Modeling. Social and Personality Psychology Compass. 2008.
- 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 growth mixture model?
An extension of growth curve modelling allowing unobserved subgroups to follow qualitatively different average trajectories, rather than varying only in degree.
Source: Muthén & Shedden 1999
What does a growth mixture model allow different subgroups to do?
A growth mixture model extends growth curve modelling by allowing unobserved subgroups within a population to follow qualitatively different average trajectories, not merely differing in degree around one common path. It combines the idea of hidden subgroups with the idea of individual variation, so each latent class has its own typical curve and members also vary around it. This is more flexible, and more demanding, than a group-based model that fixes each individual firmly to one trajectory. Letting subgroups follow distinct curves is its purpose. Nagin (2005) discusses such models.
Source: Nagin 2005
How does a growth mixture model work?
A growth mixture model works by fitting a mixture of growth curves, estimating a set number of latent classes, each with its own average trajectory, together with individual variation around each class trajectory, and the probability that each individual belongs to each class. It thus blends finite mixture modelling with growth curve modelling. So a growth mixture model works by identifying subgroups that follow different patterns of change while allowing individuals to vary within each subgroup, estimating the class trajectories, the within-class variation, and class membership probabilities, which captures heterogeneity in change both between and within the latent subgroups.
Source: Muthén & Shedden 1999
How does a growth mixture model differ from a group-based trajectory model?
A growth mixture model allows individual variation around each subgroup's trajectory, so members of a group are similar but not identical, while a group-based trajectory model, in its basic form, assumes individuals within a group share the same trajectory with no within-group variation. The growth mixture model is thus more flexible but more complex. So the two differ in whether within-group variation is allowed, with the growth mixture model permitting individuals to vary around their group's trajectory and the group-based trajectory model treating group members as following a common path, and the choice depends on whether within-subgroup differences in trajectory are expected and the complexity that can be supported.
Source: Muthén & Shedden 1999
When is a growth mixture model used?
A growth mixture model is used when a population is thought to contain distinct subgroups following different patterns of change over time, and when individuals within those subgroups also vary in their trajectories, so that both between- and within-subgroup heterogeneity are of interest. So a growth mixture model is used to identify and describe latent classes with different developmental patterns while allowing individual variation within them, which is valuable for longitudinal data where change is heterogeneous in a structured way, such as different courses of a condition, providing a richer representation than either a single growth curve or groups with no internal variation.
Source: Muthén & Shedden 1999
What are the challenges of growth mixture models?
The challenges of growth mixture models include the difficulty of choosing the number of classes; the risk of identifying spurious classes, since the models can extract classes even when none are real; sensitivity to assumptions and starting values; and computational complexity and convergence problems. So growth mixture models are applied and interpreted with considerable caution, since the latent classes they identify require validation and may not reflect genuine subgroups, and the results depend heavily on modelling choices and assumptions, which is why the number, plausibility, and replicability of the classes are scrutinised rather than accepted from the model fit alone.
Source: Muthén & Shedden 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-074
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