Trajectory Analysis

Concept

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Theoretically, Trajectory Analysis is a statistical modelling approach used to identify distinct patterns of change in an outcome over time within a population. It assumes that the study population comprises one or more latent subgroups, each following a characteristic developmental trajectory. The method is founded on finite mixture modelling and latent class theory and is widely used in longitudinal health research to characterise heterogeneous disease progression, healthcare utilisation, treatment adherence and cost trajectories.

Mathematically, trajectory analysis represents longitudinal observations using a finite mixture of probability distributions, with each latent trajectory described by a parametric function of time. Model parameters are estimated by maximum likelihood, typically using the Expectation-Maximisation algorithm or direct numerical optimisation. Polynomial functions of time are commonly used to represent trajectory shapes, while posterior probabilities quantify the probability of class membership for each individual.

In practice, trajectory analysis is implemented using repeated longitudinal observations collected from individuals over multiple time points. Competing models with different numbers of trajectory groups are compared using information criteria such as the Bayesian Information Criterion (BIC), and classification quality is assessed using posterior probabilities and entropy. In health economics, trajectory analysis is applied to identify patterns of healthcare expenditure, treatment persistence, quality-of-life progression and resource utilisation across patient populations.

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Purpose

Used to identify and characterise distinct longitudinal patterns within heterogeneous populations, supporting patient stratification, outcome prediction, resource planning and evaluation of interventions in health economic and outcomes research.

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Mathematical Formulae

Primary Formula

L = ?? ?? �? f(y? | ??)

where:

• �? = probability of membership in trajectory group j
• f(y? | ??) = likelihood of the observed trajectory for individual i given trajectory parameters ??

Supporting Formulae

Posterior probability of group membership:

P(j | y?) = �? f(y? | ??) / ?? �? f(y? | ??)

Polynomial trajectory model:

E(Y?? | j) = ??? + ???t + ???t� + ?

Bayesian Information Criterion:

BIC = ?2ln(L) + k ln(n)

Related Mathematical Methods

• Finite Mixture Models
• Latent Class Analysis
• Group-Based Trajectory Modelling
• Growth Mixture Modelling
• Maximum Likelihood Estimation
• Expectation-Maximisation Algorithm
• Bayesian Information Criterion

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Example

A health economist analyses annual healthcare costs for 4,000 patients over five years. A trajectory analysis identifies three latent groups: a stable low-cost group (68%), a gradually increasing-cost group (24%), and a persistently high-cost group (8%). The three-group model has the lowest BIC and average posterior probabilities exceeding 0.85, indicating good classification accuracy.

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Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(B2:F2,$B$1:$F$1,TRUE,TRUE)	Estimate polynomial trends for exploratory longitudinal analysis.
INDEX	=INDEX($H$2:$H$4001,MATCH(MAX(I2:K2),I2:K2,0))	Assign individuals to the trajectory group with the highest posterior probability.
MAX	=MAX(I2:K2)	Identify the highest posterior probability for each individual.
IF	=IF(MAX(I2:K2)>=0.70,"Well classified","Review")	Flag classification quality using posterior probabilities.

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VBA (Optional)

Automate the import, summarisation and classification of longitudinal patient records following estimation in specialised trajectory modelling software.

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Sources

• Nagin DS. Group-Based Modeling of Development. Harvard University Press.
• Nagin DS, Odgers CL. Group-based trajectory modeling in clinical research. Annual Review of Clinical Psychology.
• Jones BL, Nagin DS. A note on a Stata plugin for estimating group-based trajectory models. Sociological Methods & Research.
• Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.