Latent Growth Model

Concept

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Theoretically, Latent Growth Model (LGM) is a structural equation modelling technique used to estimate patterns of individual change over time while simultaneously characterising variability in growth trajectories across a population. The method assumes that repeated measurements are driven by one or more unobserved latent growth factors, typically representing the initial status (intercept) and rate of change (slope). Latent Growth Models exist to quantify longitudinal change, explain between-individual heterogeneity and evaluate factors influencing developmental or clinical trajectories.

Mathematically, a Latent Growth Model represents repeated observations as functions of latent intercept and slope variables, together with measurement error. Growth parameters are estimated using maximum likelihood or Bayesian methods within the structural equation modelling framework. More complex models may incorporate quadratic growth, piecewise growth, multiple latent factors or time-varying covariates. Model adequacy is commonly evaluated using likelihood-based methods and structural equation model fit indices.

In practice, Latent Growth Models are fitted to longitudinal datasets containing repeated measurements collected at multiple time points. In health economics, they are used to analyse changes in health-related quality of life, healthcare costs, disease progression, treatment response and patient-reported outcomes. Estimated growth trajectories support subgroup identification, long-term outcome prediction and economic evaluation of interventions with evolving treatment effects.

Purpose

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Used to model individual and population-level longitudinal change, estimate growth trajectories, explain heterogeneity in repeated measurements and evaluate predictors of change in health economic and clinical research.

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Mathematical Formulae

Primary Formula

Y?t = ??? + ?t??? + �?t

where:

• Y?t = observed outcome for individual i at time t
• ??? = latent intercept
• ??? = latent slope
• ?t = factor loading representing time
• �?t = residual error

Supporting Formulae

Latent intercept:

??? = ?? + ???

Latent slope:

??? = ?? + ???

Covariance matrix:

� = ???? + �

where:

• ? = factor loading matrix
• ? = covariance matrix of latent growth factors
• � = residual covariance matrix

Related Mathematical Methods

• Structural Equation Modelling
• Confirmatory Factor Analysis
• Maximum Likelihood Estimation
• Growth Curve Modelling
• Mixed-Effects Models
• Latent Class Growth Analysis
• Growth Mixture Modelling

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Example

A health economist evaluates EQ-5D utility scores for 600 patients measured at baseline, 6 months, 12 months and 24 months following initiation of a new treatment.

A Latent Growth Model estimates:

• Mean latent intercept (??) = 0.64
• Mean latent slope (??) = 0.05 utility units per year
• Significant variance in both intercept and slope

The results indicate that patients experience an average improvement in health-related quality of life over time, while substantial variation between individual trajectories suggests that treatment effectiveness differs across patients. These estimated trajectories are subsequently incorporated into a long-term economic evaluation.

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Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(B2:E2,$B$1:$E$1,TRUE,TRUE)	Estimate individual linear growth trajectories from repeated measurements.
SLOPE	=SLOPE(B2:E2,$B$1:$E$1)	Calculate the rate of change for an individual patient.
INTERCEPT	=INTERCEPT(B2:E2,$B$1:$E$1)	Estimate the baseline value for a patient's trajectory.
AVERAGE	=AVERAGE(F2:F601)	Summarise estimated intercepts or slopes across the study population.
Solver	Optimise model parameters by minimising the discrepancy between observed and model-implied covariance matrices.	Illustrate simplified estimation of growth parameters.

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VBA (Optional)

A VBA routine can automate estimation of individual growth parameters, summarise longitudinal trajectories and prepare repeated-measures datasets for structural equation modelling software.

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Sources

• Bollen KA, Curran PJ. Latent Curve Models: A Structural Equation Perspective. Wiley.
• Duncan TE, Duncan SC, Strycker LA. An Introduction to Latent Variable Growth Curve Modeling. Routledge.
• Kline RB. Principles and Practice of Structural Equation Modeling. Guilford Press.
• Little TD. Longitudinal Structural Equation Modeling. Guilford Press.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Good Practice Reports.