Mixed Effects Model

Concept

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Theoretically, a Mixed Effects Model is a statistical model that combines fixed effects, representing population-average relationships, with random effects, representing subject-specific or cluster-specific variation. The method extends classical regression by explicitly modelling the hierarchical or correlated structure of repeated measurements and clustered data. Mixed effects models exist to produce valid statistical inference when observations are not independent, allowing both within-group and between-group sources of variability to be estimated simultaneously.

Mathematically, a mixed effects model represents the outcome as the sum of fixed-effect predictors, random-effect components and residual error. Fixed effects estimate average relationships across the population, while random effects are assumed to arise from specified probability distributions, typically multivariate normal distributions. Model parameters are commonly estimated using Maximum Likelihood (ML) or Restricted Maximum Likelihood (REML), with covariance structures describing the dependence among repeated observations.

In practice, mixed effects models are fitted to longitudinal, clustered and multilevel datasets. In health economics, they are widely used to analyse repeated measurements of healthcare costs, quality-adjusted life years, patient-reported outcomes and clinical endpoints collected over time. They are particularly valuable for analysing incomplete longitudinal datasets under the Missing at Random assumption because all available observations contribute to parameter estimation without requiring complete follow-up for every participant.

Purpose

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Used to model correlated and hierarchical data, estimate both population-average and subject-specific effects, analyse longitudinal outcomes and account for within-cluster dependence in health economic and clinical research.

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Mathematical Formulae

Primary Formula

General linear mixed model:

Y = X? + Zb + �

where:

• Y = vector of observed outcomes
• X = fixed-effects design matrix
• ? = vector of fixed-effect parameters
• Z = random-effects design matrix
• b = vector of random effects
• � = residual error

Random effects:

b ~ N(0, G)

Residual errors:

� ~ N(0, R)

Supporting Formulae

Marginal distribution:

Y ~ N(X?, ZGZ? + R)

Restricted Maximum Likelihood objective:

REML = arg max L?(?, G, R)

Intraclass Correlation Coefficient:

ICC = ��b / (��b + ���)

Related Mathematical Methods

• Linear Mixed Models
• Generalised Linear Mixed Models
• Maximum Likelihood Estimation
• Restricted Maximum Likelihood
• Random Effects Models
• Longitudinal Data Analysis
• Hierarchical Linear Models

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Example

A health economist analyses EQ-5D utility scores measured at baseline, 6 months, 12 months and 24 months for 900 patients participating in a clinical trial.

A mixed effects model includes:

• Fixed effects for treatment, time and treatment-by-time interaction.
• A random intercept for each patient.

The estimated treatment effect at 24 months is:

?? = 0.047

95% CI = 0.028 to 0.066

The random intercept variance is:

��b = 0.081

Residual variance:

��� = 0.119

The Intraclass Correlation Coefficient is:

ICC = 0.081 / (0.081 + 0.119)

ICC = 0.405

This indicates that approximately 40.5% of the variability in utility scores is attributable to differences between patients, while the remaining variability reflects within-patient change over time.

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Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(B2:B901,C2:F901,TRUE,TRUE)	Estimate simplified fixed effects for exploratory analyses.
MMULT	=MMULT(A2:C901,D2:F4)	Perform matrix calculations used in mixed-model estimation.
MINVERSE	=MINVERSE(MMULT(TRANSPOSE(A2:C901),A2:C901))	Compute matrix inverses required in estimation algorithms.
Solver	Maximise the likelihood or restricted likelihood function by varying variance components.	Estimate fixed and random effect parameters.
AVERAGEIFS	=AVERAGEIFS(B:B,A:A,PatientID)	Calculate subject-level summaries during exploratory longitudinal analyses.

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VBA (Optional)

A VBA routine can organise repeated-measures datasets, estimate simplified mixed-model parameters through external routines and generate longitudinal model summary reports.

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Sources

• Laird NM, Ware JH. Random-Effects Models for Longitudinal Data. Biometrics. 1982;38(4):963?974.
• Pinheiro JC, Bates DM. Mixed-Effects Models in S and S-PLUS. Springer.
• Verbeke G, Molenberghs G. Linear Mixed Models for Longitudinal Data. Springer.
• Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis. Wiley.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Good Practice Reports.