Concept Architecture
Concept
Theoretically, a Multilevel Model is a hierarchical statistical model that accounts for data structured at multiple levels, such as patients nested within hospitals or repeated observations nested within individuals. The method extends conventional regression by allowing variation to be partitioned across hierarchical levels through the inclusion of both fixed effects and random effects. Multilevel models exist to produce valid statistical inference when observations within the same cluster are correlated and therefore violate the independence assumption of ordinary regression.
Mathematically, a multilevel model expresses the outcome variable as the sum of fixed effects, random effects associated with one or more hierarchical levels and residual error. Random effects capture between-cluster variation, while fixed effects estimate population-average relationships. Parameters are typically estimated using Maximum Likelihood (ML) or Restricted Maximum Likelihood (REML), and variance components quantify the proportion of total variability attributable to each hierarchical level.
In practice, multilevel models are used whenever data exhibit nested or clustered structures. In health economics, they are applied to analyse patients within hospitals, providers within healthcare systems, repeated measurements within individuals and multicentre clinical trials. These models enable estimation of both individual-level and cluster-level effects while appropriately accounting for intraclass correlation and hierarchical variation in healthcare outcomes and costs.
Purpose
Used to analyse hierarchical and clustered data, estimate effects operating at multiple organisational levels, account for within-cluster correlation and partition variation across levels in health economic research.
Mathematical Formulae
Primary Formula
Two-level linear model:
Y?? = ?? + ??X?? + u? + �??
where:
- Y?? = outcome for individual i in cluster j
- ?? = intercept
- ?? = fixed-effect coefficient
- X?? = explanatory variable
- u? = cluster-level random effect
- �?? = individual-level residual error
Random effect:
u? ~ N(0, ��u)
Residual error:
�?? ~ N(0, ���)
Supporting Formulae
Intraclass Correlation Coefficient:
ICC = ��u / (��u + ���)
General matrix form:
Y = X? + Zu + �
Related Mathematical Methods
- Mixed Effects Models
- Hierarchical Linear Models
- Linear Mixed Models
- Generalised Linear Mixed Models
- Restricted Maximum Likelihood
- Maximum Likelihood Estimation
- Random Effects Models
Example
A health economist analyses annual healthcare costs for 5,000 patients treated across 80 hospitals.
A two-level multilevel model includes:
- Patient-level predictors: age, sex and comorbidity score.
- Hospital-level random intercepts.
Estimated variance components are:
- Hospital-level variance (��u) = 220,000
- Patient-level variance (���) = 880,000
The Intraclass Correlation Coefficient is:
ICC = 220,000 / (220,000 + 880,000)
ICC = 0.20
This indicates that 20% of the variability in annual healthcare costs is attributable to differences between hospitals, while the remaining 80% reflects differences between individual patients.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B5001,C2:F5001,TRUE,TRUE) | Estimate simplified fixed effects for exploratory analyses. |
| AVERAGEIFS | =AVERAGEIFS(B:B,A:A,HospitalID) | Calculate cluster-level summary statistics. |
| VAR.S | =VAR.S(B2:B5001) | Estimate variability before multilevel modelling. |
| MMULT | =MMULT(A2:C5001,D2:F4) | Perform matrix calculations used in estimation algorithms. |
| Solver | Maximise the likelihood by estimating variance components and fixed effects. | Illustrate multilevel model estimation. |
VBA (Optional)
A VBA routine can summarise clustered datasets, calculate intraclass correlation coefficients and prepare hierarchical data structures for multilevel modelling software.
Sources
- Goldstein H. Multilevel Statistical Models. Wiley.
- Snijders TAB, Bosker RJ. Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. Sage.
- Raudenbush SW, Bryk AS. Hierarchical Linear Models. Sage.
- Gelman A, Hill J. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR Good Practice Reports.
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 multilevel model?
A model representing nested data, such as patients within hospitals within regions, allowing relationships to vary at each level while sharing strength.
Source: Goldstein 2011
How does a multilevel model let relationships vary across a hierarchy?
A multilevel model represents data arranged in a hierarchy, such as patients within hospitals within regions, and it lets relationships differ from one group to the next rather than forcing a single common pattern. Through random intercepts it allows each group its own baseline, and through random slopes it allows the effect of a predictor to vary between groups, while still borrowing strength so sparse groups lean on the overall average. Ignoring the hierarchy would misstate the uncertainty and hide these differences. Letting effects vary across levels is its purpose. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How does a multilevel model work?
A multilevel model works by including random effects for the units at each level, so that, for example, each hospital and each region has its own random deviation, and the model estimates the fixed effects of predictors together with the variance at each level. This accounts for the correlation within units and shares information across them. So a multilevel model works by modelling variation at multiple levels through random effects, which handles the nested structure, estimates how much outcomes vary at each level, and borrows strength so that estimates for units with little data are informed by the overall pattern, improving their stability and validity.
Source: Goldstein 2011
What are the levels in a multilevel model?
The levels in a multilevel model correspond to the units of the data's hierarchy, such as individuals at the lowest level nested within groups, which may themselves be nested within larger units, for example patients within hospitals within regions. Each level can have its own predictors and variation. So the levels in a multilevel model reflect the nested structure of the data, with random effects capturing the variation at each higher level, which allows relationships and variation to be examined at every level and predictors measured at different levels to be included, giving a full account of how outcomes depend on factors and vary across the hierarchy.
Source: Goldstein 2011
What are random intercepts and random slopes in a multilevel model?
In a multilevel model, a random intercept allows each higher-level unit to have its own baseline level of the outcome, capturing variation in the average outcome between units, while a random slope allows the effect of a predictor to vary between units, capturing variation in relationships. So random intercepts and random slopes represent different kinds of between-unit variation, with random intercepts modelling differences in overall level and random slopes modelling differences in how a predictor relates to the outcome across units, and including them lets a multilevel model capture both varying baselines and varying effects, giving a flexible representation of hierarchical data.
Source: Goldstein 2011
Why use a multilevel model instead of ignoring the hierarchy?
A multilevel model is used instead of ignoring the hierarchy because analysing nested data as if independent gives incorrect standard errors, understating uncertainty and inflating false positives, and misses the variation between units. So a multilevel model is preferred because it accounts for the within-unit correlation, giving valid inference, and it estimates the between-unit variation and shares strength across units, which single-level analyses cannot do, meaning that ignoring the hierarchy risks misleading conclusions and loses information, whereas the multilevel model handles the structure correctly and reveals how outcomes vary across the levels of the data.
Source: Goldstein 2011
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 19 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/multilevel-model
- Term code
- HE-ES-SA-131
Stable URI · Machine-readable · Resolvable · CC BY 4.0