Concept Architecture
Concept
Theoretically, Unstructured Correlation is the most general covariance structure used for correlated or repeated measures data, in which every variance and covariance is estimated freely without imposing any predetermined pattern. It makes no assumptions regarding the relationship between repeated observations and therefore provides maximum flexibility when modelling within-subject correlation. The concept is founded on multivariate normal theory and covariance matrix estimation and is widely used in mixed-effects models, repeated measures analysis and longitudinal health economic studies.
Mathematically, an unstructured correlation model estimates a unique covariance parameter for every pair of repeated measurements together with a separate variance for each measurement occasion. The resulting covariance matrix is symmetric and positive definite, with no restrictions placed on individual variances or correlations. Parameters are typically estimated using maximum likelihood or restricted maximum likelihood methods within linear mixed-effects models.
In practice, an unstructured correlation structure is selected when sufficient data are available and there is no theoretical justification for assuming a simpler covariance pattern such as compound symmetry or autoregressive correlation. Model selection commonly involves comparing covariance structures using likelihood ratio tests or information criteria. In health economics, unstructured correlation is frequently applied in mixed model repeated measures analyses of longitudinal costs, quality-of-life measures and clinical outcomes.
Purpose
Used to model correlated repeated measurements without imposing assumptions about the pattern of variances or correlations, allowing maximum flexibility in longitudinal and repeated measures analyses.
Mathematical Formulae
Primary Formula
� =
[�?? �?? ? �??]
[�?? �?? ? �??]
[ ? ? ? ? ]
[�?? �?? ? �??]
where � is a fully unstructured covariance matrix.
Supporting Formulae
Correlation coefficient:
??? = �?? / �(�??�??)
Number of covariance parameters:
p(p + 1) / 2
where p is the number of repeated measurement occasions.
Related Mathematical Methods
- Linear Mixed-Effects Model
- Mixed Model Repeated Measures
- Restricted Maximum Likelihood
- Maximum Likelihood Estimation
- Covariance Matrix
- Compound Symmetry
- Autoregressive Correlation Structure
- Likelihood Ratio Test
Example
A health economist analyses EQ-5D utility scores collected at baseline, 3, 6 and 12 months following treatment. Rather than assuming equal correlations across follow-up visits, an unstructured correlation matrix is specified so that each variance and each pairwise covariance is estimated separately. Model fit is superior to compound symmetry according to the Akaike Information Criterion, indicating that the more flexible covariance structure better represents the observed longitudinal data.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| COVARIANCE.S | =COVARIANCE.S(B2:B101,C2:C101) | Estimate covariance between two repeated measurements. |
| CORREL | =CORREL(B2:B101,C2:C101) | Estimate pairwise correlation between repeated observations. |
| TRANSPOSE | =TRANSPOSE(B2:E5) | Assist in constructing symmetric covariance matrices. |
| MINVERSE | =MINVERSE(B2:E5) | Invert covariance matrices for exploratory matrix calculations. |
VBA (Optional)
Automate the construction and export of covariance and correlation matrices from repeated measures datasets for use in mixed-effects modelling software.
Sources
- Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data.
- Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis.
- Verbeke G, Molenberghs G. Linear Mixed Models for Longitudinal Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
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 unstructured correlation?
A correlation structure for repeated measures allowing the correlation between every pair of time points to be estimated freely and independently.
Source: Liang & Zeger 1986
What flexibility does an unstructured correlation allow for repeated measures?
An unstructured correlation allows the correlation between every pair of time points to be estimated separately, imposing no pattern on how the repeated measurements relate. This gives the greatest flexibility, since it can capture any arrangement of correlations the data show, unlike structures that assume, say, a steady decline with time. The cost is that it must estimate a great many parameters, which demands ample data and can become unwieldy when the time points are numerous. Letting every pair correlate freely is its defining feature. Kirkwood and Sterne (2003) describe such structures.
Source: Kirkwood & Sterne 2003
When is unstructured correlation used?
Unstructured correlation is used when the pattern of correlation among repeated measurements is unknown or complex and does not fit a simple structure, and when there are enough data to estimate the many correlation parameters reliably. So unstructured correlation is used where flexibility is needed to capture an arbitrary correlation pattern, such as with a modest number of time points and ample data, since it makes no assumptions about the correlations, though it requires estimating a correlation for every pair, which becomes demanding with many time points, so it is used when the data support it and a structured form would be too restrictive.
Source: Liang & Zeger 1986
What are the advantages of unstructured correlation?
The advantages of unstructured correlation include maximum flexibility, since it estimates each pairwise correlation freely without imposing a pattern, so it can represent any correlation structure and avoids bias from a misspecified structure. So unstructured correlation is advantageous when the true correlation pattern is unknown or irregular, as it makes no assumptions and fits whatever pattern the data show, which is why it is used when a correct but simple structure cannot be assumed, though this flexibility comes at the cost of estimating many parameters, which requires sufficient data and can reduce efficiency compared with a correctly specified simpler structure.
Source: Liang & Zeger 1986
What are the limitations of unstructured correlation?
The limitations of unstructured correlation include the large number of parameters to estimate, one correlation for every pair of time points, which grows rapidly with more time points and requires ample data; the risk of estimation problems or instability with limited data; and reduced efficiency compared with a correct simpler structure. So unstructured correlation is limited by its parameter demands, which make it impractical with many time points or small samples, and by potential loss of efficiency, which is why a structured form such as exchangeable or autoregressive is preferred when it adequately describes the correlation, reserving the unstructured form for situations with few time points and sufficient data.
Source: Liang & Zeger 1986
How does unstructured correlation differ from other structures?
Unstructured correlation differs from other structures in imposing no pattern, estimating every pairwise correlation freely, whereas exchangeable correlation assumes all pairs are equally correlated and autoregressive correlation assumes the correlation declines with the time between observations. Unstructured is the most flexible but most parameter-heavy, while the structured forms are more parsimonious but assume a specific pattern. So unstructured correlation differs by making no assumptions and using many parameters, which is why it suits situations where the correlation pattern is unknown and data are ample, while structured forms are used when a simpler pattern is plausible, trading flexibility for parsimony and efficiency.
Source: Liang & Zeger 1986
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-230
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