Correlation

Correlation in Health Economic Evidence and Models

Correlation describes whether two quantities tend to rise and fall together, and in health economics it matters wherever two variable or uncertain quantities are analysed jointly, for example costs and QALYs from the same trial patients, two inputs to a decision model, or a biomarker and survival. Handled badly, it distorts the uncertainty around incremental net benefit, lets a probabilistic analysis draw implausible input combinations, and invites causal readings of confounded associations. Its sign and size can change the reported uncertainty of a result without moving the point estimate. The measures themselves (Pearson, Spearman and Kendall) and their calculation are on the Correlation Coefficient page.

What correlation describes and what it leaves out

Bewick and colleagues, in a statistics review, describe correlation as quantifying the strength of the linear relationship between a pair of variables, whereas regression expresses the relationship as an equation. A positive correlation means high values of one variable tend to accompany high values of the other, and a negative correlation means the reverse. A value near zero indicates no linear relationship, although the review notes that a nonlinear relationship may still exist and go undetected.

Correlation has no units, so it says nothing about how much one variable changes per unit of the other; a regression slope carries that, and the covariance is in the product of the two variables' units.

$$\operatorname{Cov}(X, Y) = \rho_{XY} \, \sigma_X \, \sigma_Y$$

where $\operatorname{Cov}(X, Y)$ is their covariance, $\rho_{XY}$ is their correlation and $\sigma_X$ and $\sigma_Y$ are their standard deviations. With several variables these form a correlation matrix or a variance-covariance matrix.

The level at which two quantities are correlated

Correlation arises at distinct levels, and an estimate from one level cannot be reused at another. Individual-level correlation links values measured on the same patients, such as costs and QALYs. In cluster randomised trials, Gomes and colleagues separate this from cluster-level correlation between cluster mean costs and outcomes, and the intraclass correlation describes how alike patients in one cluster are. In evidence synthesis, NICE Decision Support Unit Technical Support Document (TSD) 20 models within-study correlation between treatment effects on two outcomes (measured on the same patients) alongside between-studies correlation. Parameter-level correlation belongs to the uncertainty in estimates, such as two coefficients from one regression.

Correlated costs and outcomes in trial-based evaluation

A scoping review of trial-based economic evaluation methods by El Alili and colleagues states that costs and effects are generally correlated, so it is inappropriate to assume they are uncorrelated. It groups the methods for handling this into seemingly unrelated regression (separate cost and effect equations with correlated residuals), bootstrapping costs and effects in pairs, and Bayesian bivariate models.

The correlation feeds into the precision of the incremental net benefit. Everest and colleagues, writing on power calculations for trial-based cost-effectiveness analyses, give its variance as

$$\sigma^2_{b(\lambda)} = \frac{1}{n_0}\left(\lambda^2 \sigma^2_{E_0} + \sigma^2_{C_0} - 2\lambda\rho_0\sigma_{E_0}\sigma_{C_0}\right) + \frac{1}{n_1}\left(\lambda^2 \sigma^2_{E_1} + \sigma^2_{C_1} - 2\lambda\rho_1\sigma_{E_1}\sigma_{C_1}\right)$$

where $\sigma^2_{b(\lambda)}$ is the variance of the incremental net benefit at willingness to pay $\lambda$ and, for arm $j$ (0 for control, 1 for the experimental arm), $n_j$ is the number of patients, $\sigma_{E_j}$ and $\sigma_{C_j}$ are the standard deviations of effectiveness and cost, and $\rho_j$ is their correlation. A positive correlation reduces the variance, because patients who gain more health also cost more and the two partly offset in net benefit; a negative correlation increases it.

In four Canadian Cancer Trials Group trials examined by Everest and colleagues, allowing for the observed correlation cut the variance of the incremental net benefit by 2% to 36% compared with assuming zero correlation, at the authors' main willingness-to-pay value. Assuming zero correlation gave slightly wider confidence intervals, and assuming perfect positive correlation gave intervals that were too narrow. In the cluster trial case study of Gomes and colleagues, ignoring clustering underestimated statistical uncertainty and the value of further research.

Correlated parameters in probabilistic sensitivity analysis

A probabilistic sensitivity analysis draws uncertain inputs from distributions, and the ISPOR-SMDM parameter estimation report (Briggs and colleagues) notes that these parameters are typically not all independent. Its example is a progression probability with and without treatment, which share uncertainty about natural history, so independent sampling would be wrong. The report instead assigns distributions to the baseline risk and the relative risk, which can plausibly be treated as independent, and derives the on-treatment probability as their product.

Jointly estimated parameters such as regression coefficients carry direct evidence on correlation in their covariance matrix, and the report regards multivariate normality as appropriate for the linear predictor; a Cholesky decomposition can generate such draws. For separately estimated parameters, the report warns that this should not necessarily lead to an assumption of independence, and suggests adding a correlation coefficient as an uncertain parameter or re-parameterising the model. For probabilities of mutually exclusive events, Briggs, Ades and Price propose the Dirichlet distribution, the multivariate equivalent of the beta, so that sampled probabilities still sum to one.

The NICE technology appraisal manual (PMG36) asks for evidence on correlation between parameters to be reflected in the probabilistic analysis, for ordered parameters to avoid both artificially restricted distributions and unsupported perfect correlation, and for correlation assumptions to be clearly presented (section 4.7.19). It also notes that diagnostic sensitivity and specificity are usually correlated (4.7.18) and that threshold analysis does not suit parameters highly correlated with other influential parameters (4.7.23). In value of information analysis, the ISPOR-SMDM report adds that correlated parameters may be better reported as groups in EVPPI.

Worked example of correlation changing decision uncertainty

All numbers are illustrative. A two-arm trial randomises 200 patients to each arm and values health at GBP 20,000 per QALY. In both arms the standard deviation of patient QALYs is 0.20 and of patient costs GBP 3,000, and the mean incremental net benefit is GBP 800. Only the within-arm correlation $\rho$ between patient costs and QALYs changes.

1. Per-patient net benefit variance. The three terms of the formula above are $\lambda^2\sigma_E^2 = 4{,}000^2 = 16{,}000{,}000$, $\sigma_C^2 = 3{,}000^2 = 9{,}000{,}000$ and $\rho \times 2\lambda\sigma_E\sigma_C = 24{,}000{,}000\rho$. The per-patient variance is 25,000,000 at zero correlation, 15,400,000 at 0.4 and 34,600,000 at -0.4.

2. Standard error of the incremental net benefit. Each per-patient variance is multiplied by 1/200 + 1/200 = 0.01 before taking the square root.

$$\text{SE}_{0} = \sqrt{0.01 \times 25000000} = \sqrt{250000} = 500$$

where $\text{SE}_{0}$ is the standard error in GBP at zero correlation. The same steps give $\sqrt{154{,}000} = 392.4$ at a correlation of 0.4 and $\sqrt{346{,}000} = 588.2$ at -0.4.

3. Interval and probability of cost-effectiveness. With a normal approximation, the 95% interval is GBP 800 plus or minus 1.96 standard errors.

4. Interpretation. The point estimate never moves, yet the interval excludes zero only when the positive correlation is allowed for. Assuming independence would report the first row whatever the data showed.

5. The same logic inside a model. A treatment keeps patients in a mild state (utility 0.80, standard error 0.05) instead of a severe state (0.60, standard error 0.05) for 2 years, giving 0.40 incremental QALYs. Sampled independently, the difference has a standard deviation of $\sqrt{0.0025 + 0.0025} = 0.0707$, the incremental QALYs 0.141, and about 0.23% of normal draws put the severe state above the mild one. If both utilities come from one regression with a correlation of 0.8, the variance of the difference falls to 0.005 minus 2 × 0.8 × 0.0025, or 0.001, a standard deviation of 0.0316, so the incremental QALYs have a standard deviation of 0.063 and reversed draws virtually vanish. Ignoring it overstates uncertainty and allows implausible orderings.

Correlation, causation and surrogate outcomes

Bewick and colleagues name failure to consider a third variable related to both as one of the most common errors and state that correlation does not imply causation; checking other underlying variables and other populations strengthens a causal case. This applies when, for example, spending and outcomes move together across hospitals or regions. Estimating what a change would do is the task of causal inference, with confounding, confounding by aggregation and the Simpson paradox as the usual traps.

Surrogate outcomes show the same distinction. The NICE manual lists three levels of evidence for surrogate relationships, in which level 2 is a consistent association between the surrogate and final outcomes, usually from observational studies, and level 1 is randomised evidence that the technology's effect on the surrogate corresponds to a commensurate effect on the final outcome (PMG36, 4.6.6). Validation needs evidence that the relative effect on the surrogate predicts the relative effect on the final outcome, preferably from meta-analysis of trials using the bivariate methods in TSD 20 (4.6.7). A strong patient-level correlation between a biomarker and survival falls short of validating a surrogate endpoint, and validation is specific to the population and technology type (4.6.8).

Boundaries with related measures

Neighbouring ideas answer different questions. The table separates them.

Agreement matters when two instruments, or a mapping function and observed values, are compared; Bewick and colleagues note that a high correlation is sometimes wrongly read as agreement, and that analysing paired differences is more appropriate.

Where correlation assumptions go wrong in evaluations

The errors below come from an unexamined default rather than a wrong calculation. Each one below changes the reported uncertainty or the estimate itself.

• Independence by default. Sampling separately estimated parameters as independent is itself an assumption, which the ISPOR-SMDM report warns against making automatically.
• Perfect correlation as a fix. Forcing ordered utilities to move in lockstep removes reversed draws but, under PMG36, needs evidence.
• Baseline imbalance. Manca, Hawkins and Sculpher note that baseline utility is likely to be highly correlated with QALYs over follow-up, so unadjusted imbalance between arms can give a misleading ICER, and they show that regression can adjust for it.

Sources

• Bewick V, Cheek L, Ball J. Statistics review 7: correlation and regression. Critical Care. 2003;7(6):451-459. https://doi.org/10.1186/cc2401
• Briggs AH, Ades AE, Price MJ. Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework. Medical Decision Making. 2003;23(4):341-350. https://doi.org/10.1177/0272989X03255922
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6. Value in Health. 2012;15(6):835-842. https://doi.org/10.1016/j.jval.2012.04.014
• Bujkiewicz S, Achana F, Papanikos T, Riley RD, Abrams KR. NICE DSU Technical Support Document 20: multivariate meta-analysis of summary data for combining treatment effects on correlated outcomes and evaluating surrogate endpoints. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; Version 2, 22 October 2019. https://doi.org/10.15131/shef.data.33848854
• El Alili M, van Dongen JM, Esser JL, Heymans MW, van Tulder MW, Bosmans JE. A scoping review of statistical methods for trial-based economic evaluations: the current state of play. Health Economics. 2022;31(12):2680-2699. https://doi.org/10.1002/hec.4603
• Everest L, Chen BE, Hay AE, Cheung MC, Chan KKW. Power and sample size calculation for incremental net benefit in cost effectiveness analyses with applications to trials conducted by the Canadian Cancer Trials Group. BMC Medical Research Methodology. 2023;23:179. https://doi.org/10.1186/s12874-023-01956-y
• Gomes M, Ng ES, Grieve R, Nixon R, Carpenter J, Thompson SG. Developing appropriate methods for cost-effectiveness analysis of cluster randomized trials. Medical Decision Making. 2012;32(2):350-361. https://doi.org/10.1177/0272989X11418372
• Manca A, Hawkins N, Sculpher MJ. Estimating mean QALYs in trial-based cost-effectiveness analysis: the importance of controlling for baseline utility. Health Economics. 2005;14(5):487-496. https://doi.org/10.1002/hec.944
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 4.6.6 to 4.6.8, 4.7.18, 4.7.19 and 4.7.23. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2