Concept Architecture
Concept
Theoretically, Partial Correlation is a statistical measure of the strength and direction of the linear association between two variables after removing the linear effects of one or more control variables from both. It represents the residual relationship between the variables of interest that remains after adjustment for specified covariates. Partial correlation exists to distinguish an observed association from variation attributable to measured third variables, although it does not by itself establish causality or eliminate unmeasured confounding.
Mathematically, partial correlation is calculated as the Pearson correlation between the residuals obtained by separately regressing each variable of interest on the control variables. When controlling for a single variable, it can be calculated directly from the three pairwise correlation coefficients. The coefficient ranges from ?1 to +1, with its magnitude indicating the strength of the adjusted linear association and its sign indicating direction.
In practice, partial correlation is estimated using regression residuals, correlation matrices or inverse covariance matrices. In health economics, it is used to examine associations between costs, health outcomes, resource utilisation and patient characteristics after adjusting for variables such as age, baseline severity or comorbidity. Its interpretation depends on appropriate covariate selection, linearity and the absence of severe measurement error or multicollinearity.
Purpose
Used to quantify the linear association between two variables after controlling for measured covariates, assess adjusted relationships and support exploratory analysis in health economic research.
Mathematical Formulae
Primary Formula
For variables X and Y controlling for Z:
r??�?? = (r?? ? r???r???) / �[(1 ? r???�)(1 ? r???�)]
where:
- r??�?? = partial correlation between X and Y controlling for Z
- r?? = correlation between X and Y
- r??? = correlation between X and Z
- r??? = correlation between Y and Z
Supporting Formulae
Residual representation:
r??�?? = Corr(e?, e?)
where:
- e? = residuals from regressing X on Z
- e? = residuals from regressing Y on Z
Test statistic:
t = r�[(n ? k ? 2) / (1 ? r�)]
with:
t ~ t?????
where:
- r = estimated partial correlation
- n = sample size
- k = number of control variables
Inverse correlation matrix representation:
r??�rest = ?p?? / �(p??p??)
where P is the inverse of the correlation matrix.
Related Mathematical Methods
- Pearson Correlation
- Multiple Linear Regression
- Semi-Partial Correlation
- Residual Analysis
- Covariance Matrix
- Precision Matrix
- Hypothesis Testing
Example
A health economist examines the relationship between annual healthcare costs and EQ-5D utility while controlling for patient age.
The pairwise correlations are:
- Cost and utility: r?? = ?0.42
- Cost and age: r??? = 0.35
- Utility and age: r??? = ?0.28
The partial correlation is:
r??�?? = [?0.42 ? (0.35 ? ?0.28)] / �[(1 ? 0.35�)(1 ? (?0.28)�)]
r??�?? = ?0.322 / �(0.8775 ? 0.9216)
r??�?? � ?0.36
After controlling for age, annual healthcare costs remain moderately negatively associated with EQ-5D utility. Patients with higher costs tend to have lower utility values, although the adjusted association is weaker than the unadjusted correlation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =(CORREL(B2:B501,C2:C501)-CORREL(B2:B501,D2:D501)*CORREL(C2:C501,D2:D501))/SQRT((1-CORREL(B2:B501,D2:D501)^2)*(1-CORREL(C2:C501,D2:D501)^2)) | Calculate the partial correlation between healthcare costs and utility while controlling for one covariate. |
| LINEST | =LINEST(B2:B501,D2:F501,TRUE,FALSE) | Estimate regression coefficients used to remove the effects of multiple control variables. |
| TREND | =TREND(B2:B501,D2:F501,D2:F501) | Calculate fitted values before deriving residuals. |
| CORREL | =CORREL(G2:G501,H2:H501) | Calculate the correlation between residualised cost and outcome variables. |
| T.DIST.2T | =T.DIST.2T(ABS(J2),COUNT(B2:B501)-K2-2) | Calculate the two-sided p-value for a partial correlation test statistic. |
VBA (Optional)
A VBA routine can residualise multiple variable pairs against selected covariates and produce partial correlation coefficients, test statistics and p-values.
Sources
- Fisher RA. The Distribution of the Partial Correlation Coefficient. Metron. 1924;3:329?332.
- Anderson TW. An Introduction to Multivariate Statistical Analysis. Wiley.
- Cohen J, Cohen P, West SG, Aiken LS. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences. Routledge.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models. McGraw-Hill.
- Altman DG. Practical Statistics for Medical Research. Chapman & Hall.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
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 partial correlation?
A measure of the relationship between two variables after statistically controlling for the influence of one or more additional variables.
Source: Yule 1907
What does partial correlation reveal once other variables are held constant?
Partial correlation measures the relationship between two variables after statistically removing the influence of one or more others. It reveals whether the two are still associated once a third factor is held constant, which can expose an apparent link as the work of a shared cause, or uncover a real one that a lurking variable had masked. This makes it a way to probe whether an association survives adjustment for possible confounders. The link between two variables net of others is what it captures. Kirkwood and Sterne (2003) describe this measure.
Source: Kirkwood & Sterne 2003
How is partial correlation calculated?
Partial correlation is calculated by removing the linear influence of the controlled variables from each of the two variables of interest, for example by taking the residuals from regressing each on the controlled variables, and then correlating those residuals; formulas also express it directly from the simple correlations among the variables. So partial correlation is calculated as the correlation between the parts of the two variables not explained by the controlled variables, which quantifies their association with the others held constant, and this can be obtained either through residuals or through algebraic formulas relating it to the pairwise correlations among all the variables involved.
Source: Yule 1907
Why is partial correlation useful?
Partial correlation is useful because a simple correlation between two variables may be driven by their common relationship with other variables, and controlling for those variables reveals whether a direct association remains. It thus helps distinguish genuine relationships from spurious ones due to confounding. So partial correlation is useful for examining the association between two variables free of the influence of others, which can uncover that an apparent relationship disappears or strengthens once confounders are controlled, making it a tool for exploring relationships while accounting for other variables, though, like all correlations, it does not by itself establish causation.
Source: Yule 1907
How does partial correlation differ from simple correlation?
Partial correlation measures the association between two variables while controlling for one or more other variables, whereas simple correlation measures their association without any such control. The two can differ markedly: a strong simple correlation may vanish when a confounder is controlled, or a weak one may strengthen. So partial correlation differs from simple correlation in accounting for the influence of other variables, which reveals the relationship that remains once those are held constant, and comparing the two shows how much of a simple correlation is due to the controlled variables, helping to interpret whether an association is direct or mediated by other factors.
Source: Yule 1907
What are the limitations of partial correlation?
The limitations of partial correlation include that it controls only for the variables included, so uncontrolled confounders can still distort it; that it measures linear association and can be affected by non-linearity; and that, like any correlation, it does not establish causation. So partial correlation is interpreted with awareness that it accounts only for the specified variables and captures linear relationships, which is why it does not prove that a remaining association is causal, since other unmeasured variables could be responsible, and it is used as a tool for exploring relationships while controlling for measured variables rather than as definitive evidence of a direct or causal effect.
Source: Yule 1907
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 22 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/partial-correlation
- Term code
- HE-ES-SA-153
Stable URI · Machine-readable · Resolvable · CC BY 4.0