Partial Correlation

Partial Correlation in Health Economic Analysis and Evidence Synthesis

A partial correlation coefficient shows how closely two quantities move together once one or more other quantities have been held fixed, so the link between a patient's hospital costs and their quality of life can be judged net of comorbidity or age. It lies between −1 and +1 like an ordinary correlation coefficient, and setting the two side by side shows how much of an observed association runs alongside the variables controlled for. Health economists meet it in exploratory work on patient-level data and as the partial rank correlation coefficient used to rank the uncertain inputs of a decision model. In evidence synthesis it serves as a standardised effect size recovered from reported regression results. Which variables to control for remains a judgement about the confounding structure of the data.

Removing the controlled variables from both quantities

The idea is to strip out of each variable whatever the control variables can predict linearly, and then correlate what is left. Marino and colleagues define the partial correlation as the correlation between the residuals from linear regressions of each variable on the remaining ones.

$$r_{xy \cdot z} = \operatorname{corr}\left(x - \hat{x},\, y - \hat{y}\right)$$

where $r_{xy \cdot z}$ is the partial correlation between $x$ and $y$ controlling for $z$, $\hat{x}$ and $\hat{y}$ are the fitted values from linear regressions of $x$ and of $y$ on $z$, and $\operatorname{corr}$ is the ordinary Pearson correlation of the two sets of residuals.

With a single control variable the same quantity follows from the three pairwise correlations. The NIST Dataplot reference manual gives this first-order formula, and Kim notes that the residual definition is equivalent to it.

$$r_{xy \cdot z} = \frac{r_{xy} - r_{xz}\, r_{yz}}{\sqrt{\left(1 - r_{xz}^{2}\right)\left(1 - r_{yz}^{2}\right)}}$$

where $r_{xy}$, $r_{xz}$ and $r_{yz}$ are the simple Pearson correlations between each pair of variables. The numerator takes away the part of $r_{xy}$ that the two links through $z$ would produce on their own, and the denominator rescales the result to the variation in $x$ and $y$ that $z$ does not explain.

Several control variables and the inverse covariance matrix

Applied work usually adjusts for several variables at once, such as age, sex and baseline health. Kim states, citing Whittaker, that higher-order partial correlations can be read from the inverse of the variance-covariance matrix, which is far cheaper than applying the first-order formula recursively.

$$r_{ij \cdot S} = -\frac{d_{ij}}{\sqrt{d_{ii}\, d_{jj}}}$$

where $r_{ij \cdot S}$ is the partial correlation between variables $i$ and $j$ given the set $S$ of all other variables, and $d_{ij}$, $d_{ii}$ and $d_{jj}$ are elements of the inverse covariance matrix, often called the precision matrix. The inverse of the correlation matrix gives the same result, since rescaling a variable does not change a correlation.

Partial correlation, semi-partial correlation and the regression coefficient

Two neighbouring quantities are easily confused with the partial correlation. Kim describes the semi-partial correlation, also called the part correlation, as removing the effect of the other variables from just one of the two variables of interest.

$$r_{x(y \mid z)} = \frac{r_{xy} - r_{xz}\, r_{yz}}{\sqrt{1 - r_{yz}^{2}}}$$

where $r_{x(y \mid z)}$ is the correlation between $x$ and the part of $y$ not explained by $z$. Only the denominator differs, so the semi-partial correlation is never larger in size than the partial one, and Kim notes that the two are equal when $z$ is correlated with $y$ but not with $x$.

The second neighbour is the coefficient from a multiple regression. Van Aert and Goos show that the partial correlation can be computed from the t statistic of the coefficient of interest and the residual degrees of freedom.

$$r_p = \frac{t}{\sqrt{t^{2} + df}}$$

where $r_p$ is the partial correlation between the outcome and the predictor, $t$ is the t statistic for that predictor's coefficient, and $df = N - M - 1$, with $N$ observations and $M$ predictors. Van Aert and Goos give the size as a square root, and the sign is that of the coefficient. The coefficient states the adjusted effect in natural units, such as pounds per unit of utility, while the partial correlation states only its standardised strength. Manca, Hawkins and Sculpher note that baseline utility is likely to be highly correlated with the QALYs accrued over follow-up, and they use multiple regression to estimate differential mean QALYs while controlling for baseline imbalance.

Sampling error, tests and intervals

Fisher showed in 1924 that the sampling distribution of a partial correlation from $n$ pairs with $s$ variates eliminated is the same as that of an ordinary correlation from $n - s$ pairs. He presented this as confirming Yule's 1907 approximation, provided the number of variates eliminated is a small fraction of the sample. Kim gives the corresponding test of no partial correlation.

$$t = r_{xy \cdot S}\sqrt{\frac{N - 2 - g}{1 - r_{xy \cdot S}^{2}}}$$

where $t$ follows a t distribution with $N - 2 - g$ degrees of freedom under the null hypothesis, $N$ is the sample size and $g$ is the number of control variables. With one control variable the NIST manual expresses the same test as an F statistic on 1 and $N - 3$ degrees of freedom, the square of this t.

A confidence interval uses Fisher's z transformation. The NIST manual computes the interval for an ordinary correlation from $\operatorname{atanh}(r)$ with a standard error of one over $\sqrt{n - 3}$, and Fisher's result replaces $n$ with $n - g$.

$$z = \operatorname{atanh}\left(r_{xy \cdot S}\right), \qquad \text{limits} = \tanh\left(z \pm \frac{1.96}{\sqrt{N - 3 - g}}\right)$$

where $z$ is the transformed coefficient, 1.96 is the standard normal quantile for a 95% interval and $\tanh$ converts the limits back to the correlation scale. Fisher's derivation assumes normally distributed variates and independent observations.

Partial rank correlation in probabilistic sensitivity analysis

A probabilistic sensitivity analysis produces thousands of joint draws of the model inputs and the matching outputs, such as incremental net benefit. Correlating each input with the output across those draws, while partialling out the other inputs, ranks the inputs by influence. Marino and colleagues describe the partial rank correlation coefficient (PRCC) as a partial correlation computed after rank-transforming inputs and output, and judge it a reliable measure for nonlinear but monotonic relationships provided little or no correlation exists between the inputs. They pair it with Latin hypercube sampling as a form of global sensitivity analysis.

Menzies and colleagues used PRCCs in a cost-effectiveness analysis of Xpert MTB/RIF testing for tuberculosis in southern Africa, computing them across the simulation results of their Bayesian uncertainty analysis. The conclusions were largely consistent with their one-way sensitivity analyses. A PRCC shows how strongly the output moves with an input but not whether that movement could change the adoption decision, which is the question the expected value of partial perfect information answers.

Partial correlations as effect sizes in meta-analysis

Primary studies seldom publish full correlation matrices but routinely report regression coefficients and t statistics. Van Aert and Goos note that meta-analyses of these partial effects are regularly conducted in psychology and economics, using the partial correlation as the effect size. Inverse-variance weighting then needs a sampling variance for each study, and two estimators are in use.

$$s_1^{2} = \frac{\left(1 - r_p^{2}\right)^{2}}{df}, \qquad s_2^{2} = \frac{1 - r_p^{2}}{df}$$

where $s_1^{2}$ is the estimator derived by Olkin and Siotani and by Anderson, $s_2^{2}$ is the one given by Stanley and Doucouliagos, and $df$ is the residual degrees of freedom of the study's regression. Van Aert and Goos show that $s_2^{2}$ was derived assuming a population value of zero and is biased when the true value is not zero and samples are small, and they recommend $s_1^{2}$. In a separate paper van Aert found, in a reproduced simulation study, that pooling Fisher z transformed partial correlations gave lower bias and root mean square error, and recommends reporting such an analysis alongside any pooling of untransformed values. Each study's coefficient is conditional on its own control set, so a pooled value combines different adjustments.

Worked example with cost, utility and comorbidity in a registry

All numbers are illustrative. A registry of 200 adults with a long-term condition records annual hospital cost ($C$), EQ-5D utility ($U$) and number of comorbidities ($M$). The simple correlations are $r_{CU} = -0.40$, $r_{CM} = 0.50$ and $r_{UM} = -0.60$, so part of the cost and utility association may run through comorbidity.

1. Partial correlation. The numerator is −0.40 − (0.50 × −0.60) = −0.40 + 0.30 = −0.10. The denominator is √((1 − 0.25) × (1 − 0.36)) = √(0.75 × 0.64) = √0.48 = 0.6928. The partial correlation is −0.10 / 0.6928 = −0.144.

2. Check from the inverse correlation matrix. The determinant is 1 + 2 × (−0.40) × 0.50 × (−0.60) − 0.16 − 0.25 − 0.36, which is 1 + 0.24 − 0.77 = 0.47. The inverse elements are $d_{CC}$ = 0.64 / 0.47 = 1.3617, $d_{UU}$ = 0.75 / 0.47 = 1.5957 and $d_{CU}$ = 0.10 / 0.47 = 0.2128, so the partial correlation is −0.2128 / √(1.3617 × 1.5957) = −0.2128 / 1.4741 = −0.144.

3. Semi-partial correlation. Removing comorbidity from utility only gives −0.10 / √(1 − 0.36) = −0.10 / 0.8 = −0.125.

4. Test. With $N$ = 200 and $g$ = 1 there are 197 degrees of freedom. Since 1 − 0.1443² = 0.9792, $t$ = −0.1443 × √(197 / 0.9792) = −0.1443 × 14.184 = −2.05, a two-sided p-value of about 0.042. The unadjusted correlation gives $t$ = −0.40 × √(198 / 0.84) = −0.40 × 15.353 = −6.14.

5. Interval. atanh(−0.1443) = −0.1454 and the standard error is one over √196, which is 1/14 = 0.0714. The z-scale limits are −0.1454 ± 1.96 × 0.0714 = −0.1454 ± 0.1400, or −0.2854 and −0.0054, and back-transformed the 95% interval is −0.278 to −0.005.

6. Recovering the value from a regression. A study reporting only a regression of cost on utility and comorbidity, with $t$ = −2.047 for utility on 200 − 2 − 1 = 197 degrees of freedom, gives $r_p$ = −2.047 / √(4.190 + 197) = −2.047 / 14.184 = −0.144. For pooling, 0.9792 squared is 0.9588, so $s_1^{2}$ = 0.9588 / 197 = 0.00487 and $s_2^{2}$ = 0.9792 / 197 = 0.00497.

7. Interpretation. The correlation of −0.40 falls to −0.14 once comorbidity is held fixed, so most of the tendency for patients with lower utility to cost more travels with comorbidity count in these data. The remaining association is small and its interval nearly reaches zero. For a cost model, comorbidity is the stronger candidate predictor, since its partial correlation with cost given utility is (0.50 − 0.24) / √(0.84 × 0.64) = 0.355, though nothing here shows which variable drives which.

Choosing what to control for and common misreadings

The coefficient depends entirely on the control set. Controlling for a confounder removes association that would otherwise mislead, while controlling for a variable on the path between the two, the subject of mediation analysis, removes part of the association of interest, and controlling for a common effect of the two, a collider, can create an association that was not there. Only linear effects of measured variables are removed, so a nonlinear effect of age or an unmeasured cause such as frailty can leave spurious association behind.

• Treating a small partial correlation as no relationship. The relationship may be nonlinear, or a mediator among the controls may carry most of it.
• Comparing values across studies with different controls. The same pair of variables gives different partial correlations under different adjustment sets.
• Ranking model inputs by PRCC when inputs are correlated or relationships are not monotonic. Both conditions fall outside the method's stated scope.
• Pooling with the variance derived under a zero value. It exceeds the recommended estimator whenever the coefficient is not zero, and van Aert and Goos note that a suboptimal variance estimator biases a meta-analysis through its weights.

Sources

• Fisher RA. The distribution of the partial correlation coefficient. Metron. 1924;3:329-332. University of Adelaide Digital Library. https://digital.library.adelaide.edu.au/server/api/core/bitstreams/7cfa2da9-b52c-4cc8-98d6-7e5adc94f8fb/content
• Kim S. ppcor: an R package for a fast calculation to semi-partial correlation coefficients. Communications for Statistical Applications and Methods. 2015;22(6):665-674. https://pmc.ncbi.nlm.nih.gov/articles/PMC4681537/
• 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
• Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. Journal of Theoretical Biology. 2008;254(1):178-196. https://pmc.ncbi.nlm.nih.gov/articles/PMC2570191/
• Menzies NA, Cohen T, Lin HH, Murray M, Salomon JA. Population health impact and cost-effectiveness of tuberculosis diagnosis with Xpert MTB/RIF: a dynamic simulation and economic evaluation. PLoS Medicine. 2012;9(11):e1001347. https://doi.org/10.1371/journal.pmed.1001347
• National Institute of Standards and Technology. Dataplot Reference Manual, entries PARTIAL CORRELATION (2018) and CORRELATION CONFIDENCE LIMITS (2023). https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/partcorr.htm and https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/corrconf.htm
• van Aert RCM. Meta-analyzing partial correlation coefficients using Fisher's z transformation. Research Synthesis Methods. 2023;14(5):768-773. https://doi.org/10.1002/jrsm.1654
• van Aert RCM, Goos C. A critical reflection on computing the sampling variance of the partial correlation coefficient. Research Synthesis Methods. 2023;14(3):520-525. https://doi.org/10.1002/jrsm.1632