Correlation Coefficient

Correlation Coefficient in Health Economic Analysis

A correlation coefficient reduces a set of paired observations, such as each trial participant's cost and quality-adjusted life years, to one number whose sign gives the direction of association and whose size gives its strength. Pearson's coefficient measures linear association, while Spearman's and Kendall's coefficients work on ranks and suit skewed or ordinal data. In health economics the statistic matters because costs and health effects measured on the same people tend to move together, and because uncertain model parameters are often linked, so ignoring correlation can misstate the uncertainty around a cost-effectiveness result. In a trial-based evaluation the coefficient links costs and effects measured on the same patients, and in a decision model it links uncertain parameters in the probabilistic sensitivity analysis. The wider idea of variables varying together is covered on the Correlation page.

Pearson's product-moment coefficient

Pearson's coefficient uses the raw values of both variables. The NIST Dataplot reference manual describes it as a measure of the linear relationship between two variables and computes it from sums of squared and cross-multiplied deviations from the means.

$$r = \frac{S_{xy}}{\sqrt{S_{xx}}\,\sqrt{S_{yy}}}, \qquad S_{xy} = \sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}), \qquad S_{xx} = \sum_{i=1}^{n}(x_i - \bar{x})^2, \qquad S_{yy} = \sum_{i=1}^{n}(y_i - \bar{y})^2$$

where $r$ is the sample coefficient, $x_i$ and $y_i$ are the paired values for unit $i$, $\bar{x}$ and $\bar{y}$ are their sample means and $n$ is the number of pairs. Dividing the numerator and the denominator by $n - 1$ shows that $r$ is the sample covariance divided by the product of the two sample standard deviations, which is why a variance-covariance matrix can be rebuilt from a correlation matrix and the standard deviations.

Mukaka, in a guide to correlation in medical research, describes the coefficient as dimensionless and lying between −1 and +1, with zero indicating no linear relationship and −1 or +1 a perfect one. Because it is dimensionless, recording costs in pounds or pence leaves $r$ unchanged. Reversing the coding of one variable reverses the sign. In a simple linear regression with an intercept fitted to the same pairs, $r^2$ equals the share of the variance in the outcome explained by the fitted line. The coefficient says nothing about the slope in natural units, so two pairs of variables with the same $r$ can differ greatly in how much cost changes per unit of outcome.

Rank-based coefficients for skewed and ordinal data

Health economic data are often skewed (a few very expensive patients) or ordinal (health-state levels, symptom grades), and the rank-based coefficients suit such data. The NIST manual defines the Spearman rank correlation as the Pearson correlation applied to the ranks of each variable, with tied values given the average of the ranks they span, and recommends rank correlation when the data can be ranked but lack a meaningful numerical scale, when the relationship is monotonic but not linear, or when normality cannot be assumed. Mukaka adds that Spearman's coefficient is more robust to extreme values than Pearson's.

$$r_s = 1 - \frac{6\sum_{i=1}^{n} d_i^2}{n(n^2 - 1)}$$

where $r_s$ is Spearman's coefficient, $d_i$ is the difference between the two ranks of unit $i$ and $n$ is the number of pairs. This shortcut matches the Pearson formula on ranks only when there are no ties.

Kendall's tau instead compares every pair of units. The NIST manual calls a pair concordant when the two variables move in the same direction between the units and discordant when they move in opposite directions, and gives Kendall's original tau-a as the difference between the two counts divided by the number of possible pairs.

$$\tau_a = \frac{N_c - N_d}{n(n-1)/2}$$

where $N_c$ and $N_d$ are the numbers of concordant and discordant pairs and $n(n-1)/2$ is the number of pairs among $n$ units. The manual notes that tau-b adjusts for ties and is preferred to tau-a when ties are present. The partial correlation and the intraclass correlation answer different questions and have their own pages.

Testing a coefficient and putting an interval around it

A sample coefficient is an estimate, and in small samples it can be far from the population value. The NIST manual tests whether the correlation differs from zero with an F statistic on 1 and $n - 2$ degrees of freedom, which is the square of the more familiar t statistic.

$$t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$$

where $t$ follows a t distribution with $n - 2$ degrees of freedom under the null hypothesis of no linear association. The resulting p-value addresses only that null hypothesis. A confidence interval is usually more informative, and the NIST manual computes one with Fisher's normal approximation, which transforms $r$ onto a scale where its sampling distribution is close to normal.

$$z = \operatorname{atanh}(r) = \frac{1}{2}\ln\frac{1 + r}{1 - r}, \qquad \text{limits} = \tanh\!\left(z \pm \frac{z_{1-\alpha/2}}{\sqrt{n - 3}}\right)$$

where $z$ is the transformed coefficient, $z_{1-\alpha/2}$ is the standard normal quantile (1.96 for a 95% interval) and $n$ is the number of pairs. Both the test and the interval treat the pairs as independent and rely on approximate normality, which skewed cost data rarely meet (Mukaka and the NIST manual advise rank correlation when normality cannot be assumed). Several records from the same patient, or patients clustered within practices, make the interval too narrow unless the method respects that structure.

Worked example with costs and QALYs from six trial participants

All numbers are illustrative. Six participants in one arm of a trial have the following annual costs and QALYs.

1. Pearson's coefficient. The mean cost is 30/6 = 5 (GBP 5,000) and the mean QALY is 4.12/6 = 0.6867. The cost deviations are −3, −2, −1, 0, 1 and 5, so $S_{xx}$ = 9 + 4 + 1 + 0 + 1 + 25 = 40. The QALY deviations are 0.1133, 0.0633, 0.0133, 0.0333, −0.0867 and −0.1367, giving $S_{yy}$ = 0.04433 and $S_{xy}$ = −1.25. Then $r$ = −1.25 / √(40 × 0.04433) = −1.25 / 1.3317 = −0.939.

2. Spearman's coefficient. There are no ties, so the shortcut applies. The rank differences are −5, −3, 0, 0, 3 and 5, and the sum of squared differences is 25 + 9 + 0 + 0 + 9 + 25 = 68. With 6 × 68 = 408 and 6 × 35 = 210, $r_s$ = 1 − 408/210 = −0.943.

3. Kendall's tau-a. Of the 15 pairs of participants, only participants 3 and 4 move in the same direction on both variables, giving 1 concordant and 14 discordant pairs and $\tau_a$ = (1 − 14)/15 = −0.867.

4. Test and interval. With $n$ = 6, $t$ = −0.939 × √4 / √(1 − 0.881) = −5.44 on 4 degrees of freedom, a two-sided p-value of about 0.006. Fisher's transformation gives $z$ = atanh(−0.9387) = −1.727 with a standard error of 0.577, the reciprocal of the square root of 3. The 95% limits are tanh(−2.858) and tanh(−0.595), or −0.993 to −0.534.

5. One unusual participant. Suppose participant 6 instead had a QALY of 0.85, an expensive patient whose costly treatment worked. Recomputing gives $S_{xy}$ = 0.25 and $S_{yy}$ = 0.03733, so $r$ = 0.25 / √(40 × 0.03733) = 0.205. The QALY ranks become 5, 4, 2, 3, 1 and 6, the squared rank differences sum to 38, and $r_s$ = 1 − 228/210 = −0.086.

6. Interpretation. In the original data cost and QALYs are strongly and negatively associated, the pattern expected when sicker patients both cost more and do worse, and even the wide interval from six people excludes zero. Changing one participant moved Pearson's coefficient from −0.94 to +0.21, reversing its sign, because that participant's cost lies far from the others and dominates the cross-products. Spearman's coefficient also fell, because the change moved that participant's QALY rank from lowest to highest; rank methods limit the pull of extreme magnitudes but still respond to a change in ordering. A coefficient from a handful of patients describes those patients only and should be read with the scatter plot.

Correlation in trial-based and model-based economic evaluation

The coefficient enters economic evaluation at two levels. In a trial-based evaluation it links costs and effects measured on the same patients, and in the REALISE and HypoAware trials, re-analysed by Mutubuki and colleagues, the correlation between costs and effects was −0.42 and −0.44 respectively, which the authors read as individuals with worse outcomes incurring higher costs. Ben and colleagues list bootstrapping costs and effects in pairs among the methods that keep this correlation. In a decision model the coefficient links uncertain parameters, and the NICE technology appraisal manual (PMG36, section 4.7.19) asks for evidence on the extent of correlation between parameters to be reflected in the probabilistic sensitivity analysis, where several coefficients together form a correlation matrix. How each kind of correlation changes decision uncertainty is set out on the Correlation page.

Reading the size of a coefficient

Verbal labels for size are conventions. Mukaka reproduces a rule of thumb from Hinkle and colleagues in which 0.90 to 1.00 is very high, 0.70 to 0.90 high, 0.50 to 0.70 moderate, 0.30 to 0.50 low and below 0.30 negligible, with the same bands for negative values. A modest correlation between two influential model parameters can still move a cost-effectiveness result.

Mukaka also sets two firm limits. The coefficient describes association and gives no information on whether one variable responds to the other, so a correlation between treatment intensity and cost cannot separate the effect of treatment from confounding by severity. It must not be used to assess agreement between methods of measurement, such as patient-reported and record-based resource use, for which Bland-Altman plots are used.

Limitations and common misreadings

Most errors come from reading the coefficient without the data behind it. The following misreadings recur in economic analysis.

• Curved relationships. Mukaka points out that a quadratic relationship can be predicted exactly and still give a Pearson coefficient that is neither −1 nor +1, so a small $r$ does not mean the variables are unrelated.
• Influential points. One high-cost patient can reverse the sign, as the worked example shows.
• Wrong level of correlation. A patient-level correlation between cost and QALYs, the correlation between incremental estimates and the correlation between two model parameters come from different units of analysis and are used in different places.

Sources

• Ben ÂJ, van Dongen JM, El Alili M, Esser JL, Broulíková HM, Bosmans JE. Conducting trial-based economic evaluations using R: a tutorial. PharmacoEconomics. 2023;41(11):1403-1413. https://doi.org/10.1007/s40273-023-01301-7
• Mukaka MM. Statistics corner: a guide to appropriate use of correlation coefficient in medical research. Malawi Medical Journal. 2012;24(3):69-71. https://pmc.ncbi.nlm.nih.gov/articles/PMC3576830/
• Mutubuki EN, El Alili M, Bosmans JE, Oosterhuis T, Snoek FJ, Ostelo RWJG, van Tulder MW, van Dongen JM. The statistical approach in trial-based economic evaluations matters: get your statistics together! BMC Health Services Research. 2021;21:475. https://doi.org/10.1186/s12913-021-06513-1
• 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. Section 4.7.19. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• National Institute of Standards and Technology. Dataplot Reference Manual, entries CORRELATION (2018), CORRELATION CONFIDENCE LIMITS (2023), KENDALLS TAU (2026) and RANK CORRELATION (1997). Gaithersburg, MD: NIST Statistical Engineering Division. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/correlat.htm