Concept Architecture
Concept
Theoretically, Spearman Correlation (Spearman's rank correlation coefficient) is a non-parametric measure of the strength and direction of a monotonic association between two variables. Developed by Charles Spearman, it assesses whether higher values of one variable tend to be associated with higher or lower values of another without assuming a linear relationship or Normal distribution. Unlike Pearson Correlation, Spearman Correlation is calculated using the ranks of observations rather than their original values, making it robust to outliers and suitable for ordinal data.
Mathematically, Spearman Correlation is calculated as the Pearson correlation coefficient between the ranked values of two variables. When there are no tied ranks, the coefficient can be computed directly from the squared differences between paired ranks. The resulting statistic, denoted by ? or r?, ranges from ?1 to +1, where +1 indicates a perfect increasing monotonic relationship, ?1 indicates a perfect decreasing monotonic relationship, and 0 indicates no monotonic association.
In practice, Spearman Correlation is estimated by ranking observations and calculating the correlation between the ranks. In health economics it is commonly used to evaluate relationships involving healthcare costs, quality-of-life scores, disease severity scales, patient-reported outcome measures, and resource utilisation when variables are ordinal, skewed, or violate the assumptions required for Pearson Correlation.
Purpose
Used to quantify the strength and direction of monotonic relationships between variables, particularly when data are ordinal, skewed, non-Normally distributed, or contain influential outliers, thereby supporting exploratory and inferential analyses in health economics.
Mathematical Formulae
Primary Formula
For data without tied ranks:
r? = 1 ? [6?d?� � n(n� ? 1)]
where:
- r? = Spearman rank correlation coefficient
- d? = difference between paired ranks
- n = number of paired observations
Supporting Formulae
Rank difference:
d? = R(x?) ? R(y?)
General formulation:
r? = Corr(R(X), R(Y))
Test statistic:
t = r? �[(n ? 2) � (1 ? r?�)]
Related Mathematical Methods
- Pearson Correlation
- Kendall's Tau
- Rank Correlation
- Non-Parametric Statistics
- Monotonic Association
- Correlation Matrix
- Hypothesis Testing
Example
A health economist examines the relationship between disease severity rankings and annual healthcare cost rankings for 12 patients.
The sum of squared rank differences is:
?d� = 28
The Spearman correlation coefficient is:
r? = 1 ? [6 ? 28 � (12 ? (12� ? 1))]
= 1 ? (168 � 1,716)
= 0.902
This indicates a strong positive monotonic relationship, suggesting that patients with greater disease severity generally incur higher healthcare costs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RANK.AVG | =RANK.AVG(B2,$B$2:$B$101) | Rank healthcare costs or outcome values. |
| RANK.AVG | =RANK.AVG(C2,$C$2:$C$101) | Rank disease severity or utility scores. |
| CORREL | =CORREL(D2:D101,E2:E101) | Calculate Spearman correlation from ranked variables. |
| POWER | =POWER(D2-E2,2) | Calculate squared rank differences for manual computation. |
| SUM | =SUM(F2:F101) | Sum squared rank differences. |
VBA (Optional)
Automate ranking of variables, calculation of Spearman correlation coefficients, hypothesis tests, and correlation matrices for multiple health economic outcomes.
Sources
- Spearman C. The Proof and Measurement of Association Between Two Things. The American Journal of Psychology. 1904.
- Conover WJ. Practical Nonparametric Statistics.
- Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods.
- Altman DG. Practical Statistics for Medical Research.
- 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.
- NICE Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the Spearman correlation?
A non-parametric measure of the monotonic relationship between two ranked variables, calculated from data ranks rather than raw values.
Source: Spearman 1904
What relationship does the Spearman correlation capture?
The Spearman correlation captures the strength and direction of a monotonic relationship between two variables, whether or not that relationship is a straight line. It works from the ranks of the values rather than the values themselves, so it asks only whether higher values of one variable tend to go with higher values of the other. This makes it a non-parametric alternative to the Pearson correlation, suited to skewed data, ordinal measures, or relationships that are consistent but curved. Whether two variables move together in rank is what it captures. Kirkwood and Sterne (2003) describe this measure.
Source: Kirkwood & Sterne 2003
How is the Spearman correlation calculated?
The Spearman correlation is calculated by ranking the values of each variable and then computing the Pearson correlation between the ranks, or equivalently through a formula based on the differences between the ranks of paired observations. So the Spearman correlation is calculated from the ranks rather than the actual values, which is why it captures monotonic rather than specifically linear relationships and is robust to outliers, since converting to ranks limits the influence of extreme values, and this rank-based computation distinguishes it from the Pearson correlation, which uses the raw values and measures linear association.
Source: Spearman 1904
When is the Spearman correlation used?
The Spearman correlation is used when the data are ordinal, when the relationship between two variables is monotonic but not necessarily linear, or when the data contain outliers or are not normally distributed, situations where the Pearson correlation may be inappropriate. So the Spearman correlation is used as a robust, non-parametric alternative to the Pearson correlation, suitable for ranked data and monotonic relationships, which is why it is preferred when the assumptions of the Pearson correlation do not hold, providing a valid measure of association based on ranks that is less affected by outliers and does not require a linear relationship.
Source: Spearman 1904
How does the Spearman correlation differ from the Pearson correlation?
The Spearman correlation measures the strength of a monotonic relationship using the ranks of the data, while the Pearson correlation measures the strength of a linear relationship using the raw values. The Spearman is more robust to outliers and suits ordinal or non-linear monotonic data, whereas the Pearson is more powerful for genuinely linear relationships in continuous data. So the two differ in using ranks versus values and in the type of relationship they capture, which is why the Spearman correlation is chosen for ordinal, non-normal, or monotonic but non-linear data and the Pearson for linear relationships, according to the data and the assumptions.
Source: Pearson 1896
How is the Spearman correlation interpreted?
The Spearman correlation is interpreted by its sign and magnitude: a positive value means that as one variable increases the other tends to increase, a negative value that one tends to decrease as the other increases, and the magnitude, from zero to one in absolute terms, indicates the strength of the monotonic association. So the Spearman correlation is interpreted as the direction and strength of a monotonic relationship based on ranks, with values near one indicating a strong monotonic association, though, like all correlations, it does not imply causation, and its rank basis means it captures monotonic rather than specifically linear patterns.
Source: Spearman 1904
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
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