Concept Architecture
Concept
Theoretically, Pearson Correlation is a statistical measure of the strength and direction of the linear relationship between two continuous variables. Developed by Karl Pearson, it is based on covariance and standardisation, producing a dimensionless coefficient that ranges from ?1 to +1. The coefficient quantifies the extent to which changes in one variable are associated with proportional changes in another, assuming a linear relationship.
Mathematically, Pearson Correlation is defined as the covariance between two variables divided by the product of their standard deviations. The resulting correlation coefficient (r) is scale-independent and measures the degree of linear association. Values of +1 indicate perfect positive linear correlation, ?1 indicates perfect negative linear correlation, and 0 indicates no linear correlation.
In practice, Pearson Correlation is estimated from paired observations and is widely used in health economics to examine associations between costs, utilities, clinical outcomes, healthcare utilisation, and model parameters. It is commonly applied during exploratory data analysis, model validation, variable selection, and assessment of relationships prior to regression modelling.
Purpose
Used to quantify the strength and direction of the linear association between two continuous variables, supporting exploratory analysis, model development, variable selection, and interpretation of relationships within health economic data.
Mathematical Formulae
Primary Formula
r = ?[(x? ? x?)(y? ? ?)] � �[?(x? ? x?)� ? ?(y? ? ?)�]
Supporting Formulae
Cov(X,Y) = (1/(n?1)) ? ?(x? ? x?)(y? ? ?)
�? = �[(1/(n?1)) ? ?(x? ? x?)�]
�? = �[(1/(n?1)) ? ?(y? ? ?)�]
r = Cov(X,Y) � (�? ? �?)
t = r ? �[(n ? 2) � (1 ? r�)]
Related Mathematical Methods
- Covariance
- Partial Correlation
- Spearman Rank Correlation
- Linear Regression
- Correlation Matrix
- Fisher's z Transformation
- Hypothesis Testing for Correlation
Example
A health economist investigates the relationship between annual healthcare costs and EQ-5D utility scores for 250 patients.
Mean annual cost = �9,800
Mean EQ-5D utility = 0.74
The calculated covariance and standard deviations produce:
r = ?0.62
This indicates a moderately strong negative linear association, suggesting that higher healthcare costs are associated with lower health-related quality of life.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =CORREL(B2:B251,C2:C251) | Calculate Pearson correlation between healthcare costs and QALYs. |
| PEARSON | =PEARSON(B2:B251,C2:C251) | Alternative calculation of Pearson correlation coefficient. |
| COVARIANCE.S | =COVARIANCE.S(B2:B251,C2:C251) | Calculate sample covariance between two variables. |
| STDEV.S | =STDEV.S(B2:B251) | Estimate sample standard deviation for correlation calculations. |
| T.TEST | =T.TEST(B2:B251,C2:C251,2,2) | Perform supporting statistical comparisons where appropriate. |
VBA (Optional)
Automate calculation of Pearson correlation coefficients and generate correlation matrices across multiple clinical, cost, and utility variables.
Sources
- Pearson K. Notes on Regression and Inheritance in the Case of Two Parents. Proceedings of the Royal Society of London. 1895.
- Altman DG. Practical Statistics for Medical Research.
- Agresti A. Foundations of Linear and Generalized Linear Models.
- 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.
- ISPOR Good Practice Reports.
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 the Pearson correlation?
A measure quantifying the strength and direction of the linear relationship between two continuous variables, ranging from negative one to positive one.
Source: Pearson 1896
What relationship does the Pearson correlation measure?
The Pearson correlation measures the strength and direction of the straight-line relationship between two continuous variables, giving a value between negative one and positive one. A figure near positive one means they rise together in near-perfect linear step, near negative one that one falls as the other rises, and near zero that no linear pattern is present. Because it captures only linear association, it can miss a strong but curved relationship, and it assumes roughly normal, evenly spread data. The linear tie between two continuous variables is what it captures. Kirkwood and Sterne (2003) describe this measure.
Source: Kirkwood & Sterne 2003
How is the Pearson correlation calculated?
The Pearson correlation is calculated as the covariance of the two variables divided by the product of their standard deviations, which standardises the covariance to lie between negative one and positive one. It thus reflects how the variables vary together relative to their individual variability. So the Pearson correlation is calculated by standardising the covariance between the two variables, giving a scale-free measure of linear association, and this computation uses the actual values of the variables, which is why it captures linear relationships and is sensitive to outliers, unlike rank-based correlations that use only the order of the values.
Source: Pearson 1896
What are the assumptions of the Pearson correlation?
The Pearson correlation assumes that the relationship between the two variables is linear, that the variables are continuous, and, for inference such as significance testing, that they are approximately normally distributed; it is also sensitive to outliers. So the Pearson correlation rests on assumptions of linearity and, for testing, normality, and its interpretation can be distorted by non-linearity or outliers, which is why the data are examined, often with a scatterplot, before relying on it, since a Pearson correlation near zero indicates weak linear association but not the absence of a non-linear relationship, and outliers can inflate or deflate the coefficient.
Source: Pearson 1896
How is the Pearson correlation interpreted?
The Pearson correlation is interpreted by its sign and magnitude: a positive value means the variables tend to increase together, a negative value that one decreases as the other increases, and the magnitude, from zero to one in absolute terms, indicates the strength of the linear relationship. So the Pearson correlation is interpreted as the direction and closeness of a linear association, with values near one being strong and near zero weak, though it does not imply causation and captures only linear relationships, so it is interpreted alongside a scatterplot and with awareness that a low value does not rule out a strong non-linear relationship.
Source: Pearson 1896
How does the Pearson correlation differ from rank correlations?
The Pearson correlation measures the strength of a linear relationship between two continuous variables using their actual values, while rank correlations such as Spearman's and Kendall's measure monotonic association using the ranks, making them suitable for ordinal data or non-linear monotonic relationships and more robust to outliers. So the Pearson correlation differs from rank correlations in using values rather than ranks and in assuming linearity, which makes it powerful for linear relationships in continuous data but less robust to outliers and non-linearity, whereas rank correlations trade some of that for robustness and applicability to ordinal and monotonic but non-linear data.
Source: Pearson 1896
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 22 Dec 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/pearson-correlation
- Term code
- HE-ES-SA-155
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