Correlation

 One of the terms most frequently misused in medical publications is ‘correlation’. It is often used to describe any kind of relationship or interaction between variables. However, the term is very specific and using it arbitrarily may be misleading.

Apart from non-parametric alternatives and minor variants, two major types of correlation coefficients have been developed: Pearson's correlation coefficient and the intraclass correlation coefficient. The first one measures the linear association between two variables, the second one the resemblance of measurements made within the same unit (repeated measurements) or cluster.

Pearson's correlation coefficient

Pearson's correlation coefficient (ρ) for the two variables X and Y is the covariance of the two variables divided by the product of their standard deviations:

ρ = cov(X,Y)/(σXσY)

Pearson's correlation coefficient ranges from −1 to 1. The value 1 implies that a linear equation describes the relationship between X and Y perfectly, with all observations on a line. Y increases as X increases. If the coefficient is -1, the slope of the line is negative, i.e. Y decreases while X increases.

The intraclass correlation coefficient

When quantitative measurements are made repeatedly on subjects or units that are organised into groups, the intraclass correlation coefficient (ICC) measures the resemblance of measurements made within the same subject or group.

Several ICC estimators have been suggested, but most of them can be defined in terms of a simple random effects model:

Yij = μ + αj + εij

where Yij is the ith observation in the jth subject or group, μ is an overall mean value, αj is a random effect shared by all values in group j, and εij is an unobserved error term. The correlation between observations on the same subject or group is then:

ICC = σα²/(σα² + σε²)

The ICC ranges from 0 to 1. In a sample, negative ICC values may exist, but the population ICC cannot be negative. Therefore, negative ICCs are usually interpreted and treated as 0 (1).

References

1. Chen G, Taylor PA, Haller SP, Kircanski K, Stoddard J, Pine DS, Leibenluft E, Brotman MA, Cox RW. Intraclass correlation: Improved modeling approaches and applications for neuroimaging. Hum Brain Mapp. 2018 Mar;39(3):1187-1206. doi: 10.1002/hbm.23909. Epub 2017 Dec 7. PMID: 29218829; PMCID: PMC5807222.

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