13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.2 Pearson and Spearman correlation
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Written by Resident Doctor Committee · Published
Choosing Pearson or Spearman
A correlation coefficient measures how two paired measures co-vary. It does not test a difference between groups.
Feature | Pearson | Spearman |
|---|---|---|
Symbol | r | rho |
Data | Two continuous, normally distributed variables | Ranked, ordinal, skewed or non-normal data |
Relationship measured | Linear (straight-line trend) | Monotonic: consistently rising or falling, including curves |
Method | Uses the raw values | Applies Pearson's formula to the ranks |
Outliers | Strongly influenced | Less influenced, though not immune |
Spearman's is used when the data break the parametric assumptions that Pearson's requires.
The choice of coefficient follows from the data (distribution, level of measurement, linearity) and says nothing about how strong the association turns out to be.
Kappa is Cohen's chance-corrected measure of agreement, not a correlation coefficient.
Range, sign and strength
Both r and rho are unitless and lie between -1 and +1; no value can fall outside that range.
The sign gives the direction: positive when both variables rise together, negative when one falls as the other rises.
The magnitude shows how closely the points follow a straight line, and reaches 1 only when every point lies on one.
A perfect falling straight line gives r = -1.
An r of 0 means no linear correlation, but it does not rule out a U-shaped relationship in which opposite trends cancel.
Strength is set by the absolute value of r, so a negative coefficient can be stronger than a positive one.
An absolute value of about 0.5 is a moderate correlation.
The weakest positive correlation is the smallest r above zero.
Reading scatter plots
A scatter plot shows the direction, form and approximate strength of an association, but not the exact value of the coefficient.
Feature | Read from |
|---|---|
Direction | The slope of the points |
Strength | The size of r: how tightly the points cluster around a line |
Linearity | Whether the cloud is straight and elliptical or bends |
Points drifting upward with wide scatter around a gently rising fitted line show a weak positive correlation.
Points lying on a smooth curve show a real but non-linear association, so weak, strong, positive and negative do not apply.
What correlation cannot show
Causation: a strong positive Pearson correlation shows that two variables rise together linearly, not that either causes the other. Confounding, reverse causation and chance can all produce it.
Agreement: a high correlation between two instruments measuring the same construct shows only that the scores move together. It does not show agreement, reliability, which instrument is more valid, or that one can replace the other.
Two raters who always score five points apart have r = 1.0 yet never agree.
Agreement is assessed with Bland-Altman analysis, which plots paired differences against paired means, or with an absolute-agreement intraclass correlation coefficient.
Transitivity does not hold: knowing r for A with B and for B with C leaves r for A with C undetermined in both sign and size.
Correlation and regression
A correlation coefficient measures linear association only; predicting one variable from another needs a regression equation.
The regression slope equals r multiplied by the ratio of the standard deviations, so no correlation gives a flat line with no predictive value.
Correlation is symmetrical and unitless, whereas regression gives the outcome and predictor distinct roles and keeps their units.