13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.1 Choosing a correlation coefficient
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Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
Match the coefficient to the variables
Choose the coefficient according to the type of data in each variable.
Coefficient | Variables | What it measures |
|---|---|---|
Pearson r | Two numerical variables | Linear association, using the original values |
Spearman rho | Two ordered variables | Monotonic association, using ranks |
Kendall tau | Two ordered variables | How far pairs of observations are ordered the same way on both variables |
Phi | Two binary variables | Association in a two-by-two table |
Point-biserial | One binary and one continuous variable | Association between group membership and the continuous measure |
What the coefficients describe
Linear correlation shows how closely paired observations lie along a straight line. The sign gives the direction. The absolute value (its size, ignoring the sign) gives the strength.
Monotonic association means that as one variable rises, the other keeps rising (or keeps falling), but not necessarily in a straight line. A rank coefficient is therefore useful when ordered variables have a curved relationship.
Phi is simply Pearson correlation calculated on two variables coded 0 and 1. The two names refer to the same coefficient.
Point-biserial correlation is also Pearson correlation, here with one genuinely binary variable and one continuous variable.
Interpret with care
You can calculate Pearson r without normally distributed data. The usual inferential tests on it, though, need further assumptions about how the data are distributed.
Outliers can change Pearson r a great deal. Look at the scatter plot before you interpret any coefficient.
A variable that never changes has zero variance. Its correlation is undefined, not zero.
A Pearson r of zero does not rule out a curved association.
A high correlation does not show causation. Nor does it show that two instruments agree with each other.
The way you code a binary variable sets the sign of phi and point-biserial coefficients. Reverse the coding of one variable and the sign flips, but the strength of the association stays the same.
Worked example with ranks
Four hypothetical observations all have different values, so there are no ties (no two observations share a rank).
Observation | Rank of X | Rank of Y | Squared rank difference |
|---|---|---|---|
A | 1 | 1 | 0 |
B | 2 | 3 | 1 |
C | 3 | 2 | 1 |
D | 4 | 4 | 0 |
Add up the squared rank differences: 0 + 1 + 1 + 0 = 2.
With no ties, Spearman rho = 1 minus [6 x 2 / (4 x (4^2 minus 1))] = 0.80.
Four observations make six possible pairs. Five pairs are concordant and one is discordant.
With no ties, Kendall tau = (5 minus 1) / 6 = 0.667.
Some points to take from this:
Concordant pairs are ordered the same way on X and Y. Discordant pairs are ordered in opposite ways. In this example, only the pair B and C is discordant.
Ties need proper handling. For Spearman, give tied values their average rank. For Kendall, use the tie-adjusted version, tau-b.
Different methods need not give the same value, even when they describe the same dataset. Here Spearman gives 0.80 and Kendall gives 0.667.