13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression

13.10.8 Partial and canonical correlation

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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.

Partial correlation

Partial correlation measures the linear association between two variables after taking account of other specified variables in both of them.

A residual is what is left of a variable after predicting it from the adjustment variables. The correlation between the residuals of X and the residuals of Y is the partial correlation.

Adjustment can make an association bigger or smaller, or even reverse it. It cannot remove unmeasured confounding, and it does not show cause and effect.

Illustrative calculation

Take numerical variables X and Y and one adjustment variable Z.

Sample correlation

Value

X with Y

0.60

X with Z

0.50

Y with Z

0.40

  1. Numerator = 0.60 minus (0.50 x 0.40) = 0.40.

  2. Denominator = square root of [(1 minus 0.50^2) x (1 minus 0.40^2)] = square root of 0.63 = 0.794.

  3. Partial correlation = 0.40 / 0.794 = 0.504, compared with the unadjusted 0.60.

The constructed sample has rXY = 0.60, rXZ = 0.50 and rYZ = 0.40, matching the chapter. Removing the fitted linear component of Z from both variables leaves a residual correlation of about 0.504; this adjustment does not establish causality. Original illustration drawn in Python for MRCPsychiatrist; see images/ATTRIBUTION.md

Canonical correlation

Canonical correlation looks at the association between two sets of numerical variables. It forms a weighted combination of the variables in each set and correlates the two combinations.

  • The first pair of combinations is chosen to give the highest possible correlation.

  • Later pairs describe further association. Each must be uncorrelated with the earlier pairs (orthogonality constraints).

For example, a study might relate a set of cognitive measures to a set of symptom measures, rather than picking one pair of scores.

Method

Variables on each side

Output

Ordinary correlation

One and one

One coefficient

Partial correlation

One and one, with adjustment variables

Adjusted coefficient

Canonical correlation

A set and a set

Correlations between paired weighted combinations

Traps

  • Unstable weights. Canonical weights can be unstable with small samples or strongly correlated variables. Check the loadings and stability before interpreting them.

  • A large canonical correlation does not mean every variable in one set is strongly associated with every variable in the other.

  • No causal ordering. Neither partial nor canonical correlation shows which variable comes first or causes the other.

Revision material for the MRCPsych examinations. Not clinical guidance, and not for use in patient care.