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Correlation

Partial Correlation — Controlling for a Third Variable

Does exercise still relate to blood pressure once age is controlled for? Zero-order correlations, the partial-correlation formula, and significance test (df=n-3).

📊 Partial r(9) = −.846, p = .001 (zero-order r = −.939)
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🔗 33 · Partial Correlation — Controlling for a Third Variable

Exercise and Blood Pressure, Controlling for Age
Correlation
Research Question: Weekly exercise hours (X) correlate strongly with systolic blood pressure (Y) in 12 adults — but age (Z) also correlates with both (older adults tend to exercise less AND have higher blood pressure). Does exercise still relate to blood pressure once age is statistically removed?
Participant Exercise (hrs/wk) — X Systolic BP — Y Age — Z
1114858
2214562
3215045
4313860
5414250
6413040
7512848
8613535
9612042
10711830
11811533
12911028
First-order partial correlation of X and Y, controlling for Z
r X Y . Z = r X Y − r X Z r Y Z ( 1 − r X Z 2 ) ( 1 − r Y Z 2 )
  1. 1
    Compute the three zero-order Pearson correlations:
    r X Y = − .939 , r X Z = − .869 , r Y Z = .778
    Exercise and BP are strongly negatively correlated, but exercise and age are also strongly correlated, and age and BP too — exactly the confound pattern that makes a partial correlation necessary.
  2. 2
    Plug into the partial-correlation formula:
    r X Y . Z = − .939 − ( − .869 ) ( .778 ) ( 1 − .755 ) ( 1 − .605 ) = − .263 .245 × .395 = − .263 .311 = − .846
  3. 3
    Significance test for the partial r (df = n − 3, one less than the zero-order test's df = n − 2):
    t = r d f 1 − r 2 = − .846 9 1 − .716 = − 4.765 , d f = 9 ,   p = .001
  4. 4
    Decision rule: p = .001 < .05 → reject H₀ → exercise and blood pressure remain significantly related after removing age's shared variance. Age is a genuine confound (it attenuates r from −.939 to −.846) but does not fully explain the relationship.
−.846
Partial r (df=9)
−4.765
t(9)
.001
p
−.939
Zero-order r (for contrast)
🔴 Significant — partial r(9) = −.846, p = .001, controlling for age. Age attenuates but does not eliminate the exercise–BP relationship.
APA-7
A partial correlation controlling for age showed that weekly exercise hours remained significantly associated with systolic blood pressure, r(9) = −.846, p = .001, after removing the variance shared with age (zero-order r = −.939).

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