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Partial r12.3 and multiple R1.23 (0.69, 0.64, 0.85)

Short answer

Partial correlation r12.3 = (r12 − r13 r23)/√[(1−r13²)(1−r23²)] = 0.361. Multiple correlation R1.23 = √[(r12²+r13²−2r12 r13 r23)/(1−r23²)] = 0.697.

The simple correlation coefficient between fertilizer (X1), seeds (X2) and productivity (X3) are r12 = 0.69, r13 = 0.64 and r23 = 0.85. Calculate the partial correlation r12.3 and multiple correlations R1.23.

Given: r₁₂ = 0.69, r₁₃ = 0.64, r₂₃ = 0.85 (X₁ = fertilizer, X₂ = seeds, X₃ = productivity).

Partial correlation r₁₂.₃

r₁₂.₃ = (r₁₂ − r₁₃r₂₃) / √[(1 − r₁₃²)(1 − r₂₃²)]

  • Numerator = 0.69 − (0.64)(0.85) = 0.69 − 0.5440 = 0.1460
  • 1 − r₁₃² = 0.5904; 1 − r₂₃² = 0.2775
  • Denominator = √(0.5904 × 0.2775) = 0.4048

r₁₂.₃ = 0.1460 / 0.4048 = 0.3607

Multiple correlation R₁.₂₃

R₁.₂₃² = (r₁₂² + r₁₃² − 2r₁₂r₁₃r₂₃) / (1 − r₂₃²)

  • r₁₂² + r₁₃² = 0.4761 + 0.4096 = 0.8857
  • 2r₁₂r₁₃r₂₃ = 2(0.69)(0.64)(0.85) = 0.7507
  • Numerator = 0.1350; denominator = 1 − 0.7225 = 0.2775

R₁.₂₃² = 0.4864, so R₁.₂₃ = 0.6974

Answer: r₁₂.₃ ≈ 0.361, R₁.₂₃ ≈ 0.697. R₁.₂₃ exceeds both r₁₂ and r₁₃, as it must.

✓ Answer checked by Claude Sonnet 5.5

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