Which set of simple correlations does the answer obtain?

r12 = -0.1925, r13 = 0.618, r23 = 0.3638

r12 = 0.1925, r13 = 0.618, r23 = 0.364

r12 = -0.35, r13 = 0.109, r23 = 0.12

r12 = -0.1925, r13 = 0.3638, r23 = 0.618

Which formula gives r13.2?

(r13 − r12r23)/√[(1 − r12²)(1 − r23²)]

(r13 + r12r23)/√[(1 − r12²)(1 − r23²)]

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

In R3.12² = (r13² + r23² − 2r13r23r12)/(1 − r12²), what does X3 being dependent mean for the result?

R3.12 = 0.790 measures how well X1 and X2 jointly explain X3

R3.12 = 0.753 equals the partial correlation

R3.12 = 0.624 is the correlation itself

Asked 2/17
2074 Asoj · 5 marks

Partial and multiple correlation: 8 values

Short answer

Using corrected sums and simple correlations, the partial correlation r13.2 is 0.753, showing a strong positive relation between X1 and X3 after removing X2. The multiple correlation with X3 dependent is R3.12 = 0.790.

A sample of 8 values of three variables X1, X2 and X3 were obtained as ΣX1 = 360, ΣX2 = 64, ΣX3 = 48, ΣX1² = 17172, ΣX2² = 546, ΣX3² = 320, ΣX1X2 = 2845, ΣX1X3 = 2269, ΣX2X3 = 396. Find: i) Partial correlation between X1 and X3 eliminating the effect of X2 ii) Multiple correlation between X1, X2 and X3 assuming X3 as dependent

n = 8. Corrected sums Sij = ΣXiXj − (ΣXi)(ΣXj)/n:

S₁₁ = 17172 − 360²/8 = 972.00
S₂₂ = 546 − 64²/8 = 34.00
S₃₃ = 320 − 48²/8 = 32.00
S₁₂ = 2845 − (360)(64)/8 = -35.00
S₁₃ = 2269 − (360)(48)/8 = 109.00
S₂₃ = 396 − (64)(48)/8 = 12.00

Simple correlations:

r₁₂ = -35/√(972×34) = -0.1925
r₁₃ = 109/√(972×32) = 0.6180
r₂₃ = 12/√(34×32) = 0.3638

i) Partial correlation r₁₃.₂

r₁₃.₂ = (r₁₃ − r₁₂r₂₃)/√[(1 − r₁₂²)(1 − r₂₃²)] = (0.6180 − (-0.1925)(0.3638)) / √[(1 − 0.0371)(1 − 0.1324)] = 0.6881 / 0.9140 = 0.7528

ii) Multiple correlation R₃.₁₂ (X₃ dependent)

R₃.₁₂² = (r₁₃² + r₂₃² − 2r₁₃r₂₃r₁₂)/(1 − r₁₂²) = (0.3820 + 0.1324 − 2(0.6180)(0.3638)(-0.1925)) / (1 − 0.0371) = 0.6009 / 0.9629 = 0.6240

R₃.₁₂ = 0.7900

Answer: r₁₃.₂ = 0.753, R₃.₁₂ = 0.790.

✓ Answer checked by Claude Sonnet 5.5

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Partial and multiple correlation: 8 values

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