The product of partial regression coefficient  equals to:

Explanation

The product of partial regression coefficients b_12.3b_23.1b_31.2 is equal to r_12.3r_23.1r_31.2

Partial correlation coefficient

Let x₁, x₂, x₃ be three variables

r₁₂ = Simple correlation coefficient between x₁ and x₂

r₂₃ = Simple correlation coefficient between x₂ and x₃

r₃₁ = Simple correlation coefficient between x₃ and x₁

r_12.3 = Partial correlation coefficient between x₁ and x₂ keeping x₃ as constant

r_23.1 = Partial correlation coefficient between x₂ and x₃ keeping x₁ as constant

r_13.2 = Partial correlation coefficient between x₁ and x₃ keeping x₂ as constant

⇒ r_12.3 = (r₁₂ – r₁₃. r₂₃)/[√(1-r²₁₃) × √(1-r²₂₃)]

⇒ r_13.2 = (r₁₃ – r₁₂. r₂₃)/(√1-r²₂₃) × √(1-r²₁₂)]

⇒ r_23.1 = (r₂₃ – r₁₂. R₁₃)/(√1-r²₁₃) × √(1-r²₁₂)]

⇒ b_12.3 = r_12.3 × (σ_1.23/σ_2.13)

⇒ b_13.2 = r_13.2 × (σ_2.31/σ_3.21)

⇒ b_23.1 = r_23.1 × (σ_2.31/σ_3.21)

∴ b_12.3b_13.2 × b_23.1 = r_12.3r_13.2 .r_23.1

The partial correlation is used ina situation where three or more variables are involved. The three variables may be age, height and weight. Correlation between height and weight can be computed by keeping age constant. The partial correlation is used to keep constant the effect of age.

This statistical technique is caleed as partial correlation betweenx and y and it is denottd by r_xy