The product of partial regression coefficient \(b_1{_2}{_.3}\)\(b_2{_3}{_.1}\)\(b_3{_1}{_.2}\) equals to:

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SSC CGL JSO Tier-II, 2018 (Statistics) Official Paper-III (Held On: 14 Sept, 2019)
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  1. \(\dfrac{r_1{_2}{_.3}+r_2{_3}{_.1}+r_3{_1}{_.2}}{3}\)
  2. \((r_1{_2}{_.3}r_2{_3}{_.1}r_3{_1}{_.2})^\dfrac{1}{2}\)
  3. \(\dfrac{1}{r_1{_2}{_.3}} + \dfrac{1}{r_2{_3{_.1}}} + \dfrac{1}{r_3{_1}{_.2}}\)
  4. \(r_1{_2}{_.3}r_2{_3}{_.1}r_3{_1}{_.2}\)

Answer (Detailed Solution Below)

Option 4 : \(r_1{_2}{_.3}r_2{_3}{_.1}r_3{_1}{_.2}\)
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Detailed Solution

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Explanation

The product of partial regression coefficients b12.3b23.1b31.2 is equal to r12.3r23.1r31.2

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

r12.3 = Partial correlation coefficient between x1 and x2 keeping x3 as constant

r23.1 = Partial correlation coefficient between x2 and x3 keeping x1 as constant

r13.2 = Partial correlation coefficient between x1 and x3 keeping x2 as constant

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

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

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

⇒ b12.3 = r12.3 × (σ1.232.13)

⇒ b13.2 = r13.2 × (σ2.313.21)

⇒ b23.1 = r23.1 × (σ2.313.21)

b12.3b13.2 × b23.1 = r12.3r13.2 .r23.1

 

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 rxy

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