Question
Download Solution PDFThe product of partial regression coefficient \(b_1{_2}{_.3}\)\(b_2{_3}{_.1}\)\(b_3{_1}{_.2}\) equals to:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation
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 × (σ1.23/σ2.13)
⇒ b13.2 = r13.2 × (σ2.31/σ3.21)
⇒ b23.1 = r23.1 × (σ2.31/σ3.21)
∴ 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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