For three random variables \(X_1, X_2\) and \(X_3\), correlation coefficients between pairs are \(r_{12} = \sin^2\theta\), \(r_{13} = \cos\theta\), and \(r_{23} = \sin\theta\). The partial correlation coefficient \(r_{12,3}\) is given by:
\(\tan\theta - 1\)
The partial correlation coefficient formula is:
\(r_{12,3} = \dfrac{r_{12} - r_{13}\,r_{23}}{\sqrt{(1 - r_{13}^2)(1 - r_{23}^2)}}\)
Substituting \(r_{12} = \sin^2\theta\), \(r_{13} = \cos\theta\), \(r_{23} = \sin\theta\):
Numerator: \(\sin^2\theta - \cos\theta\sin\theta = \sin\theta(\sin\theta - \cos\theta)\)
Denominator: \(\sqrt{(1 - \cos^2\theta)(1 - \sin^2\theta)} = \sqrt{\sin^2\theta\cdot\cos^2\theta} = \sin\theta\cos\theta\)
Therefore: \(r_{12,3} = \dfrac{\sin\theta(\sin\theta - \cos\theta)}{\sin\theta\cos\theta} = \dfrac{\sin\theta - \cos\theta}{\cos\theta} = \tan\theta - 1\)
Hence the correct answer is tanθ − 1.
Important Questions from Correlation and regression of two variables
In the case of two variables. the estimated regression equation is ŷ = 60 + 5x. The total sum of squares is 15730 and the sum of squares due to error is 1530. The estimated regression line based on this information is a ______.
The Regression Coefficient is independent of the change of
(A) Scale only
(B) Origin only
(C) Both Scale and Origin
(D) Neither Scale nor Origin
Choose the most appropriate answer from the options given below:
Which of following is not correct about properties of correlation coefficient?
(A) Depends on the origin.
(B) Depends on the scale.
(C) Depends on both origin and scale.
(D) Is independent with respect to origin.
(E) Is independent with respect to unit of scale.
Choose the correct answer from the options given below :
- What is true about multiple regression ?
(A) There can be more than two criterion
(B) There can be more than two predictors
(C) It indicates linear relation between one predicator and one criterion
(D) The equation for regression line contains partial regression coefficients
Choose the correct answer from the options given below :
