The product of partial regression coefficient \(b_1{_2}{_.3}\) \(b_2{_3}{_.1}\) \(b_3{_1}{_.2}\) equals to:
For three variables, the partial regression coefficients can be written in terms of variances and partial correlations as \(b_{ij.k}=r_{ij.k}\cdot\dfrac{\sigma_{i.k}}{\sigma_{j.k}}\).
Multiplying the three coefficients in cyclic order:
\[b_{12.3}\,b_{23.1}\,b_{31.2}=\left(r_{12.3}\frac{\sigma_{1.3}}{\sigma_{2.3}}\right)\left(r_{23.1}\frac{\sigma_{2.1}}{\sigma_{3.1}}\right)\left(r_{31.2}\frac{\sigma_{3.2}}{\sigma_{1.2}}\right)\]
The standard-deviation ratios cancel cyclically, leaving:
\[b_{12.3}\,b_{23.1}\,b_{31.2}=r_{12.3}\,r_{23.1}\,r_{31.2}\]
Hence the product equals the product of the corresponding partial correlation coefficients.
Similar Questions
If six hand-writings were ranked by two judges in a competition and the rankings are as follows:
Hand Writing 1 2 3 4 5 6 Judge1 6 5 4 3 2 1 Judge2 1 2 3 4 5 6
Then the value of Spearman's rank correlation coefficient isIf the difference between the rank of the 4 observations are 2.5, 0.5, -1.5, -1.5, then Spearman's rank correlation coefficient equals to:
For the data set
x: 1 2 3 4 y: 4 5 3 2 the regression coefficient b yx (y on x) equals to:
For the two variables X and Y, the following observations are tabulated.
X: 3 4 4 Y: 10 10 9 The Spearman's correlation coefficient is:
For the variables X and Y, we collect 4 observations with ∑x = 10, ∑y = 14, ∑x 2= 30, ∑y 2= 54, ∑xy = 31. The regression line y on x is
The coefficient of correlations is r between X and Y having standard deviation σx and σy. The tangent of the between two lines of regression is:
In Spearman rank correlation coefficient r s= 1 - \(\dfrac{6∑ d^2}{n(n^2-1)}\) , the maximum value of ∑d 2in case of untied rank is:
If the correlation coefficient between x and y is 0.5 and the regression coefficient of y on x is 0.5, then the regression coefficient of x on y is:
If X 1, X 2, … X nis a simple random sample without replacement of size n from a finite population of N units with mean μ and variance σ 2, the covariance of (X i, Xj ) will be:
If x = X- \(\bar{X}\) and y = Y - \(\bar{Y}\) and the number of pairs (X, Y) is n, then the Karl Pearson's coefficient of correlation is:
Important Questions from Correlation and Regression
- For testing the significance of overall regression the test to be used is
- Including relevant lagged values of the dependent variable on the right hand side of a regression equation could lead to which one of the following ?
- If OLS is used in the presence of auto-correlation, which of the following will be likely consequences ?
i. Coefficients estimates may be misleading.
ii. Hypothesis tests could reach the wrong conclusions.
iii. Forecasts made from the model could be biased.
iv. Standard errors may be inappropriate. - Regression analysis is of particular interest
- Which of the following can be linked with those providing method for estimating black money in India ?
I. Kaldor’s approach
II. NIPFP method
III. O.P. Chopra’s method
IV. Arjun Sengupta
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