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Linear regression
Statistics - Linear regression
Once the degree of relationship between variables has been estabpshed using co-relation analysis, it is natural to delve into the nature of relationship. Regression analysis helps in determining the cause and effect relationship between variables. It is possible to predict the value of other variables (called dependent variable) if the values of independent variables can be predicted using a graphical method or the algebraic method.
Graphical Method
It involves drawing a scatter diagram with independent variable on X-axis and dependent variable on Y-axis. After that a pne is drawn in such a manner that it passes through most of the distribution, with remaining points distributed almost evenly on either side of the pne.
A regression pne is known as the pne of best fit that summarizes the general movement of data. It shows the best mean values of one variable corresponding to mean values of the other. The regression pne is based on the criteria that it is a straight pne that minimizes the sum of squared deviations between the predicted and observed values of the dependent variable.
Algebraic Method
Algebraic method develops two regression equations of X on Y, and Y on X.
Regression equation of Y on X
${Y = a+bX}$
Where −
${Y}$ = Dependent variable
${X}$ = Independent variable
${a}$ = Constant showing Y-intercept
${b}$ = Constant showing slope of pne
Values of a and b is obtained by the following normal equations:
${sum Y = Na + bsum X \[7pt] sum XY = a sum X + b sum X^2 }$
Where −
${N}$ = Number of observations
Regression equation of X on Y
${X = a+bY}$
Where −
${X}$ = Dependent variable
${Y}$ = Independent variable
${a}$ = Constant showing Y-intercept
${b}$ = Constant showing slope of pne
Values of a and b is obtained by the following normal equations:
${sum X = Na + bsum Y \[7pt] sum XY = a sum Y + b sum Y^2 }$
Where −
${N}$ = Number of observations
Example
Problem Statement:
A researcher has found that there is a co-relation between the weight tendencies of father and son. He is now interested in developing regression equation on two variables from the given data:
| Weight of father (in Kg) | 69 | 63 | 66 | 64 | 67 | 64 | 70 | 66 | 68 | 67 | 65 | 71 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Weight of Son (in Kg) | 70 | 65 | 68 | 65 | 69 | 66 | 68 | 65 | 71 | 67 | 64 | 72 |
Develop
Regression equation of Y on X.
Regression equation of on Y.
Solution:
| ${X}$ | ${X^2}$ | ${Y}$ | ${Y^2}$ | ${XY}$ |
|---|---|---|---|---|
| 69 | 4761 | 70 | 4900 | 4830 |
| 63 | 3969 | 65 | 4225 | 4095 |
| 66 | 4356 | 68 | 4624 | 4488 |
| 64 | 4096 | 65 | 4225 | 4160 |
| 67 | 4489 | 69 | 4761 | 4623 |
| 64 | 4096 | 66 | 4356 | 4224 |
| 70 | 4900 | 68 | 4624 | 4760 |
| 66 | 4356 | 65 | 4225 | 4290 |
| 68 | 4624 | 71 | 5041 | 4828 |
| 67 | 4489 | 67 | 4489 | 4489 |
| 65 | 4225 | 64 | 4096 | 4160 |
| 71 | 5041 | 72 | 5184 | 5112 |
| ${sum X = 800}$ | ${sum X^2 = 53,402}$ | ${sum Y = 810}$ | ${sum Y^2 = 54,750}$ | ${sum XY = 54,059}$ |
Regression equation of Y on X
Y = a+bX
Where , a and b are obtained by normal equations
${sum Y = Na + bsum X \[7pt] sum XY = a sum X + b sum X^2 \[7pt] Where sum Y = 810, sum X = 800, sum X^2 = 53,402 \[7pt] , sum XY = 54, 049, N = 12 }$${Rightarrow}$ 810 = 12a + 800b ... (i)
${Rightarrow}$ 54049 = 800a + 53402 b ... (ii)
Multiplying equation (i) with 800 and equation (ii) with 12, we get:
96000 a + 640000 b = 648000 ... (iii)
96000 a + 640824 b = 648588 ... (iv)
Subtracting equation (iv) from (iii)
-824 b = -588
${Rightarrow}$ b = -.0713
Substituting the value of b in eq. (i)
810 = 12a + 800 (-0.713)
810 = 12a + 570.4
12a = 239.6
${Rightarrow}$ a = 19.96
Hence the equation Y on X can be written as
${Y = 19.96 - 0.713X}$Regression equation of X on Y
X = a+bY
Where , a and b are obtained by normal equations
${sum X = Na + bsum Y \[7pt] sum XY = a sum Y + b sum Y^2 \[7pt] Where sum Y = 810, sum Y^2 = 54,750 \[7pt] , sum XY = 54, 049, N = 12 }$${Rightarrow}$ 800 = 12a + 810a + 810b ... (V)
${Rightarrow}$ 54,049 = 810a + 54, 750 ... (vi)
Multiplying eq (v) by 810 and eq (vi) by 12, we get
9720 a + 656100 b = 648000 ... (vii)
9720 a + 65700 b = 648588 ... (viii)
Subtracting eq viii from eq vii
900b = -588
${Rightarrow}$ b = 0.653
Substituting the value of b in equation (v)
800 = 12a + 810 (0.653)
12a = 271.07
${Rightarrow}$ a = 22.58
Hence regression equation of X and Y is
${X = 22.58 + 0.653Y}$ Advertisements