Regression

Regression

1. REGRESSION NADEEM UDDIN ASSOCIATE PROFESSOR OF STATISTICS

2. Regression The term regression used firstly by “ Sir Frances Galton ” is used for all such problems where we have to estimate or predict one variable on the basis of another variable.     Definition A process by which we predict or estimate values of one dependent variable from known values of other independent variables is called regression.

3. Regressand and Regressor In regression process the dependent variable is called a random variable or regressand and the independent variable is called fixed variable or regressor .     Dependent Variable The variable which is to be estimated or predicted is called dependent variable.     Independent Variable The variable on the basis of which the dependent variable is to be estimated is called independent variable.

4. For Example If we want to estimate the heights of children on the basis of their ages, then the heights of children would be dependent variable and the ages of children would be independent variable. Students are some time confused with independent variable and dependent variable here are some examples of independent and dependent variables:

5. Independent Variable ( ) Dependent Variable ( ) Age of child Weight of child Temperature of a plant Height of a plant Amount of drug Reaction time Number of registered vehicles Number of road accidents Advertising Income  

6. Scatter Diagram Scatter diagram is obtained by plotting the paired values of and on a graph paper and the points so obtained are kept disjoined. In scatter diagram we take independent variable along the and the dependent variable on the . This is the simplest method of investigating the relationship between the two variables.  

7. Here below is given heights of children in inches and weights in pounds . Now we plot this data taking the independent variable (heights) on and the dependent variable (weights) on to get scatter.   Heights 58 70 74 68 61 66 70 63 Weights 160 180 176 165 150 155 169 160 58 70 74 68 61 66 70 63 160 180 176 165 150 155 169 160

8. Scatter diagram indicates a relationship between the variables. There dots shows upward trend and we say that a linear relationship exists between height and weight. The resulting curve in scatter diagram is called curve of regression or linear regression.

9. The Least Square Line After drawing scatter diagram, a free hand line can be drawn through the plotted points, which shows trend of the variables. This free hand drawing is a “Subjective Method” as it depends upon the personal judgment of the person drawing the line. We need some objective method. An “Objective Method is the method of least square”. The line obtained by this method is called “Least Square Line”.

10. Least Square Lines of Regression The equation for a straight line or linear trend will be Where dependent variable and is independent variable. “ ” and “ ” are unknown parameters determined by solving simultaneously the following normal equation.   The Values of “ ” and “ ” can be calculated by the following formulae which are obtained by solving the above equations.   or  

11. If the variable “ ” is taken as dependent variable and “ ” is taken as independent variable then the least square line is The normal equations are   By solving the above equations, the value of “ ” and “ ” can be calculated as Or  

12. Example-1: Determine (i) the regression equation of on , and estimate at .   ( ii) the regression equation of on , and estimate at y = 4 .   1 3 3 4 5 5 5 3 2 2 1 1 3 3 4 5 5 5 3 2 2 1

13. Solution : 1 5 5 1 25 3 3 9 9 9 3 2 6 9 4 4 2 8 16 4 5 25 5 1 5 25 1 ∑x=21 ∑y=13 ∑ xy =33 ∑x 2 =85 ∑y 2 =43 1 5 5 1 25 3 3 9 9 9 3 2 6 9 4 4 2 8 16 4 5 25 5 1 5 25 1 ∑x=21 ∑y=13 ∑ xy =33 ∑x 2 =85 ∑y 2 =43

14. Regression equation on .              

15.       Line of Regression      

16. When        

17. (ii) Regression equation on          

19. When        

20. Example-2:   The heights and weights of six men are given below . (i) Determine both the lines of regression. (ii) Estimate weight when height is . (iii) Estimate height when weight is . Answers ( ; ) ;   Height( ) 2.00 1.80 1.85 1.72 1.75 1.79 Weight( ) 85.0 78.0 80.0 74.0 75.0 76.0 2.00 1.80 1.85 1.72 1.75 1.79 85.0 78.0 80.0 74.0 75.0 76.0