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- 1. UNIT III REGRESSION
- 2. Meaning The dictionary meaning of regression is “the act of returning or going back”; First used in 1877 by Francis Galton; Regression is the statistical tool with the help of which we are in a position to estimate (predict) the unknown values of one variable from the known values of another variable; It helps to find out average probable change in one variable given a certain amount of change in another;
- 3. Importance
- 4. Regression lines For two variables X and Y, we will have two regression lines: 1. Regression line X on Y gives values of Y for given values of X; 2. Regression line Y on X gives values of X for given values of Y;
- 5. Regression Equation Regression equations are algebraic expressions of regression lines; Y on X Regression equation expressed as Y=a+bX Y is dependent variable X is independent variable „a‟ & „b‟ are constants/parameters of line „a‟ determines the level of fitted line (i.e. distance of line above or below origin) „b‟ determines the slope of line (i.e change in Y for unit change in X)
- 6. Regression equations are algebraic expressions of regression lines; X on Y Regression equation expressed as X=a+bY X is dependent variable Y is independent variable „a‟ & „b‟ are constants/parameters of line „a‟ determines the level of fitted line (i.e. distance of line above or below origin) „b‟ determines the slope of line (i.e change in Y for unit change in X)
- 7. Method of Least Square Constant “a” & “b” can be calculated by method of least square; The line should be drawn through the plotted points in such a manner that the sum of square of the vertical deviations of actual Y values from estimated Y values is the least i.e. ∑(Y-Ye)2 should be minimum; Such a line is known as line of best fit; with algebra & calculus: For Y on X For X on Y ∑Y=Na+b ∑X ∑X=Na+b ∑Y ∑XY=a ∑X + b ∑X2 ∑XY=a ∑Y + b ∑Y2
- 8. Multiple Regression When we use more than one independent variable to estimate the dependent variable in order to increase the accuracy of the estimate; the process is called multiple regression analysis. It is based on the same assumptions & procedure that are encountered using simple regression. The principal advantage of multiple regression is that it allows us to use more of the information available to us to estimate the
- 9. Estimating equation describing relationship among three variables Y= a+b1X1+b2X2 where, Y = estimated value corresponding to the dependent variable a= Y intercept b1 and b2 = slopes associated with X1 and X2, respectively X1 and X2 = values of the two independent variables
- 10. Normal Equations: we use three equations (which statistician call the “normal equation”) to determine the values of the constants a, b1 and b2 ∑Y=Na+b1∑X1 + b2∑X2 ∑X1Y=a ∑X1 + b1 ∑X1 2 + b2∑X1 X2 ∑X2Y=a ∑X2 + b2 ∑X2 2 + b1∑X1 X2
- 11. Difference between regression & correlation Correlation coefficient (r) between x & y is a measure of direction & degree of linear relationship between x & y; It does not imply cause & effect relationship between the variables. It indicates the degree of association bxy & byx are mathematical measures expressing the average relationship between the two variables It indicates the cause & effect relationship between variables. It is used to forecast the nature of dependent variable when the value of independent variable is Correlation Regression

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