REGRESSION

Defination and Meaning :

Regression analysis measures the nature and extent of the relation between 2 or more variables thus enables us to make predictions. 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;

Regression lines For two variables X and Y, we will have two regression lines: Regression line X on Y gives values of Y for given values of X; Regression line Y on X gives values of X for given values of Y;

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)

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)

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

Difference between Regression and Correlation :

Correlation Regression 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 know

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 dependent variable;

Estimating equation describing relationship among three variables :

where, Y = estimated value corresponding to the dependent variable 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

Normal Equations: To determine the values of the constants a, b1 and b2 ,we use three equations (which statistician call the “normal equation”) ∑Y=Na+b1∑X1 + b2∑X2 ∑X1Y=a ∑X1 + b1 ∑X12 + b2∑X1 X2 ∑X2Y=a ∑X2 + b2 ∑X22 + b1∑X1 X2

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