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Multiple Regression

If more than one predictor variable is present in the regression equation then it is called multiple regression. Let us consider two predictor variables and one regressed variable and the multiple linear regression model can be stated as follows :

To estimate a0, a1 and a2 we take (x1;, x2;, Y;), i = 1,2, ...... , n as observed data where the x's

are assumed to be known without error while the y values are random variables.

 

Let

be the sum of squares of errors. Then to minimize S, we take

From which we obtain three normal equations

By solving these equations we obtain the least square estimates of a0, a1 and a2.

Note. 1. In the above model if a2 vanishes then it is the regression line of y on x.

2. The above model represents a regression plane.

 

Example 9. Consider the following data:

 Fit a least squares regression plane.

Solution. Here n = 6. Let the regression plane be y = a0 + a1 x1 + a2 x2.

The three normal equations can be written as follows:

97 = 6a0 + 21 a1 + 14a2

363 = 21 a0 + 91 a1 +50 a2

237 = 14 a0 + 50 a1 + 44 a2.

 

By solving we obtain,

a0= 9.7, a1= 1.3, a2=0.83.