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Multiple regression occurs when there are more than one predictor variable is present in the regression equation. For example:

y = a0 + a1x1 + a2x2

Here a0, a1 and a2 we take (x1;, x2;, Y;), i = 1,2, …… , n as observed data. Note must be made upon the values of x that are errorless and values of y that are random.

So when,

n

s =∑[yi – (a0 + a1x1i + a2x2i)]2

i=1

Here you will find sum of squares of errors. To proceed in minimizing S,

αS/αa0 = 0, αS/αa1 = 0 and αS/αa2 = 0

From here on we come to three equations:

∑yi = n a0 + a1x1i + a2 x2i

∑xiyi = a0x1i + a1x1i2+ a2x1i  x2i

x2iyi = a0x2i + a1 x1i x2i + a2∑x22i

This is how we’ll come to least square estimates of a0, a1 and a2. There are few things one must remember here;this is specifically a regression plane and by any chance value of a2 disappears then the regression line will represent y upon x.

Let us consider this data chart,

x1245631
x2121352
y141617201812

 

The first thing is to fit these data into a regression plane, for example y. n = 6.

Then, y= a0 + a1 x1 + a2 x2.

x1x2x12x22x1x2x1yx2yy
2

4

5

6

3

1

1

2

1

3

5

2

1

16

25

36

9

1

1

4

1

9

25

4

2

8

5

18

15

2

28

64

85

120

54

12

14

32

17

60

90

24

14

16

17

20

18

12

211491445036323797

 

So when placing data in the above mentioned normal equations, we get:

97 = 6a0 + 21 a1 + 14a2

363 = 21 a0 + 91 a1 +50 a2

237 = 14 a0 + 50 a1 + 44 a2

So at last we come to:

a0= 9.7, a1= 1.3, a2= 0.83