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,

 x1 2 4 5 6 3 1 x2 1 2 1 3 5 2 y 14 16 17 20 18 12

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

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

 x1 x2 x12 x22 x1x2 x1y x2y y 245631 121352 116253691 1419254 28518152 2864851205412 143217609024 141617201812 ∑ 21 14 91 44 50 363 237 97

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  