Applied Statistics and Probability for Engineers

12-2.3 More about the Extra Sum of Squares Method (CD Only)

The extra sum of squares method for evaluating the contribution of one or more terms to a

model is a very useful technique. Basically, one considers how much the regression ormodel

sum of squares increases upon adding terms to a basic model. The expanded model is called

the fullmodel, and the basic model is called the reducedmodel. Although the development in

the text is quite general, Example 12-5 illustrates the simplest case, one in which there is only

one additional parameter in the full model. In this case, the partial F-test based on the extra

sum of squares is equivalent to a t-test. When there is more than one additional parameter in

the full model, the partial F-test is not equivalent to a t-test.

The extra sum of squares method is often used sequentially when fitting a polynomial

model, such as

Here would measure the contribution of the linear term over and above a model

containing only a mean would measure the contribution of the quadratic

terms over and above the linear, and would measure the contribution of the

cubic terms over and above the linear and the quadratic. This can be very useful in selecting

the orderof the polynomial to fit. Notice from Table 12-4 that Minitab automatically pro-

duces this sequential computation. Also, note that in a sequential partition of the model or re-

gression sum of squares,

However, if we consider each variable as if it were the last to be added,

As another illustration of the extra sum of squares method, consider the model

Suppose that we are uncertain about the contribution of the second-order terms. We could

evaluate this with a partial F-test by fitting the reduced model

and computing

Finally, note that we have expressed the extra sum of squares as the difference in the regres-

sion sum of squares between the full model and the reduced model:

Some authors write SSR(Extra) as the difference between error or residual sums of squares for

the two model.

SSR 1 Extra 2 SSR 1 Full Model 2 SSR 1 Reduced Model 2

SSR 1  12 , 11 , 22 0  0 , 1 , 22 SSR 1  1 , 2 , 12 , 11 , 22 0  02 SSR 1  1 , 2 |  02

y 0  1 x 1  2 x 2 

y 0  1 x 1  2 x 2  12 x 1 x 2  11 x^21  22 x 22 

SSR 1  1 , 2 , 3 0  02 SSR 1  1 0  0 , 2 , 32 SSR 1  2 0  0 , 1 , 32 SSR 1  3 0  0 , 1 , 22

SSR 1  1 , 2 , 3 , 0  02 SSR 1  1 0  02 SSR 1  2 0  0 , 12 SSR 1  3 0  0 , 1 , 22.

SSR 1  30  0 , 1 , 22

1  02 , SSR 1  20  0 , 12

SSR 1  10  02

y 0  1 x 1  2 x 2  3 x 3 

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