Applied Statistics and Probability for Engineers

The test involves partitioning the error or residual sum of squares into the following

components:

(S11-1)

where SSPEis the sum of squares attributable to pure error,and SSLOFis the sum of squares at-

tributable to the lack of fitof the model. To compute SSPE, we must have repeated observations

on the response Yfor at least one level of x. Suppose we have ntotal observations such that

Note that there are mdistinct levels of x. The contribution to the pure-error sum of squares at

x 1 (say) would be

(S11-2)

where represents the average of all n 1 repeat observations on the response yat x 1. The total sum

of squares for pure error would be obtained by summing Equation S11-2 over all levels of xas

(S11-3)

There are degrees of freedom associated with the pure-error

sum of squares. The sum of squares for lack of fit is simply

(S11-4)

with n 2 npem2 degrees of freedom. The test statistic for lack of fit would then be

(S11-5)

and we would reject the hypothesis that the model adequately fits the data if f 0 f,m2,nm.

F 0 

SSLOF 1 m 22

SSPE 1 nm 2



MSLOF

MSPE

SSLOFSSESSPE

npe gmi 1 1 ni 12 nm

SSPE a

m

i 1

a

ni

u 1

1 yiuyi 22

y 1

a

n 1

u 1

1 y 1 uy 122

y 11 , y 12 ,p, y 1 n 1 repeated observations at x 1

y 21 , y 22 ,p, y 2 n 2 repeated observations at x 2

#

#

#

ym 1 , ym 2 ,p, ymnm repeated observations at xm

SSESSPESSLOF

11-2

Figure S11-1 A

regression model

displaying lack of fit. x

y

^y = 
^ 0 + 
^ 1 x

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