Applied Statistics and Probability for Engineers

This test procedure may be easily introduced into the analysis of variance conducted for

the significance of regression. If the null hypothesis of model adequacy is rejected, the model

must be abandoned and attempts must be made to find a more appropriate model. If H 0 is not

rejected, there is no apparent reason to doubt the adequacy of the model, and MSPEand MSLOF

are often combined to estimate ^2.

EXAMPLE S11-1 Consider the data on two variables yand xshown below. Fit a simple linear regression model

and test for lack of fit, using 0.05.

xy xy

1.0 2.3, 1.8 5.6 3.5, 2.8, 2.1

2.0 2.8 6.0 3.4, 3.2

3.3 1.8, 3.7 6.5 3.4

4.0 2.6, 2.6, 2.2 6.9 5.0

5.0 2.0

The regression model is 1.6970.259x, and the regression sum of squares is SSR3.4930.

The pure-error sum of squares is computed as follows:

yˆ

11-3

Level of x Degrees of Freedom

1.0 0.1250 1

3.3 1.8050 1

4.0 0.0166 2

5.6 0.9800 2

6.0 0.0200 1

Total 3.0366 7

a

ni

u 1

1 yiuyi 22

The analysis of variance is summarized in Table S11-1. Since the lack-of-fit F-statistic is

f 0 1.42, which has a P-value of P0.3276, we cannot reject the hypothesis that the tentative

model adequately describes the data. We will pool lack-of-fit and pure-error mean squares to form

the residual mean square that is the denominator mean square in the test for significance of

regression. In addition, since the P-value for the statistic f 0 6.66 with 1 and 14 degrees of free-

dom associated with significance of regression is P0.0216, we conclude that    1

 0.

In fitting a regression model to experimental data, a good practice is to use the lowest

degree model that adequately describes the data. The lack-of-fit test may be useful in this

Table S11-1 Analysis of Variance for Example S11-1

Source of Sum of Degrees of Mean

Variation Squares Freedom Square f 0 P-value

Regression 3.4930 1 3.4930 6.66 0.0218

Residual 7.3372 14 0.5241

(Lack of fit) 4.3005 7 0.6144 1.42 0.3276

(Pure error) 3.0366 7 0.4338

Total 10.8300 15

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