Applied Statistics and Probability for Engineers

12-6 ASPECTS OF MULTIPLE REGRESSION MODELING 459

Backward Elimination

The backward eliminationalgorithm begins with all Kcandidate regressors in the model. Then

the regressor with the smallest partial F-statistic is deleted if this F-statistic is insignificant, that

is, if f fout. Next, the model with K1 regressors is fit, and the next regressor for potential

elimination is found. The algorithm terminates when no further regressor can be deleted.

Table 12-19 shows the Minitab output for backward elimination applied to the wine quality

data. The value for removing a variable is 0.10. Notice that this procedure removes Body at

step 1 and then Clarity at step 2, terminating with the three-variable model found previously.

Some Comments on Final Model Selection

We have illustrated several different approaches to the selection of variables in multiple linear

regression. The final model obtained from any model-building procedure should be subjected

to the usual adequacy checks, such as residual analysis, lack-of-fit testing, and examination of

the effects of influential points. The analyst may also consider augmenting the original set of

candidate variables with cross-products, polynomial terms, or other transformations of the

original variables that might improve the model. A major criticism of variable selection meth-

ods such as stepwise regression is that the analyst may conclude there is one “best” regression

equation. Generally, this is not the case, because several equally good regression models can

Table 12-19 Minitab Backward Elimination Output for the

Wine Quality Data

Stepwise Regression: Quality versus Clarity, Aroma,...

Backward elimination. Alpha-to-Remove: 0.1

Response is Quality on 5 predictors, with N = 38

Step 1 2 3

Constant 3.997 4.986 6.467

Clarity 2.3 1.8

T-Value 1.35 1.12

P-Value 0.187 0.269

Aroma 0.48 0.53 0.58

T-Value 1.77 2.00 2.21

P-Value 0.086 0.054 0.034

Body 0.27

T-Value 0.82

P-Value 0.418

Flavor 1.17 1.26 1.20

T-Value 3.84 4.52 4.36

P-Value 0.001 0.000 0.000

Oakiness 0.68 0.66 0.60

T-Value 2.52 2.46 2.28

P-Value 0.017 0.019 0.029

S 1.16 1.16 1.16

R-Sq 72.06 71.47 70.38

R-Sq(adj) 67.69 68.01 67.76

C–p 6.0 4.7 3.9

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