Applied Statistics and Probability for Engineers

458 CHAPTER 12 MULTIPLE LINEAR REGRESSION

type I error rates) in several different runs to see if this substantially affects the choice of the

final model.

EXAMPLE 12-14 Table 12-18 gives the Minitab stepwise regression output for the wine quality data. Minitab uses

fixed values of for entering and removing variables. The default level is 0.15 for both de-

cisions. The output in Table 12-18 uses the default value. Notice that the variables were entered in

the order Flavor (step 1), Oakiness (step 2), and Aroma (step 3) and that no variables were re-

moved. No other variable could be entered, so the algorithm terminated. This is the three-variable

model found by all possible regressions that results in a minimum value of Cp.

Forward Selection

The forward selectionprocedure is a variation of stepwise regression and is based on the

principle that regressors should be added to the model one at a time until there are no remain-

ing candidate regressors that produce a significant increase in the regression sum of squares.

That is, variables are added one at a time as long as their partial F-value exceeds fin. Forward

selection is a simplification of stepwise regression that omits the partial F-test for deleting

variables from the model that have been added at previous steps. This is a potential weakness

of forward selection; that is, the procedure does not explore the effect that adding a regressor

at the current step has on regressor variables added at earlier steps. Notice that if we were to

apply forward selection to the wine quality data, we would obtain exactly the same results as

we did with stepwise regression in Example 12-14, since stepwise regression terminated

without deleting a variable.

Table 12-18 Minitab Stepwise Regression Output for the

Wine Quality Data

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

Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15

Response is Quality on 5 predictors, with N 38

Step 1 2 3

Constant 4.941 6.912 6.467

Flavor 1.57 1.64 1.20

T-Value 7.73 8.25 4.36

P-Value 0.000 0.000 0.000

Oakiness 0.54 0.60

T-Value 1.95 2.28

P-Value 0.059 0.029

Aroma 0.58

T-Value 2.21

P-Value 0.034

S 1.27 1.22 1.16

R-Sq 62.42 66.11 70.38

R-Sq(adj) 61.37 64.17 67.76

C–p 9.0 6.8 3.9

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