Applied Statistics and Probability for Engineers

536 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

feed, depth, and angle, each having a single degree of freedom, giving the total 3 in the col-

umn headed “DF.’’The column headed “Seq SS’’(an abbreviation for sequential sum of

squares) reports how much the model sum of squares increases when each group of terms is

added to a model that contains the terms listed abovethe groups. The first number in the “Seq

SS’’column presents the model sum of squares for fitting a model having only the three main

effects. The row labeled “2-Way Interactions’’refers to AB, AC, and BC, and the sequential

sum of squares reported here is the increase in the model sum of squares if the interaction

terms are added to a model containing only the main effects. Similarly, the sequential sum of

squares for the three-way interaction is the increase in the model sum of squares that results

from adding the term ABCto a model containing all other effects. The column headed “Adj

SS’’(an abbreviation for adjusted sum of squares) reports how much the model sum of squares

increases when each group of terms is added to a model that contains allthe other terms. Now

since any 2kdesign with an equal number of replicates in each cell is an orthogonal design, the

adjusted sum of squares will equal the sequential sum of squares. Therefore, the F-tests for

each row in the Minitab analysis of variance table are testing the significance of each group of

terms (main effects, two-factor interactions, and three-factor interactions) as if they were the

last terms to be included in the model. Clearly, only the main effect terms are significant. The

t-tests on the individual factor effects indicate that feed rate and depth of cut have large main

effects, and there may be some mild interaction between these two factors. Therefore, the

Minitab output is in agreement with the results given previously.

Residual Analysis

We may obtain the residuals from a 2kdesign by using the method demonstrated earlier for the 2^2

design. As an example, consider the surface roughness experiment. The three largest effects are

A,B, and the ABinteraction. The regression model used to obtain the predicted values is

where x 1 represents factor A, x 2 represents factor B, and x 1 x 2 represents the ABinteraction. The

regression coefficients  1 ,  2 , and  12 are estimated by one-half the corresponding effect esti-

mates, and  0 is the grand average. Thus

Note that the regression coefficients are presented by Minitab in the upper panel of Table 14-18.

The predicted values would be obtained by substituting the low and high levels of Aand Binto

this equation. To illustrate this, at the treatment combination where A, B, and Care all at the low

level, the predicted value is

Since the observed values at this run are 9 and 7, the residuals are 9 9.250.25 and

7 9.252.25. Residuals for the other 14 runs are obtained similarly.

A normal probability plot of the residuals is shown in Fig. 14-20. Since the residuals lie

approximately along a straight line, we do not suspect any problem with normality in the data.

There are no indications of severe outliers. It would also be helpful to plot the residuals ver-

sus the predicted values and against each of the factors A, B, and C.

yˆ11.0651.6875 1  12 0.8125 1  12 0.6875 1  121  12 9.25

11.06251.6875x 1 0.8125x 2 0.6875x 1 x 2

yˆ11.0625a

3.375

2

b x 1 a

1.625

2

b x 2 a

1.375

2

b x 1 x 2

Y 0  1 x 1  2 x 2  12 x 1 x 2  

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