14-5 GENERAL FACTORIAL EXPERIMENTS 521For example, consider the three-factor-factorial experiment,with underlying model(14-10)Notice that the model contains three main effects, three two-factor interactions, a three-factor
interaction, and an error term. Assuming that A, B, and Care fixed factors, the analysis of vari-
ance is shown in Table 14-9. Note that there must be at least two replicates (n2) to compute
an error sum of squares. The F-test on main effects and interactions follows directly from
the expected mean squares. These ratios follow Fdistributions under the respective null
hypotheses.EXAMPLE 14-2 A mechanical engineer is studying the surface roughness of a part produced in a metal-cutting
operation. Three factors, feed rate (A), depth of cut (B), and tool angle (C), are of interest. All
three factors have been assigned two levels, and two replicates of a factorial design are run.
The coded data are shown in Table 14-10.
The ANOVA is summarized in Table 14-11. Since manual ANOVA computions are
tedious for three-factor experiments, we have used Minitab for the solution of this problem. 1 (^2) ijk ijkl μ
i1, 2,p, a
j1, 2,p, b
k1, 2,p, c
l1, 2,p, n
Yijklijk 1 (^2) ij 1 (^2) ik 1 (^2) jk
Table 14-9 Analysis of Variance Table for the Three-Factor Fixed Effects Model
Source of Sum of Degrees of Expected
Variation Squares Freedom Mean Square Mean Squares F 0
ASSA a 1 MSA
BSSB b 1 MSB
CSSC c 1 MSC
AB SSAB 1 a 121 b 12 MSAB
AC SSAC 1 a 121 c 12 MSAC
BC SSBC 1 b 121 c 12 MSBC
ABC SSABC 1 a 121 b 121 c 12 MSABC
Error SSE abc 1 n 12 MSE
Total SST abcn 1
2
MSABC
MSE
2
n 1 (^2) ijk^2
1 a 121 b 121 c 12
MSBC
MSE
2
an 1 (^2) jk^2
1 b 121 c 12
MSAC
MSE
2
bn 1 (^2) ik^2
1 a 121 c 12
MSAB
MSE
2
cn 1 (^2) ij^2
1 a 121 b 12
MSC
MSE
2
abn k^2
c 1
MSB
MSE
2
acn j^2
b 1
MSA
MSE
2
bcn ^2 i
a 1
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