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248 A Glossary for Design of Experiments with Examples
Table 16.7 Result of Example 1 with zero noise.
Run A B C Y 1 Y 2 Y 3
1 −1 −1 +1 0 −2 −1
2 +1 −1 −1 −2 0 −1
3 −1 +1 −1 −2 0 −1
4 +1 +1 +1 4 2 3

Given the perfect case where noise is zero (so as to eliminate any further error in con-
clusion), Table 16.7 shows the results obtained. Standard analysis yields the following
response functions:

(a) Y 1 = A+ B + 2C,
(b) Y 2 = A+ B,
(c) Y 3 = A+ B + C.

It may be observed that when a factor and its alias are synergistic, the effect of the
factor is overestimated. On the other hand, if a factor is in ‘conflict’ with its alias,
its effect is underestimated. Hence, when a factor is found to be significant, is it the
effect of the factor that is significant, the effect of the alias that is significant, or the
combined effect of the factor and alias that is significant? Similarly, when a factor is
found to be insignificant, is it because the factor is truly insignificant, or the factor is
in conflict with its alias? In short, in the presence of confounding, we can be sure of
nothing. This uncertainty grows if the AC and/or BC interactions is/are also present.

16.5.3 Example 2
A2 7−2 fractional factorial (1/4 fractional) design has the following alias structure:

I + CEFG + ABCDF + ABDEG,
A+ BCDF + BDEG + ACEFG,

B + ACDF + ADEG + BCEFG,
C + EFG + ABDF + ABCDEG,
D + ABCF + ABEG + CDEFG,
E + CFG + ABDG + ABCDEF,

F + CEG + ABCD + ABDEFG,
G + CEF + ABDE + ABCDFG,
AB + CDF + DEG + ABCEFG,
AC + BDF + AEFG + BCDEG,