Applied Statistics and Probability for Engineers

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN 553

In this design, the main effects are aliased with the three-factor interactions; note that the

alias of Ais

or

and similarly B ACD,C ABD, and D ABC.

The two-factor interactions are aliased with each other. For example, the alias of ABis CD:

or

The other aliases are AC BDand AD BC.

The estimates of the main effects and their aliases are found using the four columns of

signs in Table 14-25. For example, from column Awe obtain the estimated effect

The other columns produce

Clearly, and are large, and if we believe that the three-factor interactions are negligible,

the main effects A(gap) and D(power setting) significantly affect etch rate.

The interactions are estimated by forming the AB, AC, and ADcolumns and adding them

to the table. For example, the signs in the ABcolumn are , , , , , , , , and this

column produces the estimate

From the ACand ADcolumns we find

The estimate is large; the most straightforward interpretation of the results is that since A

and Dare large, this is the ADinteraction. Thus, the results obtained from the 2^4 ^1 design

agree with the full factorial results in Example 14-5.

/AD

/ACACBD 25.50 and /ADADBC197.50

/ABABCD^141550  749  1052  650  1075  642  601  7292  10

/A /D

/BBACD4.00 /CCABD11.50 and /DDABC290.50

127.00

/AABCD^141  550  749  1052  650  1075  642  601  7292

AB # IAB # ABCD ABA^2 B^2 CDCD

A # IA # ABCD AA^2 BCDBCD

abcd = 729

cd = 1075

bd = 1052

ad = 749

bc = 601

ac = 642

ab = 650

(1) = 550

A

C

B

• D +

Figure 14-29 The 2^4 ^1 design for the experiment of Example 14-7.

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