Applied Statistics and Probability for Engineers

14-7 2kFACTORIAL DESIGNS 533

main effect row by row. For example, the signs in the ABcolumn are the products of the A

and Bcolumn signs in each row. The contrast for any effect can easily be obtained from this

table.

Table 14-15 has several interesting properties:

1. Except for the identity column I, each column has an equal number of plus and minus
   signs.


2. The sum of products of signs in any two columns is zero; that is, the columns in the
   table are orthogonal.


3. Multiplying any column by column Ileaves the column unchanged; that is, Iis an
   identity element.


4. The product of any two columns yields a column in the table, for example AB
   AB, and ABABCA^2 B^2 CC, since any column multiplied by itself is the
   identity column.
   The estimate of any main effect or interaction in a 2kdesign is determined by multiplying
   the treatment combinations in the first column of the table by the signs in the corresponding
   main effect or interaction column, by adding the result to produce a contrast, and then by di-
   viding the contrast by one-half the total number of runs in the experiment. For any 2kdesign
   with nreplicates, the effect estimates are computed from

Table 14-15 Algebraic Signs for Calculating Effects in the 2^3 Design

Treatment Factorial Effect

Combination I A B AB C AC BC ABC

112 

a 

b 

ab 

c 

ac 

bc 

abc 

Effect (14-22)

Contrast

n 2 k^1

SS (14-23)

1 Contrast 22

n 2 k

and the sum of squares for any effect is

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