Method for 2**2 design generalizes to 2**k (k>2).
k=3 case: Model is:
y = q_0 + q_A * x_A + q_B * x_B + q_C * x_C +
q_AB * x_A * x_B + q_BC * x_B * x_BC + q_AC * x_A * x_C
q_ABC * x_A * x_B * x_C
The sign table generalizes to allow computation of the q's.
General case: Analysis produces 2**k effects:
k ( ) two factor interactions (AB, AC, ...) 2
k ( ) three factor interactions (ABC, ...) 3
Consider a workstation design with three factors:
Factor Level -1 Level 1 ---------------------------------------------------- Memory size, A 4 Mbytes 16 Mbytes Cache size, B 1 kbytes 2 kbytes Number processors, C 1 2
The experiment results, measuring throughput in MIPS, are:
Cache 4Mbytes 16 Mbytes Size One Two One Two (kbytes) Procs Procs Procs Procs ---------------------------------------------------------- 1 14 46 22 58 2 10 50 34 86
The sign table is:
Experiment I A B C AB AC BC ABC y ------------------------------------------------------------- 1 1 -1 -1 -1 1 1 1 -1 14 2 1 1 -1 -1 -1 -1 1 1 22 3 1 -1 1 -1 -1 1 -1 1 10 4 1 1 1 -1 1 -1 -1 -1 34 5 1 -1 -1 1 1 -1 -1 1 46 6 1 1 -1 1 -1 1 -1 -1 58 7 1 -1 1 1 -1 -1 1 -1 50 8 1 1 1 1 1 1 1 1 86 ------------------------------------------------------------- Totals: 320 80 40 160 40 16 24 8 Totals/8: 40 10 5 20 5 2 3 1
Thus:
q_0 = sample mean = 40 q_A = 10 q_B = 5 q_C = 20 q_AB = 5 q_AC = 2 q_BC = 3 q_ABC = 1
The variation is
SST = SSA + SSB + SSC + SSAB + SSAC + SSBC + SSABC
= 2**3 * q_A**2 + ... + 2**3 * q_ABC**2
= 4512
variation due to A = SSA/SST
= (2**3 * q_A**2)/SST
= 800/4512
= 18%
variation due to B = 4%
variation due to C = 71%
variation due to AB = 4%
variation due to AC = 1%
variation due to BC = 2%
variation due to ABC = 0%