TABLE OF CONTENTS As the number of factors to be considered in a factorial experiment increases, the number of treatment combinations increases very rapidly. This can be seen with a 2 f factorial where f = 5 requires 32 experiments for one replication, f = 6 requires 64, f = 7 requires 128, and so on. Along with this increase in the amount or experimentation comes an increase in the number of high-order interactions. Some of these high-order interactions may be used as error, since those above second order (three-way) would be difficult to explain if found significant. Table 13.1 gives some idea of the number of main effects, first-order, second-order, , interactions that can be recovered if a complete 2 f factorial can be run.
| f | 2 f | Main | Order of Interaction | ||||||
|---|---|---|---|---|---|---|---|---|---|
| 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | |||
| 5 | 34 | 5 | 10 | 10 | 5 | 1 | |||
| 6 | 64 | 6 | 15 | 20 | 15 | 6 | 1 | ||
| 7 | 128 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | |
| 8 | 256 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 |
Considering f = 7, there will be 7 df for the seven main effects, 21 df for the 21 first-order interactions, and 35 df for the second-order interactions, leaving
for an error estimate, assuming no blocking and no interactions above second order. Even if this experiment were confounded in blocks, there is still a...
