The default calculations are for the largest model that can be fit. Calculations for models with the maximum number of terms are conservative for reduced models. For example, the power for the unreduced model is less than the power for the reduced model, with all other properties the same. To obtain more accurate calculations of power, number of replicates, effect size, or number of center points for reduced models, omit the appropriate number of terms from the model.
To omit terms, use the Design
subdialog of Power and Sample Size for a 2-Level Factorial Design
. Basic calculations for the maximum number of terms follow:
- Maximum number of terms in the model
- Unblocked design or design with complete blocks
- Number of unique corner points in the base design – 1
- For example, for a design with 3 factors and 8 corner points, the 7 terms that represent factors are A, B, C, AB, AC, BC, and ABC. The maximum number of terms for this design is 7. If you omit 1 term, the calculations are correct for a model that fits main effects and 2-factor interactions. If you omit 4 terms, then the calculations are correct for a model with the 3 main effects.
- Design with incomplete blocks and 1 replicate
- Number of unique corner points in the base design – 1 – (Number of blocks – 1)
- A complete block contains all the corner points in the design in equal numbers. If the design contains incomplete blocks, then the number of blocks decreases the maximum number of terms. The decrease is the number of terms that are aliased with blocks. For example, for a design with 3 factors, 8 corner points, 1 replicate and 2 blocks, the 3-factor interaction is aliased with blocks. Thus, the maximum number of terms in the model is 6 instead of 7. For a design with 3 factors, 8 corner points, 1 replicate and 4 blocks, 3 two-way interactions are confounded with blocks. Thus, the maximum number of terms in the model is 4 instead of 7.
Use Minitab to determine the number of terms to omit
To determine the number of terms to omit in complicated cases, create the design in Minitab Statistical Software and examine the alias table. The maximum number of terms is the number of rows for terms that are not aliased with blocks or the identity term (I). For example, this alias table is for a design with 5 factors, 16 runs, 2 replicates and 4 blocks. The alias table has 14 rows for factorial terms that are not aliased with blocks or the identity. Thus, 14 is the maximum number of terms in the model. To do calculations for a main effects model, you omit 9 terms. Nine is the number of rows for terms that are in the model that are not aliased with main effects.
* NOTE * Blocks are confounded with two-way interactions.
Design Generators: E = ABCD
Block Generators: AB, replicates
I + ABCDE
Blk = AB + CDE
A + BCDE
B + ACDE
C + ABDE
D + ABCE
E + ABCD
AC + BDE
AD + BCE
AE + BCD
BC + ADE
BD + ACE
BE + ACD
CD + ABE
CE + ABD
DE + ABC