Power Calculation
User Tip: Evaluate the power of a factorial design against a Main Effects Model. If you are evaluating a Res III design for the use of ruggedness testing, put only 1 main effect in the model.
The Bottom Line: To get more confidence that you will detect a certain size effect – add more runs!!
Two-Level factorial designs:
The basis for the power calculations for factorials is the difference between the average at +1 and the average at –1. This difference is equal to the effect. For a two-level factorial a 2 standard deviation effect is a 1 standard deviation change in the coefficient. Power for unreplicated factorials can only be estimated if you designate some effects as error.
General factorial designs:
For general factorials, power is defined as the probability of finding a difference separating the two most extreme means in a group of means. (For example, it calculates the probability of finding a difference of (by default) ½, 1, or 2 standard deviations separating the two most extreme means within a main effect.) For unbalanced designs the minimum power for each group of means is reported.
RSM designs:
Note
Power is not an effective measurement for evaluating the capability of a response surface design to model the data.
The basis for the power calculations for response surface designs is the difference between the value at +1 and the value at –1 for a linear effect. These limits on the linear effects define the range for all higher order effects. Therefore the range for a pure quadratic (squared) effect is from 0 to +1 when the linear effect range is from –1 to +1. An effect of ½, 1, or 2 standard deviations implies there is a difference of that many standard deviations separating the high and low values (see table below) of that effect in the model.
Effect |
Low |
High |
Effect |
Low |
High |
|
|---|---|---|---|---|---|---|
Linear |
-1 |
+1 |
Quartic A4 |
0 |
+1 |
|
2FI |
-1 |
+1 |
Quartic A3B |
-1 |
+1 |
|
Quadratic |
0 |
1 |
Quartic A2B2 |
0 |
+1 |
|
Cubic A3 |
-1 |
+1 |
Quartic A2BC |
-1 |
+1 |
|
Cubic A2B |
-1 |
+1 |
Quartic ABCD |
-1 |
+1 |
|
Cubic ABC |
-1 |
+1 |
Effect Ranges for Calculating Power of Response Surface Designs
Mixture designs:
Note
Power is not an effective measurement for evaluating the capability of a mixture design to model the data. See Power for Mixture Designs.
The basis for the power calculations for mixture designs is the difference between the value at 0 and the value at +1 for a linear effect. These limits on the linear effects define the range for all higher order effects. Therefore the range for a quadratic effect is from 0 to +¼ (the maximum product of A*B) when the linear effect range is from 0 to +1. Similar rules apply to the higher order mixture effects. An effect of either ½, 1 or 2 standard deviations implies that there is a difference of that many standard deviations separating the high and low values (see table below) of that effect in the model.
Effect |
Low |
High |
|---|---|---|
Linear A |
0 |
+1 |
Quadratic AB |
0 |
+1/4 |
Special Cubic ABC |
0 |
+1/27 |
Cubic AB(A-B) |
-3/32 |
+3/32 |
Effect Ranges for Calculating Power of Mixture Designs
Combined designs:
Note
Power is not an effective measurement for evaluating the capability of a combined design to model the data.
The basis for the power calculations for mixture designs is the difference between the value at 0 and the value at +1 for a linear mixture effect. These limits on the linear effects define the range for all higher-order effects. The basis for the power calculations for response surface designs is the difference between the value at +1 and the value at –1 for a linear effect. The basis for the power calculations for a crossed effect is the difference based on the range of the crossed effect. The range for a linear mixture by linear process effect is the range you get by multiplying (0, 1) times (–1, +1) or from –1 to +1 0.The range for a linear mixture by quadratic process effect is the range you get by multiplying (0, 1) times (0, +1) or from 0 to +1 1. Similar rules apply to the higher order crossed effects. An effect of either ½, 1 or 2 standard deviations implies that there is a difference of that many standard deviations separating the high and low values (see table below) of that effect in the model.
- 0
The range of (0, 1) times (–1, +1) = Range[(0 times –1 = 0), (0 times +1 =0), (1 times –1 = –1) and (1 times +1 = +1)] = Range[0, 0, –1, +1] = –1 to +1.
- 1
The range of (0, 1) times (0, +1) = Range[(0 times 0 = 0), (0 times +1 =0), (1 times 0 = 0) and (1 times +1 = +1)] = Range[0, 0, 0, +1] = 0 to +1.
Note
Power for any designs that contain categorical factors is calculated using Type II sum of squares. Designs that have only continuous variables have their power calculated using Type III sum of squares.
See Also: