The model implicit in the 2 k factorial design is
| Y pred = | b 0 | constant term | |||
| + b 1 X 1 | + b 2 X 2 | + ... | +b k X k | linear terms | |
| + b 12 X 1 X 2 | + b 13 X 1 X 3 | + ... | + b k-1,k X k-1 X k | cross product terms | |
| + b 123 X 1 X 2 X 3 | + ... |
The unknown constants in the equation can be obtained from the factorial effects,
| b 0 | = | Ybar |
| b i | = | 0.5*(Effect for Factor X i ) |
| b ij | = | 0.5*(Interaction for Factor X i X j ) |
| b ijk | = | 0.5*(Three factor interaction for X i X j X k ) |
| and | so on. |
The coefficients are half the value of the effects because coefficients represent the slope or change in Y for a one-unit change in X (from X = 0 to X = 1), whereas the effects represent the change in Y for a two-unit change in X (from X = -1 to X= +1).
When writing the prediction equation, only the statistically significant terms (corresponding to the effects) are included.
Thus, in the earlier example, the prediction equation can be written as
Y pred = 668.6 - (33.63/2)X 1 + (25.13/2)X 1 X 3 = 668.6 - 16.81X 1 + 12.56X 1 X 3.
But when an interaction such as X 1 X 3 is included in the model, the general practice is to include both effects that contribute to the model even though both may not be significant. This is called "preserving the hierarchy of the model". Therefore, the model equation would become
Y pred = 668.6 - 16.81X 1 + 5.44X 3 + 12.56X
1 X 3.
The coded variable X could be replaced by the physical variable (uncoded), to make the model equation directly usable. For example, if the factors and levels were
| Factors | Levels | |||
|---|---|---|---|---|
| - | + | |||
| X 1 | = | Ambient temperature, o C | 22 | 32 |
| X 2 | = | Warmup time, minutes | 0.5 | 5.0 |
| X 3 | = | Time power connected, minutes | 0.5 | 5.0 |
| Y | = | measured voltage, millivolts |
Reference:
Lawson, John and Erjavec, John, "Modern Statistics for Engineering and Quality Improvement", Thomson - Duxbury, 2001. pp.254-255