5.
Process Improvement
5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.3. Full factorial designs


Description  
Graphical representation of a twolevel design with 3 factors 
Consider the twolevel, full factorial design for three factors, namely
the 2^{3} design. This implies eight runs (not counting
replications or center point runs). Graphically, we can represent
the 2^{3} design by the cube shown in Figure 3.1. The arrows
show the direction of increase of the factors. The numbers `1' through
`8' at the corners of the design box reference the `Standard Order' of
runs (see Figure 3.1).


The design matrix 
In tabular form, this design is given by:


Standard Order for a 2^{k} Level Factorial Design  
Rule for writing a 2^{k} full factorial in "standard order"  We can readily generalize the 2^{3} standard order matrix to a 2level full factorial with k factors. The first (X1) column starts with 1 and alternates in sign for all 2^{k} runs. The second (X2) column starts with 1 repeated twice, then alternates with 2 in a row of the opposite sign until all 2^{k} places are filled. The third (X3) column starts with 1 repeated 4 times, then 4 repeats of +1's and so on. In general, the ith column (X_{i}) starts with 2^{i1} repeats of 1 folowed by 2^{i1} repeats of +1.  
Example of a 2^{3} Experiment  
Analysis matrix for the 3factor complete factorial 
An engineering experiment called for running three factors; namely,
Pressure (factor X1), Table speed (factor X2) and Down
force (factor X3), each at a `high' and `low' setting, on a
production tool to determine which had the greatest effect on product
uniformity. Two replications were run at each setting. A (full
factorial) 2^{3} design with 2 replications calls for
8*2=16 runs.
The block with the 1's and 1's is called the Model Matrix or the Analysis Matrix. The table formed by the columns X1, X2 and X3 is called the Design Table or Design Matrix. 

Orthogonality Properties of Analysis Matrices for 2Factor Experiments  
Eliminate correlation between estimates of main effects and interactions 
When all factors have been coded so that the high value is "1" and
the low value is "1", the design matrix for any full (or suitably
chosen fractional) factorial experiment has columns that are all
pairwise orthogonal
and all the columns (except the "I" column) sum to 0.
The orthogonality property is important because it eliminates correlation between the estimates of the main effects and interactions. 