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


A Full Factorial Design Example  
An example of a full factorial design with 3 factors 
The following is an example of a full factorial design with 3 factors
that also illustrates
replication,
randomization, and
added center points.
Suppose that we wish to improve the yield of a polishing operation. The three inputs (factors) that are considered important to the operation are Speed (X_{1}), Feed (X_{2}), and Depth (X_{3}). We want to ascertain the relative importance of each of these factors on Yield (Y). Speed, Feed and Depth can all be varied continuously along their respective scales, from a low to a high setting. Yield is observed to vary smoothly when progressive changes are made to the inputs. This leads us to believe that the ultimate response surface for Y will be smooth. 

Table of factor level settings 


Factor Combinations  
Graphical representation of the factor level settings 
We want to try various combinations of these settings so as to
establish the best way to run the polisher. There are eight different
ways of combining high and low settings of Speed, Feed, and Depth.
These eight are shown at the corners of the following diagram.


2^{3} implies 8 runs  Note that if we have k factors, each run at two levels, there will be 2^{k} different combinations of the levels. In the present case, k = 3 and 2^{3} = 8.  
Full Model 
Running the full complement of all possible factor combinations means
that we can estimate all the main and interaction effects. There are
three main effects, three twofactor interactions, and a threefactor
interaction, all of which appear in the full model as follows:


Standard order  
Coded variables in standard order  The numbering of the corners of the box in the last figure refers to a standard way of writing down the settings of an experiment called `standard order'. We see standard order displayed in the following tabular representation of the eightcornered box. Note that the factor settings have been coded, replacing the low setting by 1 and the high setting by 1.  
Factor settings in tabular form 


Replication  
Replication provides information on variability  Running the entire design more than once makes for easier data analysis because, for each run (i.e., `corner of the design box') we obtain an average value of the response as well as some idea about the dispersion (variability, consistency) of the response at that setting.  
Homogeneity of variance  One of the usual analysis assumptions is that the response dispersion is uniform across the experimental space. The technical term is `homogeneity of variance'. Replication allows us to check this assumption and possibly find the setting combinations that give inconsistent yields, allowing us to avoid that area of the factor space.  
Factor settings in standard order with replication 
We now have constructed a design table for a twolevel full factorial
in three factors, replicated twice.


Randomization  
No randomization and no center points 
If we now ran the design as is, in the order shown, we would have two
deficiencies, namely:


Randomization provides protection against extraneous factors affecting the results 
The more freely one can randomize experimental runs, the more insurance
one has against extraneous factors possibly affecting the results, and
hence perhaps wasting our experimental time and effort. For example,
consider the `Depth' column: the settings of Depth, in standard order,
follow a `four low, four high, four low, four high' pattern.
Suppose now that four settings are run in the day and four at night, and that (unknown to the experimenter) ambient temperature in the polishing shop affects Yield. We would run the experiment over two days and two nights and conclude that Depth influenced Yield, when in fact ambient temperature was the significant influence. So the moral is: Randomize experimental runs as much as possible. 

Table of factor settings in randomized order 
Here's the design matrix again with the rows randomized. The old standard order column is also shown
for comparison and for resorting, if desired, after the runs are in.


Table showing design matrix with randomization and center points 
This design would be improved by adding at least 3 centerpoint runs
placed at the beginning, middle and end of the experiment. The final
design matrix is shown below:
