Statistical design is an alternative to the scientific method of checking one variable while all others are held constant that saves time in multivariable systems. Given a system where all variables (called factors) can be considered independent of each other, an orthogonal array can be set up that reduces the number of trials in a seven-factor system with three different values (called levels) for each factor from 37 trials (if every combination is tested) to around twenty trials. Consider the following example of chemical vapor deposition (CVD) for a polysilica film:
| Factor | 1 | 2 | 3 |
|---|---|---|---|
| Temperature (A) | T - 25 | T | T + 25 |
| Pressure (B) | P - 200 | P | P + 200 |
| Set Time (C) | t | t + 8 | t + 16 |
| Cleaning (D) | None | Chemical 1 | Chemical 2 |
We first set up an orthogonal array where each factor is combined with each level of each factor compared with another factor of another level only once. In the following matrix, A-B-C-D represent the four factors in the experiment (temperature, pressure, set time, and cleaning chemical) and 1-2-3 are the different levels for each of these factors.
| Trial | A | B | C | D | Result (signal/noise) |
|---|---|---|---|---|---|
| #1 | 1 | 1 | 1 | 1 | -20 |
| #2 | 1 | 2 | 2 | 2 | -10 |
| #3 | 1 | 3 | 3 | 3 | -30 |
| #4 | 2 | 1 | 2 | 3 | -25 |
| #5 | 2 | 2 | 3 | 1 | -45 |
| #6 | 2 | 3 | 1 | 2 | -65 |
| #7 | 3 | 1 | 3 | 2 | -45 |
| #8 | 3 | 2 | 1 | 3 | -65 |
| #9 | 3 | 3 | 2 | 1 | -70 |
Once results of each trial is obtained (which we just call the signal/noise ratio above), a table of the averaged values of each factor and level must be constructed:
| Factor | 1 | 2 | 3 |
|---|---|---|---|
| A | -20 | -45 | -60 |
| B | -30 | -40 | -55 |
| C | -50 | -35 | -40 |
| D | -45 | -40 | -40 |
Each cell in the above table is an average of the results of the three trials which contain its row and column. For example, cell A1's value (-20) was obtained by averaging the result of trials #1, #2, and #3 since all three of those trials had factor A (temperature) set at level 1 (T - 25).
If a less negative (higher) signal/noise ratio is preferable, one can just select the least negative values for each factor from this last table. Clearly, for factor A's optimal value occurred under level 1 conditions; factor B's optimal value was reached while at level 1; C's at level 2; D's at 2 or 3. Thus, the optimal experimental conditions would be
A = 1, B = 1, C = 2, D = 2
or
A = 1, B = 1, C = 2, D = 3
Notice that this combination of levels was not an experiment actually conducted; the ideal conditions were inferred using this method of statistical design. Furthermore, these conditions were determined after only nine experiments as opposed to the eighty-one (34) required if the exhaustive approach was taken.
The result of these optimal conditions can also be estimated using the relationship:
optimal signal/noise = m + (mA - m) + (mB - m) + (mC - m) + (mD - m)
or
optimal signal/noise = m + (mA - m) + (mB - m)
where m is the mean of all results, mA is the optimal value of factor A, mB is the optimal value of factor B, and so on. To be truest to the statistics behind this model, terms involving small values of (mX - m) are ignored and are considered as error; otherwise, the optimal value will inaccurately high.
Other analyses that can be done with these results include examining the standard deviation of each factor in the third table above; this is an indication of which variables are least and most sensitive to change. Various plots can express this sensitivity to variability. These factor-averages can also be compared with each other to determine if they interact to any degree and whether this correlation (if any) is synergistic or antisynergistic.
For more in-depth information on this topic, you may wish to read Quality Engineering Using Robust Design by M.S. Phadke. The third chapter of that book details statistical design using orthogonal arrays.