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 3^{7} 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 (3^{4}) required if the exhaustive approach
was taken.

The result of these optimal conditions can also be estimated using the relationship:

optimal signal/noise = m + (m_{A} - m) + (m_{B} - m) + (m_{C} - m) + (m_{D} - m)

or

optimal signal/noise = m + (m_{A} - m) + (m_{B} - m)

where m is the mean of all results, mA is the optimal value of factor A,
m_{B} is the optimal value of factor B, and so on. To be truest to the
statistics behind this model, terms involving small values of (m_{X} -
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.