Design of Experiments (DOE) has become one of the most popular statistical techniques of the 1990's. DOE originated in 1920 by a British scientist, Sir R. A. Fisher, as a method to maximize the knowledge gained from experimental data and it has evolved over the last 70 years. Unfortunately, most of the development of DOE was mathematically complex and thus, its use has been restricted to those well versed in mathematics. Recent widespread popularity of DOE is associated with the works of Taguchi [2], a Japanese engineer, who focused on the practical use versus mathematical perfection of the technique. In short, Taguchi's work began a revolution in the presentation of DOE material where mathematical theory is downplayed in order to enhance the clarity and practicality of the subject. Thus, scientists, engineers, technicians, and managers who are not mathematical experts are now becoming experimental design practitioners. However, close scrutiny of the Taguchi method [3] revealed limitations, which has led to the most recent evolution in DOE, a blended approach of Taguchi and classical techniques.

The gap between the current level of knowledge and nature's perfect knowledge tends to widen as industry today seeks to understand more complex processes.

The typical engineering, scientific, or physics approach is to spend years of research trying to narrow the gap utilizing theoretical knowledge. DOE, on the other hand, will allow one to quickly narrow the gap through proper planning, designing, data collection, analysis, and confirmation.

Consider again nature's perfect knowledge of any process (activity or specified area of science etc.) where there is a true mathematical relationship that can describe process outputs as a function of all the process inputs, e.g. Newton's law of motion F = MA. If nature knows this relationship but we don't, how can we obtain the relationship quickly without getting tied up in years of theoretical research as Newton did.

A simple DOE for 2 inputs such as mass (M) and acceleration (A), each tested at 2 levels, can be set up as the four experimental runs (or trials) of low (Lo) and High (Hi) settings shown in Table 1.

To obtain the numbers in red (last three rows), you have to do the following for each of the M, A and MxA columns: find the average of the output when the effect column values are 1, find the average of the output when the effect column values are +1, and then find the difference of the two effect averages (Delta). For example, when the M effect column is at 1 (runs 1 and 2) the output is 500 and 1000 which averages to 750.

The model generated from the DOE is built using least squares regression which can be simplified for 2 level designs as shown below:

    F hat is the predicted average for Force
    Mc is the coded variable for mass
    Ac is the coded variable for acceleration
    Avg(Avg. F) is the grand experimental mean
    Delta(M) is the size of the linear effect for M, etc.

(2) Thus, our actual model becomes

where

    Ma = actual setting value of mass
    Mc = coded setting value of mass
    MH = experimental high setting of actual values
    ML = experimental low setting of actual values

For our example

Since DOE was not developed until the 1900's, Newton never had access to the tool. Newton, however, was not opposed to the use of new tools. For example, when he ran into a road block in his research he developed calculus as a new tool to assist him. It is, therefore, logical to speculate that Newton, if he was living today, would eagerly embrace new tools such as DOE to assist him in gaining knowledge from experimental data.

The previous example used data generated from an F = MA model instead of a physical experiment. In our next example, we will tackle a similar problem using actual experimental data. Suppose we go back in time to Germany during the early 1800 s. In 1826, German scientist George Simon Ohm determined the formula now known as Ohm s law. Ohm's law establishes the mathematical relationship between voltage (V), current (I), and resistance (R) as V = IR. Today, we know that this law applies to both direct and alternating currents, but the formula, as given in its original simple form, applied only to steady DC situations. It is known that Ohm conducted much careful experimental work. Unfortunately, he did not have access to DOE, or his work might have been shortened considerably.

What if we pretend to be ignorant of Ohm's law, and see how Ohm might have proceeded with even the most basic of today's DOE tools. Of course, nature knows those factors affecting voltage, so we will interrogate nature in a methodical way. First, however, we have to establish those experimental variables to include in a test using DOE. Specifically, we need to identify any factors we believe may affect the response of interest in this case "voltage." Suppose the factors in Table 5 are felt to be important; each is addressed as to how it will be incorporated into the DOE:

We all know that wattage of the resistor (W) should have no effect, but remember that we are pretending to be ignorant of Ohm's law; we also know that room temperature and humidity should have a negligible or imperceptible effect under normal room conditions. The other factors are "real" considerations worthy of treatment as described above.

A full factorial is chosen for this experiment. In a full factorial, all possible combinations of experimental levels of all variables are run. Since there are only three experimental variables of interest, the number of runs in the full factorial equals 23 = 8 which is certainly feasible. This will permit us to evaluate all possible effects and interactions of the three experimental variables (I, R, and W) and to write a mathematical model including any and all contributing terms. The experimental array is shown in Table 6. Note that availability of resistors at particular wattages, safety through low voltages and currents, the need to be within the wattage ratings of all resistors, and instrumentation ranges and precision were major factors used in the selection of experimental variable settings.

Table 7 presents the array of coded input values, actual responses, and averages used in the modeling analysis of the DOE. The coded values of I, R, and W are set at 1 for low settings of each experimental variable and at +1 for high settings. The 1 and +1 values in the interaction columns do not reflect experimental settings, but are determined from the product of the coded values in their component columns and are used to analyze the strength of the interactive effect of the variables.

Multiple replications (reps) of the experiment provide five different voltage measurements. These voltages vary somewhat due to the "noise" always involved in experimental research. The "noise" is due to the continued re-establishment of the operating conditions at each of the experimental run settings (different resistors were used for each rep, and current was re-established for each rep). In this experiment, each of the runs was performed in a random order for the first rep. Then, each was performed again in a different random order for the second rep. This was continued throughout the five reps. The term represents the average of the five reps for each of the eight experimental runs. Appended to the bottom of Table 7 (in the shaded area) are the relevant average calculations, similar to those illustrated in Table 4.

   V hat is the predicted average for V
   Ic is the coded variable for current
   Rc is the coded variable for resistance
   Avg. (Avg. V) is the grand experimental mean
   Delta (I) is the size of the linear effect for I, etc.

(2) The actual model becomes

       = - 0.019 + 3.013Ia + 0.000Ra + 0.994IaRa

Under actual experimental conditions, we got a model very close to nature's V = IR. Note that the wattage of the resistor (W) dropped out as insignificant during the analysis and does not appear in the model. Of course, we knew this should happen, but it is always comforting to know that if we had not known that W is not an affecting factor, DOE would have told us! The coefficient for Ia of 3.013 appears large relative to the coefficient of IaRa. However, you must remember that the actual range of Ia is .005 to .020. Thus, the prediction of V is only changing from about .015 to .060 over the range of Ia. Therefore, the most likely reason for a non-zero coefficient for Ia is the low test values (.005 and .020) used in the DOE.

If we were to now use the model to predict the voltage across a resistor of 220 ohms when the current is 0.012 amps, the model would predict V hat = 2.641 volts. Nature (using Ohm's law) knows that the voltage would be V = IR = 2.640 volts. Even though the experimental model did not yield a perfect V = IR, knowing that the voltage is in large part determined by the interaction or combined effect of I and R might have reduced Ohm's research time significantly.

Obviously, not all models in nature are as simple as F = MA and V = IR, however the purpose here is to show how DOE can enable today's scientist, engineers, and physicists to close the gap quickly with a model from experimental data. For more information on evaluating non-linear relationships and/or large numbers of input variables, see a good text on DOE!