The FACTEX Procedure

Replication

In quality improvement applications, it is often important to analyze both the mean response of a process and the variability around the mean. To study variability with an experimental design, you must take several measurements of the response for each different combination of the factors of interest; that is, you must replicate the design runs.

Replicating a Fixed Number of Times

A simple method of replication is to take a given number of measurements for each combination of factor levels in the basic design. You can replicate runs in the design by specifying numbers for the POINTREP= and DESIGNREP= options in the OUTPUT statement. For example, the following code constructs a full $2^2$ design and uses both of these options to replicate the design three times: 

proc factex;
   factors A B;
   output out=one pointrep =3;
run;
   output out=two designrep=3;
run;

The output data sets ONE and TWO have the same 12 runs, but they are in different orders. In the data set ONE, the POINTREP= option causes all three replications of each run to occur together, as shown in Output 7.41.

Output 7.41: Four-Run Design Replicated Using the POINTREP= Option

\[ \begin{array}{ll} \left.\begin{array}{ccc} OBS & A & B \end{array}\right. & \\ \ \\ \left.\begin{array}{rrr} 1 & -1 & -1 \\ 2 & -1 & -1 \\ 3 & -1 & -1 \end{array}\right\} & \emph{three replicates of run 1} \\ \ \\ \left.\begin{array}{rrr} 4 & -1 & 1 \\ 5 & -1 & 1 \\ 6 & -1 & 1 \end{array}\right\} & \emph{three replicates of run 2} \\ \ \\ \left.\begin{array}{rrr} 7 & 1 & -1 \\ 8 & 1 & -1 \\ 9 & 1 & -1 \end{array}\right\} & \emph{three replicates of run 3} \\ \ \\ \left.\begin{array}{rrr} 10 & 1 & 1 \\ 11 & 1 & 1 \\ 12 & 1 & 1 \end{array}\right\} & \emph{three replicates of run 4} \end{array} \]


On the other hand, in the data set TWO, the DESIGNREP= option causes all four runs of the design to occur together three times, as shown in Output 7.42.

Output 7.42: Four-Run Design Replicated Using the DESIGNREP= Option

\[ \begin{array}{rr}& \left. \begin{array}{ccc} OBS & A & B \end{array}\right. \\ \ \\ \emph{Replicate 1} & \left\{ \begin{array}{rrr} 1 & -1 & -1 \\ 2 & -1 & 1 \\ 3 & 1 & -1 \\ 4 & 1 & 1 \end{array}\right. \\ \ \\ \emph{Replicate 2} & \left\{ \begin{array}{rrr} 5 & -1 & -1 \\ 6 & -1 & 1 \\ 7 & 1 & -1 \\ 8 & 1 & 1 \end{array}\right. \\ \ \\ \emph{Replicate 3} & \left\{ \begin{array}{rrr} 9 & -1 & -1 \\ 10 & -1 & 1 \\ 11 & 1 & -1 \\ 12 & 1 & 1 \end{array}\right. \\ \end{array} \]


Replicating with an Outer Array

Another method of design replication considers the range of environmental conditions over which the process should maintain consistency. This method distinguishes between control factors and noise factors. Control factors are factors that are under the control of the designer or the process engineer. Noise factors cause the performance of a product to vary when the nominal values of the control variables are fixed (noise factors are controllable for the purposes of experimenting with the process). Typical noise factors are variations in the manufacturing environment or the customer’s environment due to temperature or humidity. The object of experimentation is to find the best settings for the control factors for a variety of settings for the noise factors. In other words, the goal is to develop a process that runs well in a variety of environments. Refer to Dehnad (1989) and Phadke (1989) for further discussion.

To achieve this goal, a collection of environmental conditions (settings for the noise factors) is determined. This collection is called the outer array. Each run in the control factor design (inner array) is replicated within each of these environments. The mean and variance of the process over the outer array are computed for each run in the inner array. Either the outer array or the inner array might consist of all possible different settings for the associated factors, or they might be fractions of all possible settings.

You can replicate designs in this way by using data set names for the POINTREP= and DESIGNREP= options in the OUTPUT statement. If you construct a design for your control factors and you want to run a noise factor design for each run in the control factor design, specify the data set that holds the noise factor design (that is, the outer array) with the POINTREP= option in the OUTPUT statement. See Example 7.14 for an example.