by Manikandan Jayakumar, guest blogger

We use Design of Experiments (DOE) to optimize the value of a response (Y) by simultaneously changing the values of several factors (X’s). The response will often be a continuous variable, but in some scenarios you need to optimize an attribute or categorical response (Pass/Fail, Accept/Reject, etc.).

Collecting the Data for an Attribute Response DOE

Let’s see how we can use DOE to optimize an attribute response, using data from the manufacturing sector. We are looking at 3 factors with 2 levels each, which are coded as -1 and +1. The total number of treatment combinations is LF => 23 = 8.

To create this experimental design in Minitab, we can choose Stat > DOE > Factorial > Create Factorial Design…. Select 3 as the number of the factors, then click on “Options” and uncheck “Randomize runs” to reproduce the experimental design as shown below—a nonrandomized design has been used here to more clearly show the 8 different factor level settings in the experiment.

Running more replicates in an experiment gives more accuracy, so we will repeat (or replicate) the experiment 10 times to capture the variance at each given combination of factor levels. Now the total number of runs is LF x Replicates => 23 X 10 = 80.

Here are the results of the experiment:

Now we can convert the captured OK/NOK attribute response for all the replicates into a proportion, as shown below:

Using the Arcsine Transformation

Percentages follow a binomial rather than a normal distribution, which violates basic assumptions for most statistical tests—especially for smaller or greater percentages (< 30% or > 70%) which constrain the limit of variability. To restore normality to the data distribution, we need to apply an arcsine (or inverse sine) transformation.

This transformation assists when working with proportions and percentages by stabilizing variation for the binomial distribution. The proportion p can be made nearly "normal" if the square root of p is used with the arcsine transformation. The arcsine transformation is then computed as a function of the proportion, p --> (arcsine (SQRT (p)).

Note that arcsine transformation is appropriate for data obtained from a count and expressed as decimal fractions or percentages. There is a tendency to transform any percentage using arcsine transformation. But only percentages derived from count data, such as % Rejections (which is derived from the ratio of the number of Rejected parts to the Total number of parts) should be transformed. You should not transform percentage data such as % protein or % carbohydrates, which are not derived from count data.

We can perform the arcsine transformation using Minitab’s calculator in two steps. First, convert the calculated proportion (p) using the expression SQRT(Proportion(p)) and store it in a separate column named Sqrt(p). Now apply the arcsine function to this column, using the expression ASIN(Sqrt(p)). If you prefer, you can perform the transformation in a single step by nesting the commands, like this: ASIN(SQRT(Proportion(p))).

The values calculated using the arcsine transformation are shown below. By default, Minitab will calculate to five decimal places, but we have rounded our data to two decimal places here:

Now that we have created the design, collected the data and transformed it, we are ready to conduct the analysis and determine which factors are significant. These steps and the outcome will be provided in the next post.