One of the issues that is addressed more carefully in our new class than in our old is transformations. Here's a review for those of you that might have found them puzzling, as I did when I first encountered them.
These questions are answered in terms of using our software, STRATEGY, but you should be able to apply the answers to any modern Design of Experiments software. If you would like to see one of our regression output reports, download alumina.out, a regression report in ASCII format.
A transformation is a mathematical calculation that is performed on each value in an Factor or Response column. (Transformations of responses are more common than transformations of factors.) Adding 1 to every value in a particular response column is a transformation; taking a square root is another.
You probably never WANT to make a transformation, but sometimes you should anyway. You should make a transformation when the s's for different trials are not estimates of the same s.
How can this happen? Perhaps your instument is not very precise when making measurements for a small response, but is very precise when measuring a large response. A trial with a small response will then have a large s and a trial with a large response will have a small s. You cannot combine, or "pool," these s's because they are different. Of course STRATEGY relies on pooling the s's. A transformation will make the s's consistent so that pooling is legitimate.
STRATEGY provides four indicators in the regression report that tell you when a transformation is necessary. They are:
If you check the box for "cues" before running the regression you will be provided with instructons in the printout for interpreting these tests.
Each test for consistency, or "uniformity," of the s's is imperfect. Each can give false alarms and false comfort. If two of the three plots indicate the need for a transformation, you should make one.
Yes, but this is less common. You also want to transform your factors BEFORE running an experiment if at all possible to insure that the trials will be well-spread-out. If you have a factor with levels spanning a few orders of magnitude, consider transforming the factor with a log transformation. STRATEGY provides plots at the bottom of the regression output to determine if a factor transformation is necessary. The cues provide complete instructions.
If you have run replicates of 5 to 8 trials, you can use the slope of the Box-Cox plot to determine an appropriate transformation. First, calculate the slope of the line from the plot. Next, subtact this slope from 1. Finally, raise Y to the power of the number just calculated. Precision is not necessary here - feel free to round. Some common transformations are square root (slope = 0.5), log (slope = 1), and reciporical (slope = 2). STRATEGY also offers a wide range of alternative transformations. If you didn't collect many replicates, you can try several transformations to see if any of them helps. You can also go back to the lab and run more replicates to improve your Box-Cox plot.
You can always use our free design check service. We will help you determine the need for a transformation.