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COMPARING POWER OF THE DOE WHEN USING QUALITATIVE (NOMINAL AND ORDINAL)
VARIABLES COMPARED TO QUANTITATIVE (CONTINUOUS OR NUMERIC DISCRETE ) VARIABLES
By Arved Harding, Applied Statistics Group, Eastman Chemical Company
In this example* two factors are used with three levels each: material (A) and temperature (B). In
Experiment #1 temperature is treated as qualitative. In Experiment #2 it is treated as a numeric
continuous factor. The experiments are exactly the same, but by changing the designation of
each variable as categoric or numeric you change how Design-Expert® software treats those
variables.
Power is interpreted as the probability of seeing a change in the response of size “x” in units for
standard deviation. For Experiment # 1 the model is A, B and AB. The table for the power is
shown below.
Power at 5 % alpha level to detect signal/noise ratios of
Term
StdErr
0.5 Std. Dev.
1 Std. Dev.
1.667 Std. Dev.
A[1]
0.235702
16.4 %
53.4 %
94.2 %
A[2]
0.235702
B[1]
0.204124
16.4 %
53.4 %
94.2 %
B[2]
0.117851
9.6 %
27.3 %
68.2 %
A[1]B[1] 0.288675
A[2]B[1] 0.288675
A[1]B[2] 0.166667
A[2]B[2] 0.166667
Basis Std. Dev. = 1.0
For Categoric Terms the minimum power for each group of terms is reported.
So for the main effect factor A we have a 94.2% chance of seeing a change in the response of
1.667 standard deviations.
___________________________________________________________________________
*
Source: “General Factorial Tutorial” parts 1 and 2 for Design-Expert software posted at:
1. http://statease.info/dx8files/manual/DX8-02C-Gen2Factorial-P1.pdf and
2. http://statease.info/dx8files/manual/DX8-02D-Gen2Factorial-P2.pdf
The data used are from Design and Analysis of Experiments by Douglas C. Montgomery.
Thanks to everyone for sharing materials and knowledge!!!
7/8/2010 9:28:00 AM Z:\Publications\LDB PDFs\Comparing power of the DOE when using Qualitative Eastman.docx
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For Experiment # 2 the model is A, B, AB, B2 and AB2. The table for the power is shown below.
Power at 5 % alpha level
Term
StdErr**
0.5 Std. Dev.
1 Std. Dev.
1.667 Std. Dev.
A[1]
0.408248
16.4 %
53.4 %
94.2 %
A[2]
0.408248
B
0.204124
21.9 %
65.6 %
97.6 %
A[1]B
0.288675
12.4 %
37.6 %
81.2 %
A[2]B
0.288675
B^2
0.353553
27.6 %
77.8 %
99.5 %
A[1]B^2
0.5
7.3 %
15.0 %
35.2 %
A[2]B^2
0.5
Compare the two tables for the main effect of factor A: Notice that the power is the same. The
“A” terms are both qualitative and the experiments are the same, so they should be the same.
For the main effect of factor B, though, there is a small difference. For Experiment #1 the power
at 1.667 standard deviations is 94.2% versus 97.6% for Experiment #2.
For the AB interaction term the difference is much larger. For Experiment #1 the power at 1.667
standard deviations is 68.2% versus 81.2% for Experiment #2. This is because of the qualitative
treatment of factor B in Experiment #1, which consumes more degrees of freedom than if it had
been designated as a continuous factor.
The lesson here is if you have a factor that is continuous then you should treat it so
because you can get more statistical power to see the changes of interest.
_________________
An extra lesson. 
Notice the AB2 term in Experiment #2. The power to detect a change of 1.667 standard
deviations is only 35.2% and yet in the analysis of this dataset a difference was detected (pvalue ~ 0.01). This could be for one of two reasons:


The change was much larger than 1.667 standard deviations
Even at a probability of 35.2% of success, detecting the difference of interest could
happen. Remember that in baseball a batting average higher than .300 is considered to
be excellent and yet sometimes those players get on base anyway. 
– Your friendly, neighborhood statistician: Arved
Harding
® Design-Expert is a trademark of Stat-Ease, Inc.
7/8/2010 9:28:00 AM Z:\Publications\LDB PDFs\Comparing power of the DOE when using Qualitative Eastman.docx