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Page 1 of 2 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 Page 2 of 2 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