DOE: Full-factorial vs. Fractional-factorials (and notations)

This experiment will test 4 factors at each of 2 levels, in a half-fraction factorial (2 ⁴ would be 16 runs, this experiment is the equivalent of 2 ³ = 8 runs).

The resolution of IV means:

• Main effects are confounded with 3-way interactions (1 + 3 = 4). You have to acknowledge that any measured main effects could be influenced by 3-way interactions. Since 3-way interactions are relatively rare, attributing the measured differences to the main effects only is most often a safe assumption.

• 2-way interactions are confounded with each other (2 + 2 = 4). This design would not be a good way to estimate 2-way interactions.

Interpreting DOE results

Most statistical software packages will give you results for main effects, interactions, and standard deviations.

1. Main effects plots for mean

   • Interpretation of slopes is all relative. Lines with steeper slopes (up or down) have a bigger impact on the output means than lines with little or no slope (flat or almost flat lines).

     

   • In this example, the line for shelf placement slopes much more steeply than the others—meaning it has a bigger effect on sales than the other factors. The other lines seem flat or almost flat, so the main effects are less likely to be significant.

2. Main effects plots for standard deviation

   

   • These plots tell you whether variation changes or is the same between factor levels.

   • Again, you want to compare slopes in comparison to each other. Here, Design has much more variation one level than at the factors (so you can expect it to have much more variation at one level than at the other level).

3. Pareto chart of the means for main factor effects and higher-order interactions

   

   • You're looking for individual factors (labeled with a single letter) and interactions (labeled with multiple letters ) that have bars that extend beyond the "significance line"

   • Here, main factor A and interaction AB have significant effects, meaning placement, and interaction of placement and color have the biggest impact on sales (compare to the "main effects plot for mean," previous page).

4. Pareto chart on the standard deviation of factors and interactions

   • Same principle as the Pareto chart on means

     

   • Here, only Factor C (Design) shows a significant change in variation between levels

5. Minitab session window reports

   • Shelf Placement and the Shelf Placement* Color interactions are the only significant factors at a 90% confidence internal (if alpha were 0.05 instead of 0.10, only placement would be significant)

     Fractional Factorial Fit: Sales versus Shelf Placem, Color, Design, Text
     Term	Effect	Coef	SE Coef	T	P
     Constant	128.50	0.2500	514.00	0.001
     Shelf PI	− 38.50	− 19.25	0.2500	− 77.00	0.008
     Color	2.00	1.00	0.2500	4.00	0.156
     Design	0.50	0.25	0.2500	1.00	0.500
     Text	− 0.00	− 0.00	0.2500	− 0.00	1.000
     Shelf PI*Color	3.50	1.75	0.2500	7.00	0.090
     Shelf PI*Design	− 3.00	− 1.50	0.2500	− 6.00	0.105

     Analysis of Variance for Sales (coded units)
     Source	DF	Seq SS	Adj SS	Adj MS	F	P
     Main Effects	4	2973.00	2973.00	743.250	1E+03	0.019
     2-Way Interactions	2	42.50	42.50	21.250	42.50	0.108
     Residual Error	1	0.50	0.50	0.500
     Total	7	3016.00

   • Design is the only factor that has a significant effect on variation at the 90% confidence level

     Fractional Factorial Fit: Std Dev versus Shelf Placement, Color, …
     Term	Effect	Coef	SE Coef	T	P
     Constant	9.0000	0.2500	36.00	0.018
     Shelf PI	− 1.5000	− 0.7500	0.2500	− 3.00	0.205
     Color	− 0.0000	− 0.0000	0.2500	− 0.00	1.000
     Design	6.5000	3.2500	0.2500	13.00	0.049
     Text	1.0000	0.5000	0.2500	2.00	0.295
     Shelf PI*Color	0.5000	0.2500	0.2500	1.00	0.500
     Shelf PI*Design	0.0000	0.0000	0.2500	0.00	1.000

     Analysis of Variance for Std (coded units)
     Source	DF	Seq SS	Adj SS	Adj MS	F	P
     Main Effects	4	91.0000	91.0000	22.7500	45.50	0.111
     2-Way Interactions	2	0.5000	0.5000	0.2500	0.50	0.707
     Residual Error	1	0.5000	0.5000	0.5000
     Total	7	92.0000