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Exploring Interaction Effects in Factorial Experiments: A Statistical Approach

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This document discusses the testing of factors and their levels in factorial experiments, emphasizing the importance of randomization in selecting run combinations for testing. It highlights main effects of factors A and B, the interaction effect (AB), and provides multiple equations to quantify these effects. The comprehensive examples, including RC Circuit applications, illustrate practical implementations of these concepts in statistical experimentation, allowing for a deeper understanding of how different factors influence outcomes.

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Exploring Interaction Effects in Factorial Experiments: A Statistical Approach

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  1. Table 8.11

  2. Table 8.1

  3. Table 8.2 Note: There are (# of Treatment Combinations)! number of possible ways to test these factors and levels (“Run Combinations”). Pick one randomly to do your testing. Don’t always test in a pre-described order.

  4. Table 8.3

  5. Table 8.4

  6. Figure 8.2

  7. Eq. 1 “A main effect” The “A main effect” is the average change in Y when factor A changes from low to high (- to +) A is the average change from the bottom to the top of the square.

  8. Eq. 2 “B main effect” The “B main effect” is the average change in Y when factor B changes from low to high (- to +) B is the average change from the left to the right side of the square.

  9. Eq. 3 AB is the difference in the average of the responses across both diagonals of the square. AB is a measure of the interaction between factors A and B.

  10. Table 8.5

  11. Figure 8.3

  12. Figure 8.4

  13. Figure 8.5

  14. Eq. 4 Eq. 5

  15. Table 8.6

  16. Table 8.7 Table 8.8

  17. kI = ((1)+b+a+ab)/4 Eq. 6a kA = (-(1)-b+a+ab)/4 Eq. 6b kB = (-(1)+b-a+ab)/4 Eq. 6c kAB = ((1)-b-a+ab)/4 Eq. 6d Y = kI+kAXA+ kBXB+ kABXAXB Eq. 7

  18. Example: RC Circuit

  19. Example: RC Circuit Example: RC Circuit

  20. Tables 8.7 and 8.8 Example: RC Circuit

  21. km = ((1)+b+a+ab)/4 Eq. 6a kA = (-(1)-b+a+ab)/4 Eq. 6b kB = (-(1)+b-a+ab)/4 Eq. 6c kAB = ((1)-b-a+ab)/4 Eq. 6d Y = km+kAXA+ kBXB+ kABXAXB Eq. 7 km = ((1)+b+a+ab)/4 = (350+420+360+440)/4 = 392.5 msec. kA = (-(1)-b+a+ab)/4 = (-350-420+360+440)/4 = 7.5 msec. kB = (-(1)+b-a+ab)/4 = (-350+420-360+440)/4 = 37.5 msec. kAB = ((1)-b-a+ab)/4 = (1570+30+150+10)/4 = 2.5 msec. t = 392.5 + 7.5XA+ 37.5XB+ 2.5XAXB msec.

  22. Figure 8.6

  23. Table 8.9 km = ((1)+b+a+ab)/12 Eq. 8a kA = (-(1)-b+a+ab)/12 Eq. 8b kB = (-(1)+b-a+ab)/12 Eq. 8c kAB = ((1)-b-a+ab)/12 Eq. 8d

  24. Table 8.10

  25. km = ((1)+c+b+bc+a+ac+ab+abc))/8m Eq. 9a kA = (-(1)-c-b-bc+a+ac+ab+abc))/8m Eq. 9b kB =(-(1)-c+b+bc-a-ac+ab+abc))/8m Eq. 9c kC =(-(1)+c-b+bc-a+ac-ab+abc))/8m Eq. 9d kAB =((1)+c-b-bc-a-ac+ab+abc))/8m Eq. 9e kAC =((1)-c+b-bc-a+ac-ab+abc))/8m Eq. 9f kBC =((1)-c-b+bc+a-ac-ab+abc))/8m Eq. 9g kABC = (-(1)+c+b-bc+a-ac-ab+abc))/8m Eq. 9h Y = km+ kAXA+ kBXB+ kCXC+ kABXAXB + kACXAXC+ kBCXBXC+kABCXAXBXCEq. 10

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