Stat 602 The Design of Experiments

Transcription

1 Stat 602 The Design of Experiments Yuqing Xu Department of Statistics University of Wisconsin Madison, WI 53706, USA April 28, 2016 Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

2 Blocking in 2 k Model to 2 Blocks Steps of Blocking in 2 k Model to 2 Blocks Choose a factorial effect to confound with blocks (defining contrast). Put all rows with a plus sign on the defining contrast in one block and all the rows with a minus sign in the other block. We have zero information on that defining contrast effect. He have complete information for all unconfounded factorial effects. Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

3 Blocking in 2 k Model to 2 Blocks How to choose the defining contrast? Fact from class: At least one effect should be confounded. We will lose all information about whatever contrast is used as defining contrast. The multifactor interactions are generally of less interest. For the 2 k factorial in two blocks, the obvious defining contrast is the k-factor interaction. Another way to think about blocking Question: How many degrees of freedom we have for the block? How many confounding columns we have? Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

4 Blocking in 2 k Model to 2 Blocks Example: 2 3 in two blocks of size 4 Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

5 Example: 2 3 into 4 blocks Start by choosing two defining contrasts for confounding a 2 3 design into four blocks of size 2. We will use ABC and BC as defining contrasts. We have confounded into four blocks, so there are 3 degrees of freedom between blocks. What is the third degree of freedom that is confounded with block differences? Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

6 Example: 2 3 into 4 blocks Start by choosing two defining contrasts for confounding a 2 3 design into four blocks of size 2. We will use ABC and BC as defining contrasts. We have confounded into four blocks, so there are 3 degrees of freedom between blocks. What is the third degree of freedom that is confounded with block differences? Example: 2 3 into 4 blocks A is also confounded! Why is that? Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

7 Generalized Interaction (Block Interaction) Definition: The Generalized Interaction is the product of two effects. Example: The Generalized Interaction of ABC and BC is ABC BC = ABBCC = AB 2 C 2 = A Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

8 Generalized Interaction (Block Interaction) Definition: The Generalized Interaction is the product of two effects. Example: The Generalized Interaction of ABC and BC is Example: 2 3 into 4 blocks ABC BC = ABBCC = AB 2 C 2 = A When we confound into four blocks using two defining contrasts, we not only confound the defining contrasts with blocks, we also confound their generalized interaction. This corresponds to the degrees of freedom. Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

9 Example: 2 4 into 4 blocks In general a 2 k factorial can be confounded into 2 q blocks of size 2 k?q. Now we choose two defining contrasts for confounding a 2 4 design into four blocks of size four. We will use ABC and BCD as defining contrasts; these are good choices. The generalized interaction of ABC and BCD is ABC BCD = AB 2 C 2 D = AD If we had chosen AD and ABC as our defining contrasts, we would get the same four blocks, and the generalized interaction BCD would also be confounded with blocks. Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

10 Example: 2 4 into 4 blocks Now we have done with choosing the defining contrasts. How to split treatment levels? Example: 2 4 into 4 blocks Table! Else? Even/odd rule Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

11 Even/odd rule Look at the letters of a factor-level combination contain an even or odd number of letters from the defining contrast. Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10

12 Fraction Factorial Idea of Fraction Factorial A fractional-factorial design allows us to get information on main effects and low-order interactions without having to run the full factorial design. A fractional factorial is to confound the factorial into blocks, but only run one of the blocks. e.g. When we have 2 4 design with 4 blocks, we only use block with ABC+ and BCD-. Yuqing Xu (UW-Madison) Stat 602 Week 14 April 28, / 10