Stat 470-13

1. Stat 470-13 • Today: Finish Chapter 3 • Assignment 3: 3.14 a, b (do normal qq-plots only),c, 3.16, 3.17 • Additional questions: 3.14 b (also use the IER version of Lenth’s method and compare to the qq-plot conclusions), 3.19 • Important Sections in Chapter 3: • 3.1-3.7 (Read these) • 3.8 (don’t bother reading), • 3.11-3.13 (Read these)

2. Analysis of Unreplicated 2k Factorial Designs • For cost reasons, 2k factorial experiments are frequently unreplicated • Can assess significance of the factorial effects using a normal or half-normal probability plot • May prefer a formal significance test procedure • Cannot use an F-test or t-test because there are no degrees of freedom for error

3. Lenth’s Method • Situation: • have performed an unreplicated 2k factorial experiment • have 2k-1 factorial effects • want to see which effects are significantly different from 0 • If none of the effects is important, the factorial effects are an iid sample of size n= 2k-1 from a N( , ) • Can use this fact to develop an estimator of the effect variance based on the median of the absolute effects

4. Lenth’s Method • s0= • PSE= • PSE=“pseudo standard error” • tPSE,i=

5. Lenth’s Method • Both s0 and the PSE are estimates of the variance of an effect • Use s0 to screen out important effects from the calculation of the PSE • Once you have an estimate of the standard error, can construct a t-like statistic • Appendix H of text gives critical values for the t-stats • Will use the IER version of Lenth’s method

6. Example: Original Growth Layer Experiment • Effect Estimates and QQ-Plot:

7. Lenth’s Method • s0= • PSE= • tPSE,i= • Cut-off:

8. Blocking in 2k Experiments • The factorial experiment is an example of a completely randomized design • Often wish to block such experiments • As an example, you may wish to use the same paper helicopter for more than one trial • But which treatments should appear together in a block?

9. Blocking in 2k Experiments • Consider a 23 factorial experiment in 2 blocks

10. Blocking in 2k Experiments • What is the relationship between ABC interaction and block? • What if estimate block effect?

11. Blocking in 2k Experiments • Can use an interaction to determine which trials are performed in which blocks • Drawback:

12. Blocking in 2k Experiments • What do the columns in the table mean for a regression model? • If there was a column for the mean, what would it look like?

13. Blocking in 2k Experiments • What would the interaction column between the block and ABC interaction look like? • Can write as:

14. Blocking in 2k Experiments • Presumably, blocks are important. • Effect hierarchy suggest we sacrifice higher order interactions • Which is better: • b=ABC • b=AB • Can write as:

15. Blocking in 2k Experiments • Suppose wish to run the experiment in 4 blocks • b1=AB and b2=BC • These imply a third relation • Group is called the defining contrast sub-group

16. Blocking in 2k Experiments • Identifies which effects are confounded with blocks • Cannot tell difference between these effects and the blocking effects

17. Suppose wish to run the 23 experiment in 4 blocks b1=AB and b2=BC I=ABb1=BCb2=ACb1b2 Suppose wish to run the 23 experiment in 4 blocks b1=ABC and b2=BC I=ABCb1=BCb2=Ab1b2 Which design is better?

18. Ranking the Designs • Let D denote a blocking design • gi(D) is the number of i-factor interactions confounded in blocks (i=1,2,…k) • For any 2 blocking schemes (D1 and D2) , let r be the smallest i such that

19. Ranking the Designs • Effect hierarchy suggests that the design that confounds the fewest lower order terms is best • So, if then D1 has less aberration • A design has minimum aberration (MA) if no design has less aberration