Stat 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: – (Read these) –3.8 (don’t bother reading), – (Read these)

Analysis of Unreplicated 2 k Factorial Designs For cost reasons, 2 k 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

Lenth’s Method Situation: –have performed an unreplicated 2 k factorial experiment –have 2 k -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= 2 k -1 from a N(, ) Can use this fact to develop an estimator of the effect variance based on the median of the absolute effects

Lenth’s Method s 0 = PSE= PSE=“pseudo standard error” t PSE,i =

Lenth’s Method Both s 0 and the PSE are estimates of the variance of an effect Use s 0 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

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

Lenth’s Method s 0 = PSE= t PSE,i = Cut-off:

Blocking in 2 k 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?

Blocking in 2 k Experiments Consider a 2 3 factorial experiment in 2 blocks

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

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

Blocking in 2 k 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?

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

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

Blocking in 2 k Experiments Suppose wish to run the experiment in 4 blocks b 1 =AB and b 2 =BC These imply a third relation Group is called the defining contrast sub-group

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

Which design is better? Suppose wish to run the 2 3 experiment in 4 blocks b 1 =AB and b 2 =BC I=ABb 1 =BCb 2 =ACb 1 b 2 Suppose wish to run the 2 3 experiment in 4 blocks b 1 =ABC and b 2 =BC I=ABCb 1 =BCb 2 =Ab 1 b 2

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

Ranking the Designs Effect hierarchy suggests that the design that confounds the fewest lower order terms is best So, if then D 1 has less aberration A design has minimum aberration (MA) if no design has less aberration