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)

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: -

Effect

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

Estimate

0.055

0.142

-0.109

0.836

-0.032

-0.074

-0.025

0.047

0.010

-0.037

0.060

0.067

-0.056

0.098

0.036

-1 0 1 Qu a n t i l e s o f S ta n d a rd No rm a l

• s 0 = • PSE= • t PSE,i = • Cut-off:

Lenth’s Method

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

A

-1 -1 -1 -1 +1 +1 +1 +1

B

-1 -1 +1 +1 -1 -1 +1 +1

C

-1 +1 -1 +1 -1 +1 -1 +1

AB

+1 +1 -1 -1 -1 -1 +1 +1

AC

+1 -1 +1 -1 -1 +1 -1 +1

BC

+1 -1 -1 +1 +1 -1 -1 +1

ABC

-1 +1 +1 -1 +1 -1 -1 +1

BLOCK

I II II I II I I II

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=1,2,…k

)

i

-factor interactions confounded in blocks • For any 2 blocking schemes (

D 1 g r

( 

D

2 ) and

D 2

) , let

r

be the smallest

i

such

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