Lecture 10 Last day: 3.5, , 3.10 Today: , Talk about mid-term Next day: Assignment #3: Chapter 3: 3.6, 3.7, 3.14a (use a normal probability plot to detect significant effect), 3.17 and 3.20 Not to be handed in…Solutions on web by Sunday

Lecture 10 Sections covered: –Chapter 1: , 1.6, 1.7 (Tukey method only), 1.9 –Chapter 2: 2.1,2.2, 2.3 (up to only), 2.4, –Chapter 3: ,

Analysis of Location and Dispersion Effects The epitaxial growth layer experiment is a 2 4 factorial design Have looked at ways to analyze response of a factorial experiment –Plotting effects on a normal probability plot –Regression Can also model the dispersion:

Example: Original Growth Layer Experiment

Model Matrix for a single replicate:

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

Example: Original Growth Layer Experiment Regression equation for the mean response:

Example: Original Growth Layer Experiment Dispersion analysis:

Example: Original Growth Layer Experiment Regression equation for the ln(s 2 ) response:

Example: Original Growth Layer Experiment Suggested settings for the process:

Blocking in 2 k Experiments The factorial experiment is an example of a completely randomized design Often wish to block such experiments 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 Use the ABC interaction column to assign treatments to blocks What if estimate block effect?

Blocking in 2 k Experiments Presumably, blocks are important. Effect hierarchy suggest we sacrafice 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