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Lecture 7. Last day: 2.6 and 2.7 Today: 2.8 and begin 3.1-3.2 Next day: 3.3-3.5 Assignment #2: Chapter 2: 6, 15 (treat tape speed and laser power as qualitative factors), 27, 30, 32, and 36. Balanced Incomplete Block Designs. Sometimes cannot run all treatments in each block

Lecture 7

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- Last day: 2.6 and 2.7
- Today: 2.8 and begin 3.1-3.2
- Next day: 3.3-3.5
- Assignment #2: Chapter 2: 6, 15 (treat tape speed and laser power as qualitative factors), 27, 30, 32, and 36

- Sometimes cannot run all treatments in each block
- That is, block size is smaller than the number of treatments
- Instead, run sets of treatments in each block

- Experiment is run on a resistor mounted on a ceramic plate to study the impact of 4 geometrical shapes of resistor on the current noise
- Factor is resistor shape, with 4 levels (A-D)
- Only 3 resistors can be mounted on a plate
- If 4 runs of the of the plate are to be made, how would you run the experiment?

- Situation:
- have b blocks
- each block is of size k
- there are t treatments (k<t)
- each treatment is run r times

- Design is incomplete because blocks do not contain each treatment
- Design is balanced because each pair of treatments appear together the same number of times

- Randomization:

- Model:

- The analysis of a BIBD is slightly more complicated than a RCB design
- Not all treatments are compared within a block
- Can use the extra sum of squares principle (page 16-17) to help with the analysis

- Suppose have 2 models, M1 and M2, where the first model is a special case of the second
- Can use the residual sum of squares from each model to form an F-test

- Model I:
- Model II:
- Hypothesis:
- F-test:

- Similar to other cases, can do parameter estimation using the typical constraints
- Can also do multiple comparisons

- Experiment is run on a resistor mounted on a ceramic plate to study the impact of 4 geometrical shapes of resistor on the current noise
- Factor is resistor shape, with 4 levels (A-D)
- Only 3 resistors can be mounted on a plate
- If 4 runs of the of the plate are to be made, how would you run the experiment?

- Data:

- Model I:
- Model II:
- Hypothesis:
- F-test:

- Often wish to investigate impact of several (k) factors
- If each factor has ri levels, then there are possible treatments
- To keep run-size of the experiment small, often run experiments with factors with only 2-levels
- An experiment with k factors, each with 2 levels, is called a 2k full factorial design
- Can only estimate linear effects!

- In IC fabrication, grow an epitaxial layer on polished silicon wafers
- 4 factors (A-D) are thought to impact the layer growth
- Experimenters wish to determine the level settings of the 4 factors so that:
- the process mean layer thickness is as close to the nominal value as possible
- the non-uniformity of the layer growth is minimized

- A 16 run 24 experiment was performed (page 97) with 6 replicates
- Notation: