1. 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

2. Balanced Incomplete Block Designs 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

3. Example (2.31) 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?

4. Balanced Incomplete Block Design 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

5. Randomization:

6. Model:

7. Analysis 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

8. Extra Sum of Squares Principle Suppose have 2 models, M 1 and M 2, 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

9. Analysis of a BIBD Model I: Model II: Hypothesis: F-test:

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

11. Example (2.31) 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?

12. Example (2.31) Data:

13. Noise vs. Shape

14. Noise vs. Plate

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

16. Chapter 3 - Full Factorial Experiments at 2-Levels Often wish to investigate impact of several (k) factors If each factor has r i 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 2 k full factorial design Can only estimate linear effects!

17. Example - Epitaxial Layer Growth 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

18. Example - Epitaxial Layer Growth A 16 run 2 4 experiment was performed (page 97) with 6 replicates Notation: