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Lecture 7

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

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Lecture 7

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  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, 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

  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 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!

  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 24 experiment was performed (page 97) with 6 replicates • Notation:

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