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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

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Lecture 7

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Lecture 7 l.jpg

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


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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


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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?


Balanced incomplete block design l.jpg

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


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  • Randomization:


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  • Model:


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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


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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


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Analysis of a BIBD

  • Model I:

  • Model II:

  • Hypothesis:

  • F-test:


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Comments

  • Similar to other cases, can do parameter estimation using the typical constraints

  • Can also do multiple comparisons


Example 2 3111 l.jpg

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?


Example 2 3112 l.jpg

Example (2.31)

  • Data:


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Noise vs. Shape


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Noise vs. Plate


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  • Model I:

  • Model II:

  • Hypothesis:

  • F-test:


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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!


Example epitaxial layer growth l.jpg

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


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Example - Epitaxial Layer Growth

  • A 16 run 24 experiment was performed (page 97) with 6 replicates

  • Notation:


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