Transcript Balanced Incomplete Block Designs
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 • That is, block size is smaller than the number of treatments • Instead, run sets of treatments in each block
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 •
Situation:
• have
b
blocks • each block is of size
k
• there are
t
treatments (
k
) • 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:
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
Extra Sum of Squares Principle • Suppose have 2 models, M 1 case of the second and M 2 , where the first model is a special • Can use the residual sum of squares from each model to form an F-test
Analysis of a BIBD • Model I: • Model II: • Hypothesis: • F-test:
Comments • Similar to other cases, can do parameter estimation using the typical constraints • Can also do multiple comparisons
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 (2.31) •
Data: Plate/Shape 1 2 3 4 A
1.11
1.70
1.60
B
1.22
1.11
1.22
C
0.95
1.52
1.54
D
0.82
0.97
1.18
Noise vs. Shape
Noise vs. Plate A A C C B A C D D B B D
• Model I: • Model II: • Hypothesis: • F-test:
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!
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
Example - Epitaxial Layer Growth • A 16 run 2 4 experiment was performed (page 97) with 6 replicates • Notation: