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<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:
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, 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
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
1
.
6
y
1
.
4
1
.
2
1
.
0
0
.
8
1
.
01
.
52
.
02
.
53
.
03
.
54
.
0
s
h
a
p
e
Noise vs. Plate
A
A
1
.
6
C
C
y
1
.
4
B
A
1
.
0
C
0
.
8
B
D
B
1
.
2
D
D
1
.
0 1
.
5 2
.
0 2
.
5 3
.
0 3
.
5 4
.
0
p
l
a
t
e
• 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 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
• 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 24 experiment was performed (page 97) with 6 replicates
• Notation: