Transcript Balanced Incomplete Block Designs

Lecture 14
• Today:
• Next day:
• Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question
on D-optimality
Fractional Factorial Split-Plot Designs
• It is frequently impractical to perform the fractional factorial design in
a completely randomized manner
• Can run groups of treatments in blocks
• Sometimes the restrictions on randomization take place because some
factors are hard to change or the process takes place in multiple stages
• Fractional factorial split-plot (FFSP) design may be a practical option
Performing FFSP Designs
• Randomization of FFSP designs different from fractional factorial
designs
• Have hard to change factors (whole-plot or WP factors) and easy to
change factors (sub-plot or SP factors)
• Experiment performed by:
– selecting WP level setting, at random.
– performing experimental trials by varying SP factors, while keeping the
WP factors fixed.
Example
• Would like to explore the impact of 6 factors in 16 trials
• The experiment cannot be run in a completely random order because 3
of the factors (A,B,C) are very expensive to change
• Instead, several experiment trials are performed with A, B, and C
fixed…varying the levels of the other factors
Design Matrix
A
-1
B
-1
C
+1
+1
-1
-1
-1
+1
-1
+1
+1
+1
p
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
q
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
r
-1
+1
+1
-1
+1
-1
-1
+1
-1
+1
+1
-1
+1
-1
-1
+1
Impact of the Randomization Restrictions
• Two Sources of randomization  Two sources of error
– Between plot error: ew (WP error)
– Within plot error:  s (SP error)
• Model:
y  X  eW   S
• The WP and SP error terms have mutually independent normal
distributions with standard deviations σw and σs
The Design
• Situation:
– Have k factors: k1 WP factors and k2 SP factors
s
– Wish to explore impact in 2k-p trials
– Have a 2 k1-p1 fractional factorial for the WP factors
– Require p=p1+p2 generators
– Called a 2(k1+ p2)-(k1+ p2) FFSP design
Running the Design
• Randomly select a level setting of the WP factors and fix their levels
 s run experimental trials varying the level
• With the WP levels fixed,
settings of the SP factors in random order
• Can view WP as a completely randomized design
• Can view SP as a randomized block design with the blocks defined by
the WP factors
Constructing the Design
• For a 2(k1+ p2)-(k1+ p2) FFSP design, have generators for WP and SP
designs
• Rules:
– WP generators (e.g., I=ABC ) contain ONLY WP factors
– SP generators (e.g., I=Apqr ) must contain AT LEAST 2 SP factors
• Previous design: I=ABC=Apqr=BCpqr
Analysis of FFSP Designs
• Two Sources of randomization  Two sources of error
– Between plot error: σw (WP error).
– Within plot error: σs (SP error).
• WP Effects compared to: aσs2 + bσs2
• SP effects compared to : bσs2
• df for SP df for WP.
• Get more power for SP effects!!!
WP Effect or SP Effect?
• Effects aliased with WP main effects or interactions involving only WP
factors tested as a WP effect.
• E.g., pq=ABCD tested as a WP effect.
• Effects aliased only with SP main effects or interactions involving at
least one SP factors tested as a SP effect .
• E.g., pq=ABr tested as a SP effect.
Ranking the Designs
• Use minimum aberration (MA) criterion
Example
• Experiment is performed to study the geometric distortion of gear
drives
• The response is “dishing” of the gear
• 5 factors thought to impact response:
–
–
–
–
–
A: Furnace track
B: Tooth size
C: Part position
p: Carbon potential
q: Operating Mode
Example
• Because of the time taken to change the levels of some of the factors, it
is more efficient to run experiment trials keeping factors A-C fixed and
varying the levels of p and q
• A 2(3+2)-(0+1) FFSP design was run ( I=ABCpq )
Example
A
-1
-1
-1
-1
+1
+1
+1
+1
B
-1
-1
+1
+1
-1
-1
+1
+1
C
-1
+1
-1
+1
-1
+1
-1
+1
p
-1
q=ABCp
+1
Y
18.0
+1
-1
21.5
-1
-1
13.0
+1
+1
-4.5
-1
-1
22.5
+1
+1
15.0
-1
+1
0.5
+1
-1
5.5
-1
-1
27.5
+1
+1
17.0
-1
+1
17.5
+1
-1
14.5
-1
+1
19.0
+1
-1
22.0
-1
-1
24.0
+1
+1
13.5
Example
• This is a 16-run design…have 15 effects to estimate
• Which effects are test as WP effects? SP effects?
• I=ABCpq
• Have a 23 design for the WP effects: A,B,C,AB,AC,BC,ABC=pq are
tested as WP effects
• SP effects: p,q,Ap,Aq,Bp,Bq,Cp,Cq
• Need separate qq-plots for each set of effects
S
P
e
f
c
t
s
-3 -2 -1 0 1
-10 -5 WPefcts 0 5
QQ-Plots
- 1.0
00
.5
.0
0 .5
1 .0
Q u a n tile s
- 1.5
1.0
00
.5
.0
0 .5
1 .0
1 .5
o f
Q
u
aa
nr
tile
s
St
a n
d
d
N