Orthogonal Latin Square Configuration for LSI Memory Yield and ...
G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A...
-
Upload
trinhtuyen -
Category
Documents
-
view
221 -
download
0
Transcript of G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A...
![Page 1: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/1.jpg)
G. Latin Square Designs
Latin square designs are special block designs with
two blocking factors and only one treatment per block
instead of every treatment per block.
500
![Page 2: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/2.jpg)
CLASSIC AG EXAMPLE: A researcher wants to
determine the optimal seeding rate for a new variety
of wheat: 30, 80, 130, 180, or 230 pounds of seed
per acre.
The experimental plot of land available has an
irrigation source along one edge and a slope
perpendicular to the irrigation flow.
501
![Page 3: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/3.jpg)
irrigation source
A B C D E
B C D E A
C D E A B
D E A B C
E A B C D
———- slope ———->
where the five seeding rates are randomly assigned to
the five letters A, B, C, D, E.
How often does each treatment appear?
502
![Page 4: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/4.jpg)
A Latin square design does not have to correspond to
a physical layout.
EXAMPLE: In a study of a new chemotherapy treat-
ment for breast cancer, researchers wanted to control
for the effects of age and BMI. They believe the
responses of younger patients will be more like each
other than those of older patients, and likewise that
the responses of heavier patients will be more like each
other than those lighter patients.
503
![Page 5: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/5.jpg)
Age (years)[40,50) [50,60) [60,70) 70+
<20 A B C DBMI [20,25) B C D A
[25,30) C D A B30+ D A B C
504
![Page 6: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/6.jpg)
A standard Latin square has the treatment levels (A,
B, etc.) written alphabetically in the first row and
the first column. The remaining cells are filled in by
incrementing the letters by one within each row and
column.
A B C D
B
C
D
505
![Page 7: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/7.jpg)
Therefore, what restrictions are needed for an
experiment to be able to use a Latin square design?
506
![Page 8: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/8.jpg)
Randomization
Randomization is a bit complex because there are
multiple possible Latin squares. For example,
for t = 4,
A B C D
B C D A
C D A B
D A B C
A B C D
B A D C
C D B A
D C A B
507
![Page 9: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/9.jpg)
For t = 3,4,5:
1. Choose a standard Latin square at random.
2. Randomly permute (re-order) all rows but the first.
3. Randomly permute all columns.
4. Randomly assign treatments to the letters A, B,
C, etc.
508
![Page 10: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/10.jpg)
For t ≥ 6:
1. Choose a standard Latin square not at random.
2. Randomly permute all rows.
3. Randomly permute all columns.
4. Randomly assign treatments to the letters A, B,
C, etc.
509
![Page 11: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/11.jpg)
Advantages of a Latin square design:
•
•
510
![Page 12: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/12.jpg)
Disadvantages of a Latin square design:
•
•
511
![Page 13: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/13.jpg)
More disadvantages of a Latin square design:
•
•
512
![Page 14: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/14.jpg)
More disadvantages of a Latin square design:
•
•
513
![Page 15: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/15.jpg)
Model
Yij = µ + ρi + γj + τk + eij
eij ∼iid N(0, σ2e )
i = 1, . . . , t, j = 1, . . . , t, k = 1, . . . , t
with row effect ρi, column effect γj, and treatment
effect τk. We can have any combination of fixed
or random for each of these, adding constraints as
needed for fixed effects and random effects indepen-
dent of each other.
514
![Page 16: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/16.jpg)
Why is there no k subscript on Yij?
515
![Page 17: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/17.jpg)
Deviations:
With only one observation per cell, no interactions are
estimable:
Yij − Y..︸ ︷︷ ︸total
= (Yi. − Y..)︸ ︷︷ ︸row
+(Y.j − Y..)︸ ︷︷ ︸column
+(Yk − Y..)︸ ︷︷ ︸treatment
+ (Yij − Yi. − Y.j − Yk + 2Y..)︸ ︷︷ ︸error
where the error deviation comes from subtraction.
516
![Page 18: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/18.jpg)
ANOVA table:
Source df SS
Rows t − 1 t∑i(Yi. − Y..)2
Columns t − 1 t∑
j(Y.j − Y..)2
Treatment t − 1 t∑
k(Yk − Y..)2
Error (t − 1)(t − 2) by subtraction
Total t2 − 1∑i
∑j(Yij − Y..)2
517
![Page 19: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/19.jpg)
With no replication, df Error is quite small. For this
design to be effective, we need SS(Rows) and SS(Columns)
to be large.
518
![Page 20: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/20.jpg)
Source E[MS] F ∗
Rows
Columns
Treatment σ2e + t
t−1∑
k (τk)2
Error σ2e
Total
Rows, columns, and treatments can be fixed or ran-
dom as needed, which dictate the appropriate E[MS].
519
![Page 21: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/21.jpg)
Was blocking effective?
We can compare the efficiency of the Latin square
design to what we would have seen with a CRD or
with various CBDs:
Efficiency relative to a CRD:
RE =MSRows + MSColumns + (t − 1)MSError
(t + 1)MSError
520
![Page 22: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/22.jpg)
Efficiency relative to a CBD using the row blocks only:
RE =MSColumns + (t − 1)MSError
t MSError
Efficiency relative to a CBD using the column blocks
only:
RE =MSRows + (t − 1)MSError
t MSError
Each of these could be used with the df correction:
(dfError(LS) + 1)(dfError(other) + 3)
(dfError(LS) + 3)(dfError(other) + 1)RE
521
![Page 23: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/23.jpg)
Extensions
The Latin square design can be extended to include:
• replicates within square
• subsampling within square
522
![Page 24: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/24.jpg)
• replicate squares
- with no blocking factor in common across
sqaures
- with one blocking factor in common across squares
- with both blocking factors in common across
squares
523
![Page 25: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/25.jpg)
H. Latin Squares with Subsampling
Subsampling can be done within each cell of a Latin
square.
Yij` = µ + ρi + γj + τk + eij + δij`
eij ∼iid N(0, σ2e )
δij` ∼iid N(0, σ2d)
i = 1, . . . , t, j = 1, . . . , t, k = 1, . . . , t, ` = 1, . . . , n
with any combination of fixed or random, adding
constraints as needed for fixed effects and random
effects independent of each other.524
![Page 26: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/26.jpg)
ANOVA table:
Source df SS
Rows t − 1 tn∑
i(Yi.. − Y...)2
Columns t − 1 tn∑
j(Y.j. − Y...)2
Treatment t − 1 tn∑
k(Yk − Y...)2
Error (t − 1)(t − 2) by subtraction
Subsampling t2(n − 1)∑
i∑
j∑
`(Yij` − Yij·)2
Total nt2 − 1∑
i
∑
j
∑
`(Yij` − Y...)2
525
![Page 27: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/27.jpg)
Source E[MS] F ∗
Rows
Columns
Treatment σ2d + nσ2
e + tnt−1
∑k (τk)
2
Error σ2d + nσ2
e
Subsampling σ2d
Total
Rows, columns, and treatments can be fixed or ran-
dom as needed, which dictate the appropriate E[MS].
526
![Page 28: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/28.jpg)
I. Replicated Latin Squares
Often Latin square designs are replicated in their
entirety to get more error df. Two possibilities are:
...a Latin rectangle:
A B C D A B C D
B C D A B C D A
C D A B C D A B
D A B C D A B C
where the row blocks are identical across the two
squares.
527
![Page 29: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/29.jpg)
...or replicated Latin squares:
A B C D
B C D A
C D A B
D A B C
A B C D
B A D C
C D B A
D C A B
where neither the row blocks nor the column blocks
are identical across the two squares.
528
![Page 30: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/30.jpg)
For a Latin rectangle, randomization could be done:
• separately for each square (thus we have 4 columns
nested within each of 2 squares)
• across all columns at once (thus we have 8 columns).
Your analysis should match the randomization!
529
![Page 31: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/31.jpg)
For replicated Latin squares,
• randomization is done separately for each square
• we have row(square) and column(square) effects
(nesting within square).
530
![Page 32: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/32.jpg)
Replicated Latin Squares Model
Yij` = µ + ρi(`) + γj(`) + τk + κ` + eij`
eij` ∼iid N(0, σ2e )
i = 1, . . . , t, j = 1, . . . , t, k = 1, . . . , t, ` = 1, . . . , s
with any combination of fixed or random for each of
these, adding constraints as needed for fixed effects
and random effects independent of each other. A
square by treatment interaction (τκ)k` could be
considered as well.531
![Page 33: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/33.jpg)
ANOVA table:
Source df SS
Squares s − 1 t2∑
`(Y..` − Y...)2
Rows(Square) s(t − 1) t∑
i∑
`(Yi.` − Y..`)2
Columns(Square) s(t − 1) t∑
j∑
`(Y.j` − Y..`)2
Treatment t − 1 st∑
k(Yk − Y...)2
Error (t − 1)(t − 2) by subtraction
Total st2 − 1∑
i∑
j∑
`(Yij` − Y...)2
532
![Page 34: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/34.jpg)
Source E[MS] F ∗
Square
Rows(Square)
Columns(Square)
Treatment σ2e + st
t−1∑
k (τk)2
Error σ2e
Total
Rows, columns, and treatments can be fixed or ran-
dom as needed, which dictate the appropriate E[MS].
533
![Page 35: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/35.jpg)
Latin Rectangle Model 1
Yij = µ + ρi + γj + τk + eij
eij ∼iid N(0, σ2e )
i = 1, . . . , t, j = 1, . . . , st, k = 1, . . . , t
with any combination of fixed or random for each of
these, adding constraints as needed for fixed effects
and random effects independent of each other.
534
![Page 36: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/36.jpg)
ANOVA table:
Source df SS
Rows t − 1 st∑i(Yi. − Y..)2
Columns st − 1 t∑
j(Y.j − Y..)2
Treatment t − 1 st∑
k(Yk − Y..)2
Error (t − 1)(st − 2) by subtraction
Total st2 − 1∑i
∑j(Yij − Y..)2
535
![Page 37: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/37.jpg)
Source E[MS] F ∗
Rows
Columns
Treatment σ2e + st
t−1∑
k (τk)2
Error σ2e
Total
Rows, columns, and treatments can be fixed or ran-
dom as needed, which dictate the appropriate E[MS].
536
![Page 38: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/38.jpg)
Latin Rectangle Model 2
Yij` = µ + ρi + γj(`) + τk + κ` + eij`
eij` ∼iid N(0, σ2e )
i = 1, . . . , t, j = 1, . . . , t, k = 1, . . . , t, ` = 1, . . . , s
with any combination of fixed or random for each of
these, adding constraints as needed for fixed effects
and random effects independent of each other.
537
![Page 39: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/39.jpg)
ANOVA table:
Source df SS
Squares s − 1 t2∑
`(Y..` − Y...)2
Rows t − 1 st∑
i(Yi.. − Y...)2
Columns(Square) s(t − 1) t∑
j∑
`(Y.j` − Y..`)2
Treatment t − 1 st∑
k(Yk − Y...)2
Error (t − 1)(st − 2) by subtraction
Total st2 − 1∑
i∑
j∑
`(Yij` − Y...)2
538
![Page 40: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/40.jpg)
Source E[MS] F ∗
Rows
Columns
Treatment σ2e + st
t−1∑
k (τk)2
Error σ2e
Total
Rows, columns, and treatments can be fixed or ran-
dom as needed, which dictate the appropriate E[MS].
539
![Page 41: G. Latin Square Designs Latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · A Latin square design does not have to correspond to a physical layout. EXAMPLE:](https://reader031.fdocuments.us/reader031/viewer/2022021511/5b15b54a7f8b9a45448da673/html5/thumbnails/41.jpg)
How do we get from the Latin rectangle Model 2
ANOVA table to the Latin rectangle Model 1 ANOVA
table?
540