g. latin square designs latin square designs are special ...lynn/ph7406/notes/latinsquares.pdf · a...
Post on 05-Jun-2018
221 Views
Preview:
TRANSCRIPT
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
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
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
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
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
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
Therefore, what restrictions are needed for an
experiment to be able to use a Latin square design?
506
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
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
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
Advantages of a Latin square design:
•
•
510
Disadvantages of a Latin square design:
•
•
511
More disadvantages of a Latin square design:
•
•
512
More disadvantages of a Latin square design:
•
•
513
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
Why is there no k subscript on Yij?
515
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
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
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
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
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
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
Extensions
The Latin square design can be extended to include:
• replicates within square
• subsampling within square
522
• 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
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
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
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
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
...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
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
For replicated Latin squares,
• randomization is done separately for each square
• we have row(square) and column(square) effects
(nesting within square).
530
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
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
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
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
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
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
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
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
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
How do we get from the Latin rectangle Model 2
ANOVA table to the Latin rectangle Model 1 ANOVA
table?
540
top related