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 D BMI [20,25) B C D A [25,30) C D A B 30+ 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:
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510 Disadvantages of a Latin square design:
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511 More disadvantages of a Latin square design:
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512 More disadvantages of a Latin square design:
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513 Model
Yij = µ + ρi + γj + τk + eij iid 2 eij ∼ N(0, σe ) 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¯.. = (Y¯i. − Y¯..) + (Y¯.j − Y¯..) + (Y¯k − Y¯..) | total{z } | row{z } | column{z } |treatment{z } + (Yij − Y¯i. − Y¯.j − Y¯k +2Y¯..) | error{z } where the error deviation comes from subtraction.
516 ANOVA table:
Source df SS
2 Rows t − 1 t P(Y¯i. − Y¯..) i
2 Columns t − 1 t P(Y¯.j − Y¯..) j
2 Treatment t − 1 t P(Y¯k − Y¯..) k
Error (t − 1)(t − 2) by subtraction
2 2 Total t − 1 P P(Yij − Y¯..) i j
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
2 t 2 Treatment σe + t−1 Pk (τk)
2 Error σe
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: MSRows + MSColumns + (t − 1)MSError RE = d (t + 1)MSError
520 Efficiency relative to a CBD using the row blocks only: MSColumns + (t − 1)MSError RE = d tMSError
Efficiency relative to a CBD using the column blocks only: MSRows + (t − 1)MSError RE = d tMSError
Each of these could be used with the df correction: (dfError(LS) + 1)(dfError(other)+3) RE (dfError(LS) + 3)(dfError(other)+1) d
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` iid 2 eij ∼ N(0, σe ) iid 2 δij` ∼ N(0, σd ) 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
2 Rows t − 1 tn P(Y¯i.. − Y¯...) i
2 Columns t − 1 tn P(Y¯.j. − Y¯...) j
2 Treatment t − 1 tn P(Y¯k − Y¯...) k
Error (t − 1)(t − 2) by subtraction
2 2 Subsampling t (n − 1) Pi Pj P`(Yij` − Y¯ij·)
2 2 Total nt − 1 P P P(Yij` − Y¯...) i j ` 525 Source E[MS] F ∗
Rows
Columns
2 2 tn 2 Treatment σd + nσe + t−1 Pk (τk)
2 2 Error σd + nσe
2 Subsampling σd
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` iid 2 eij` ∼ N(0, σe ) 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
2 2 Squares s − 1 t P`(Y¯..` − Y¯...)
2 Rows(Square) s(t − 1) t Pi P`(Y¯i.` − Y¯..`)
2 Columns(Square) s(t − 1) t Pj P`(Y¯.j` − Y¯..`)
2 Treatment t − 1 st Pk(Y¯k − Y¯...)
Error (t − 1)(t − 2) by subtraction
2 2 Total st − 1 Pi Pj P`(Yij` − Y¯...)
532 Source E[MS] F ∗
Square
Rows(Square)
Columns(Square)
2 st 2 Treatment σe + t−1 Pk (τk)
2 Error σe
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 iid 2 eij ∼ N(0, σe ) 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
2 Rows t − 1 st P(Y¯i. − Y¯..) i
2 Columns st − 1 t P(Y¯.j − Y¯..) j
2 Treatment t − 1 st P(Y¯k − Y¯..) k
Error (t − 1)(st − 2) by subtraction
2 2 Total st − 1 P P(Yij − Y¯..) i j
535 Source E[MS] F ∗
Rows
Columns
2 st 2 Treatment σe + t−1 Pk (τk)
2 Error σe
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` iid 2 eij` ∼ N(0, σe ) 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
2 2 Squares s − 1 t P`(Y¯..` − Y¯...)
2 Rows t − 1 st Pi(Y¯i.. − Y¯...)
2 Columns(Square) s(t − 1) t Pj P`(Y¯.j` − Y¯..`)
2 Treatment t − 1 st Pk(Y¯k − Y¯...)
Error (t − 1)(st − 2) by subtraction
2 2 Total st − 1 Pi Pj P`(Yij` − Y¯...)
538 Source E[MS] F ∗
Rows
Columns
2 st 2 Treatment σe + t−1 Pk (τk)
2 Error σe
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