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.

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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.

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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?

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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.

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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

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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

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Therefore, what restrictions are needed for an

experiment to be able to use a Latin square design?

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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

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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.

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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.

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Advantages of a Latin square design:

•

•

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Disadvantages of a Latin square design:

•

•

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More disadvantages of a Latin square design:

•

•

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More disadvantages of a Latin square design:

•

•

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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.

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Why is there no k subscript on Yij?

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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.

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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

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With no replication, df Error is quite small. For this

design to be effective, we need SS(Rows) and SS(Columns)

to be large.

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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].

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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

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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

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Extensions

The Latin square design can be extended to include:

• replicates within square

• subsampling within square

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• 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

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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

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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].

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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.

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...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.

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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!

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For replicated Latin squares,

• randomization is done separately for each square

• we have row(square) and column(square) effects

(nesting within square).

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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

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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].

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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.

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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

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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].

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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.

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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

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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].

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How do we get from the Latin rectangle Model 2

ANOVA table to the Latin rectangle Model 1 ANOVA

table?

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