replicated latin squares three types of replication in traditional (1 treatment, 2 blocks) latin...
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Replicated Latin Squares
Three types of replication in traditional (1 treatment, 2 blocks) latin squares– Case study (s=square, n=# of trt levels)
Crossover designs– Subject is one block, Period is another– Yandell introduces crossovers as a
special case of the split plot design
Replicated Latin Squares
Column=Operator, Row=Batch Case 1: Same Operator, Same Batch Source df
Treatment n-1Batch n-1Operator n-1Rep s-1Error By
subtractionTotal sn2-1
Replicated Latin Squares
Case 2: Different Operator, Same Batch
Source dfTreatment n-1Batch n-1Operator sn-1
O(S) s(n-1)Square s-1
Error By subtraction
Total sn2-1
Replicated Latin Squares
Case 3: Different Operator, Different Batch
Source dfTreatment n-1Batch sn-1Operator sn-1Error By
subtractionTotal sn2-1
Replicated Latin Squares
Case 3: Different Operator, Different Batch
Montgomery’s approachSource dfTreatment n-1Batch(Square) s(n-1)Operator(Square) s(n-1)Square s-1Error By
subtractionTotal sn2-1
Crossover Design
Two blocking factors: subject and period
Used in clinical trialsSubject
1 2 3 4 5 6Period 1 A A B A B BPeriod 2 B B A B A A
Crossover Design
Rearrange as a replicated Latin Square
Subject1 3 2 5 4 6
Period 1 A B A B A BPeriod 2 B A B A B A
Crossover Designs
Yandell uses a different approach, in which – Sequence is a factor (basically the WP factor)
– Subjects are nested in sequence
1 2 3 4 5 6Period 1 A B C C A BPeriod 2 B C A B C APeriod 3 C A B A B C
Crossover Designs
Yandell uses a different approach, in which – Period is an effect (I’d call it a common SP)
– Treatment (which depends on period and sequence) is the Latin letter effect (SP factor)
– Carryover is eventually treated the same way we treat it
Crossover Designs
The replicated Latin Square is an artifice, but helps to organize our thoughts
We will assume s Latin Squares with sn subjects
If you don’t have sn subjects, use as much of the last Latin Square as possible
Crossover Designs
Example (n=4,s=2)
1 2 3 4 5 6 7 8Period 1 A B C D A B C DPeriod 2 B C D A B C D APeriod 3 C D A B C D A BPeriod 4 D A B C D A B C
Crossover Designs
This is similar to Case 2 The period x treatment interaction could be separated out as a separate test– Block x treatment interaction
Periods can differ from square to square--this is similar to Case 3
Carry-over in Crossover Designs
Effects in Crossover Designs are confounded with the carry-over (residual effects) of previous treatments
We will assume that the carry-over only persists for the treatment in the period immediately before the present period
Carry-over in Crossover Designs
In this example, we observe the sequence AB, but never observe BA
1 2 3 4 5 6 7 8Period 1 A B C D A B C DPeriod 2 B C D A B C D APeriod 3 C D A B C D A BPeriod 4 D A B C D A B C
Carry-over in Crossover Designs
A crossover design is balanced with respect to carry-over if each treatment follows every other treatment the same number of times
We can balance our example (in a single square) by permuting the third and fourth rows
Carry-over in Crossover Designs
Each pair is observed 1 time
A B C DB C D AD A B CC D A B
Carry-over in Crossover Designs
For n odd, we will need a replicated design
A B C A B CB C A C A BC A B B C A
Carry-over in Crossover Designs
These designs are not orthogonal since each treatment cannot follow itself. We analyze using Type III SS (i indexes period, j indexes treatment)
1,,1
,,1
,,1
,,1
nl
snk
nj
ni
Y ijkllkjiijkl
Carry-over in Crossover Designs
Example:A B C DB C D AD A B CC D A B
ExampleFirst Two Rows
Period Trt Subject Res Trt
Y1110 1 1 1 0
Y1220 1 2 2 0
Y1330 1 3 3 0
Y1440 1 4 4 0
Y2211 2 2 1 1
Y2322 2 3 2 2
Y2433 2 4 3 3
Y2144 2 1 4 4
ExampleNext Two Rows
Period Trt Subject Res Trt
Y3412 3 4 1 2
Y3123 3 1 2 3
Y3234 3 2 3 4
Y3341 3 3 4 1
Y4314 4 3 1 4
Y4421 4 4 2 1
Y4132 4 1 3 2
Y4243 4 2 4 3
Carry-over in Crossover Designs
The parameter go is the effect of being in the first row--it is confounded with the period 1 effect and will not be estimated
Each of these factors loses a df as a result
Carry-over in Crossover Designs
Source Usual df Type III df
Treatment n-1 n-1Period n-1 n-2Subject sn-1 sn-1Res Trt n n-1Error By
subtractionTotal sn2-1 sn2-1