Replicated Latin Squares Three types of replication in traditional (1 treatment, 2 blocks) latin squares Three types of replication in traditional (1 treatment, 2 blocks) latin squares –Case study (s=square, n=# of trt levels) Crossover designs 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 Column=Operator, Row=Batch Case 1: Same Operator, Same Batch Sourcedf Case 1: Same Operator, Same Batch Sourcedf Treatmentn-1 Batchn-1 Operatorn-1 Reps-1 ErrorBy subtraction Totalsn 2 -1

Replicated Latin Squares Case 2: Different Operator, Same Batch Case 2: Different Operator, Same Batch Sourcedf Treatmentn-1 Batchn-1 Operatorsn-1 O(S)s(n-1) Squares-1 ErrorBy subtraction Totalsn 2 -1

Replicated Latin Squares Case 3: Different Operator, Different Batch Case 3: Different Operator, Different Batch Sourcedf Treatmentn-1 Batchsn-1 Operatorsn-1 ErrorBy subtraction Totalsn 2 -1

Replicated Latin Squares Case 3: Different Operator, Different Batch Case 3: Different Operator, Different Batch Montgomery’s approach Montgomery’s approach Sourcedf Treatmentn-1 Batch(Square)s(n-1) Operator(Square)s(n-1) Squares-1 ErrorBy subtraction Totalsn 2 -1

Crossover Design Two blocking factors: subject and period Two blocking factors: subject and period Used in clinical trials Used in clinical trialsSubject Period 1AABABB Period 2BBABAA

Crossover Design Rearrange as a replicated Latin Square Rearrange as a replicated Latin SquareSubject Period 1ABABAB Period 2BABABA

Crossover Designs Yandell uses a different approach, in which Yandell uses a different approach, in which –Sequence is a factor (basically the WP factor) –Subjects are nested in sequence Period 1ABCCAB Period 2BCABCA Period 3CABABC

Crossover Designs Yandell uses a different approach, in which 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 The replicated Latin Square is an artifice, but helps to organize our thoughts We will assume s Latin Squares with sn subjects 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 If you don’t have sn subjects, use as much of the last Latin Square as possible

Crossover Designs Example (n=4,s=2) Example (n=4,s=2) Period 1ABCDABCD Period 2BCDABCDA Period 3CDABCDAB Period 4DABCDABC

Crossover Designs This is similar to Case 2 This is similar to Case 2 The period x treatment interaction could be separated out as a separate test 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 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 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 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 In this example, we observe the sequence AB, but never observe BA Period 1ABCDABCD Period 2BCDABCDA Period 3CDABCDAB Period 4DABCDABC

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 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 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 Each pair is observed 1 time ABCD BCDA DABC CDAB

Carry-over in Crossover Designs For n odd, we will need a replicated design For n odd, we will need a replicated design ABCABC BCACAB CABBCA

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) These designs are not orthogonal since each treatment cannot follow itself. We analyze using Type III SS (i indexes period, j indexes treatment)

Carry-over in Crossover Designs Example: Example: ABCD BCDA DABC CDAB

Example First Two Rows

Example Next Two Rows

Carry-over in Crossover Designs The parameter  o is the effect of being in the first row--it is confounded with the period 1 effect and will not be estimated The parameter  o 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 Each of these factors loses a df as a result

Carry-over in Crossover Designs SourceUsual dfType III df Treatmentn-1n-1 Periodn-1n-2 Subjectsn-1sn-1 Res Trtnn-1 ErrorBy subtraction Totalsn 2 -1sn 2 -1