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On the Analysis of Crossover Designs Dallas E. Johnson Professor Emeritus Kansas State University. dejohnsn@ksu.edu 785-532-0510 (Office) 785-539-0137 (Home) Dallas E. Johnson 1812 Denholm Dr. Manhattan, KS 66503-2210. Note that . and.

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slide1

On the Analysis of Crossover Designs

Dallas E. Johnson

Professor Emeritus

Kansas State University

slide2

dejohnsn@ksu.edu

785-532-0510 (Office)

785-539-0137 (Home)

Dallas E. Johnson

1812 Denholm Dr.

Manhattan, KS 66503-2210

slide27

To answer these kinds of questions, Shanga simulated two period/two treatment crossover experiments satisfying four different conditions:

(1) no carryover and equal variances (C0V0),

(2) no carryover and unequal variances(C0V1),

(3) carryover and equal variances (C1V0), and

(4) carryover and unequal variances (C1V1).

slide28

Each of 1000 sets of data under each of these conditions was analyzed four different ways assuming:

(1) no carryover and equal variances (C0V0),

(2) no carryover and unequal variances(C0V1),

(3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

slide29

PROCMIXED;

TITLE2'EQUAL VARIANCES';

CLASSES SEQ PERIOD TRT PERSON;

MODEL PEF=SEQ TRT PERIOD/DDFM=SATTERTH;

REPEATED TRT/SUBJECT=PERSON(SEQ) TYPE=CS;

LSMEANS TRT /PDIFF;

RUN;

PROCMIXED;

TITLE2'UNEQUAL VARIANCES';

CLASSES SEQ PERIOD TRT PERSON;

MODEL PEF=SEQ TRT PERIOD/DDFM=SATTERTH;

REPEATED TRT/SUBJECT=PERSON(SEQ) TYPE=CSH;

LSMEANS TRT /PDIFF;

RUN;

slide32

NOTE: Failing to assume carryover

when carryover exists invalidates

the tests for equal treatment

effects and the invalidation

generally gets worse as the

slide41

TITLE1'A THREE PERIOD/THREE TRT DESIGN';

TITLE2'ANALYSIS ASSUMES NO CARRY-OVER';

PROCMIXED;

TITLE3'ANALYSIS USING SAS-MIXED';

CLASSES SEQ PER TRT SUBJ;

MODEL Y=SEQ TRT PER/DDFM=SATTERTH;

RANDOM SUBJ(SEQ);

LSMEANS TRT PER/PDIFF;

RUN;

slide55

13

-2

1

-2

1

13

4

4

10

-2

10

-2

4

13

4

13

-5

-5

slide57

PROCGLM;

CLASSES SEQ PER TRT PRIORTRT SUBJ;

MODEL Y = SEQ SUBJ(SEQ) TRT PER PRIORTRT/EE3;

LSMEANS TRT PER PRIORTRT/PDIFFSTDERR E;

slide58

OPTIONS NODATE PAGENO=1;

TITLE1'CRSOVR EXAMPLE #3 - A THREE PERIOD/THREE TRT DESIGN';

TITLE2'ANALYSIS ASSUMES NO CARRY-OVER';

DATA ONE;

INPUT SEQ PER TRT $ PRIORTRT $ @@;

DO N=1TO6;

INPUT Y @@; OUTPUT; END;

CARDS;

1 1 A O 20.1 23.3 23.4 19.7 19.2 22.2

1 2 B A 20.3 24.8 24.8 21.3 20.9 22.0

1 3 C B 25.6 28.7 28.3 25.7 25.9 26.2

2 1 A O 24.7 23.8 23.6 20.2 19.8 21.5

2 2 C A 29.4 28.7 26.4 26.2 23.7 25.5

2 3 B C 27.5 24.1 25.0 21.4 23.3 20.8

slide59

PROCGLM;

CLASSES SEQ PER TRT PRIORTRT SUBJ;

MODEL Y = SEQ SUBJ(SEQ) TRT PER PRIORTRT/EE3;

LSMEANS TRT PER PRIORTRT/TDIFFPDIFFSTDERR E;

RUN;

slide60

Incorrect

Why 1?

slide62

W

R

O

N

G

slide63

PROCMIXED;

CLASSES SEQ PER TRT PRIORTRT SUBJ;

MODEL Y = SEQ SUBJ(SEQ) TRT PER PRIORTRT/

DDFM=SATTERTH;

LSMEANS TRT PER PRIORTRT/PDIFFSTDERR;

RUN;

slide64

ESTIMATE'A LSM DFN 2' INTERCEPT 9 PER 333 TRT 900

PRIORTRT 2223/DIVISOR=9;

ESTIMATE'B LSM DFN 2' INTERCEPT 9 PER 333 TRT 090

PRIORTRT 2223/DIVISOR=9;

ESTIMATE'C LSM DFN 2' INTERCEPT 9 PER 333 TRT 009

PRIORTRT 2223/DIVISOR=9;

slide74

Goad and Johnson (2000) showed:

(1) If  satisfies the H-F conditions, then the

traditional tests for treatment and period

effects are valid for all crossover experiments

both with and without carryover.

slide75

(2) There are cases where the ANOVA tests are valid even

when  does not satisfy the H-F conditions.

(a) In the no carryover case, tests for equal treatment

effects are valid for the six sequence three

period/three treatment crossover design when

there are an equal number of subjects assigned

to each sequence.

(b) In the no carryover case, tests for equal

period effects are valid only when the H-F conditions be satisfied

slide76

(b) The traditional tests for equal treatment effects and equal period effects are valid for a crossover design generated by t-1 mutually orthogonal tt Latin squares when there are equal numbers of subjects assigned to each sequence.

(c) The traditional tests for equal treatment effects, equal period effects, and equal carryover effects are likely to be invalid in the four period/four treatment design regardless of whether carryover exists or not.

slide77

Cases where the validity of ANOVA tests are still in doubt.

(4) When carryover exists, the tests for equal carryover effects are not valid unless E satisfies the H-F conditions.

(5) When there are unequal numbers of subjects assigned to each sequence, the ANOVA tests are unlikely to be valid unless E satisfies the H-F conditions.

slide78

Goad and Johnson (2000) provide some alternative analyses for crossover experiments.

Consider again, the three period/three treatment crossover design in six sequences.

slide79

Question: Suppose the variance of a response depends on the treatment, but that the correlation is the same between all pairs of sequence cells. That is, for Sequence 1, the covariance matrix is:

slide80

Shanga simulated three period/three treatment crossover experiments satisfying four different conditions:

(1) no carryover and equal variances (C0V0),

(2) no carryover and unequal variances(C0V1),

(3) carryover and equal variances (C1V0), and

(4) carryover and unequal variances (C1V1).

slide81

Each of 1000 sets of data under each of these conditions was analyzed four different ways assuming:

(1) no carryover and equal variances (C0V0),

(2) no carryover and unequal variances(C0V1),

(3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

slide82

TITLE1'CRSOVR EXAMPLE - A THREE PERIOD/THREE TRT DESIGN';

TITLE2'ASSUMES CARRYOVER AND UNEQUAL VARIANCES';

PROCMIXED;

CLASSES SEQ PER TRT PRIORTRT SUBJ;

MODEL Y = SEQ TRT PER PRIORTRT/DDFM=KR;

LSMEANS TRT PER PRIORTRT/PDIFF;

REPEATED TRT/SUBJECT=SUBJ TYPE=CSH;

ESTIMATE'A LSM DFN 2' INTERCEPT 9 PER 333 TRT 900

PRIORTRT 2223/DIVISOR=9;

ESTIMATE'B LSM DFN 2' INTERCEPT 9 PER 333 TRT 090

PRIORTRT 2223/DIVISOR=9;

ESTIMATE'C LSM DFN 2' INTERCEPT 9 PER 333 TRT 009

PRIORTRT 2223/DIVISOR=9;

slide88

In the three treatment/three period/six sequence crossover design, Shanga also considered testing

Shanga claimed that his tests were LRTs, but Jung (2008) has shown that they are not LRTs. Nevertheless, Shanga's tests had good power for detecting unequal variances.