On the Analysis of Crossover Designs

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On the Analysis of Crossover Designs Dallas E.
Johnson Professor Emeritus Kansas State University
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dejohnsn_at_ksu.edu 785-532-0510 (Office) 785-539-013
7 (Home) Dallas E. Johnson 1812 Denholm
Dr. Manhattan, KS 66503-2210
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Note that
and
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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).
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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).
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PROC MIXED TITLE2 'EQUAL VARIANCES' CLASSES
SEQ PERIOD TRT PERSON MODEL PEFSEQ TRT
PERIOD/DDFMSATTERTH REPEATED
TRT/SUBJECTPERSON(SEQ) TYPECS LSMEANS TRT
/PDIFF RUN PROC MIXED TITLE2 'UNEQUAL
VARIANCES' CLASSES SEQ PERIOD TRT PERSON
MODEL PEFSEQ TRT PERIOD/DDFMSATTERTH REPEATED
TRT/SUBJECTPERSON(SEQ) TYPECSH LSMEANS TRT
/PDIFF RUN
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Tests for equal treatment effects.
N 6, ?.5, ?B2 Analysis Assumptions Analysis Assumptions Analysis Assumptions Analysis Assumptions
Simulation C0V0 C0V1 C1V0 C1V1
C0V0 ?.040 ?(1) .87 ?.040 ?(1) .87 ? .050 ?(1) .38 ?.050 ?(1).38
C0V1 ?.045 ?(1) .43 ?.045 ?(1) .43 ? .050 ?(1) .18 ?.046 ?(1).17
C1V0 ?.124 ?(1) .66 ?.124 ?(1) .66 ? .050 ?(1) .38 ?.050 ?(1).38
C1V1 ?.066 ?(1) .26 ?.066 ?(1) .26 ? .050 ?(1) .18 ?.046 ?(1).17
N 12, ?.5, ?B2
C0V0 ?.048 ?(1) 1.0 ?.048 ?(1) 1.0 ? .055 ?(1) .68 ?.055 ?(1).66
C0V1 ?.055 ?(1) .79 ?.055 ?(1) .80 ? .055 ?(1) .32 ?.054 ?(1).31
C1V0 ?.214 ?(1) .95 ?.214 ?(1) .95 ? .055 ?(1) .67 ?.055 ?(1).66
C1V1 ?.102 ?(1) .54 ?.102 ?(1) .54 ? .055 ?(1) .32 ?.054 ?(1).31
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Tests for equal treatment effects.
N 18, ?.5, ?B2 Analysis Assumptions Analysis Assumptions Analysis Assumptions Analysis Assumptions
Simulation C0V0 C0V1 C1V0 C1V1
C0V0 ?.046 ?(1) 1.0 ?.046 ?(1) 1.0 ? .045 ?(1) .83 ?.045 ?(1).83
C0V1 ?.040 ?(1) .92 ?.040 ?(1) .92 ? .034 ?(1) .47 ?.034 ?(1).46
C1V0 ?.297 ?(1) .99 ?.297 ?(1) .99 ? .045 ?(1) .83 ?.045 ?(1).83
C1V1 ?.117 ?(1) .69 ?.117 ?(1) .69 ? .034 ?(1) .47 ?.034 ?(1).46
N 30, ?.5, ?B2
C0V0 ?.051 ?(1) 1.0 ?.051 ?(1) 1.0 ? .061 ?(1) .96 ?.061 ?(1).96
C0V1 ?.055 ?(1) .99 ?.055 ?(1) .99 ? .057 ?(1) .67 ?.055 ?(1).67
C1V0 ?.507 ?(1) 1.0 ?.508 ?(1) 1.0 ? .061 ?(1) .96 ?.061 ?(1).96
C1V1 ?.230 ?(1) .92 ?.230 ?(1) .92 ? .054 ?(1) .67 ?.054 ?(1).67
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NOTE Failing to assume carryover when
carryover exists invalidates the tests for equal
treatment effects and the invalidation
generally gets worse as the
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TITLE1 'A THREE PERIOD/THREE TRT DESIGN' TITLE2
'ANALYSIS ASSUMES NO CARRY-OVER' PROC MIXED
TITLE3 'ANALYSIS USING SAS-MIXED' CLASSES SEQ
PER TRT SUBJ MODEL YSEQ TRT PER/DDFMSATTERTH
RANDOM SUBJ(SEQ) LSMEANS TRT PER/PDIFF RUN
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Analyses assuming carryover.
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PROC GLM CLASSES SEQ PER TRT PRIORTRT SUBJ
MODEL Y SEQ SUBJ(SEQ) TRT PER PRIORTRT/E E3
LSMEANS TRT PER PRIORTRT/PDIFF STDERR E
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OPTIONS NODATE PAGENO1 TITLE1 'CRSOVR EXAMPLE
3 - A THREE PERIOD/THREE TRT DESIGN' TITLE2
'ANALYSIS ASSUMES NO CARRY-OVER' DATA ONE INPUT
SEQ PER TRT PRIORTRT _at__at_ DO N1 TO 6 INPUT
Y _at__at_ 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
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PROC GLM CLASSES SEQ PER TRT PRIORTRT SUBJ
MODEL Y SEQ SUBJ(SEQ) TRT PER PRIORTRT/E E3
LSMEANS TRT PER PRIORTRT/TDIFF PDIFF STDERR
E RUN
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Incorrect
Source DF Type III SS Mean Square F Value Pr gt F
SEQ 5 53.6725833 10.7345167 10.70 lt.0001
SUBJ(SEQ) 30 307.7788889 10.2592963 10.23 lt.0001
TRT 2 249.7263611 124.8631806 124.51 lt.0001
PER 1 15.1250000 15.1250000 15.08 0.0002
PRIORTRT 2 4.4492130 2.2246065 2.22 0.1168
Why 1?
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TRT Y LSMEAN LSMEAN Number
A Non-est 1
B Non-est 2
C Non-est 3
Least Squares Means for Effect TRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect TRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect TRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect TRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y
i/j 1 2 3
1 3.139398lt.0001 -11.8234lt.0001
2 -3.1394lt.0001 -14.9628lt.0001
3 11.8234lt.0001 14.9628lt.0001
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PRIORTRT Y LSMEAN LSMEAN Number
A Non-est 1
B Non-est 2
C Non-est 3
O Non-est 4
Least Squares Means for Effect PRIORTRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect PRIORTRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect PRIORTRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect PRIORTRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y Least Squares Means for Effect PRIORTRTt for H0 LSMean(i)LSMean(j) / Pr gt tDependent Variable Y
i/j 1 2 3 4
1 -0.77084lt.0001 1.312187lt.0001 ..
2 0.770836lt.0001 2.083024lt.0001 ..
3 -1.31219lt.0001 -2.08302lt.0001 ..
4 .. .. ..
W R O N G
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PROC MIXED CLASSES SEQ PER TRT PRIORTRT SUBJ
MODEL Y SEQ SUBJ(SEQ) TRT PER PRIORTRT/
DDFMSATTERTH LSMEANS TRT PER PRIORTRT/PDIFF
STDERR RUN
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ESTIMATE 'A LSM DFN 2' INTERCEPT 9 PER 3 3 3 TRT
9 0 0 PRIORTRT 2 2 2 3/DIVISOR9
ESTIMATE 'B LSM DFN 2' INTERCEPT 9 PER 3 3 3 TRT
0 9 0 PRIORTRT 2 2 2 3/DIVISOR9
ESTIMATE 'C LSM DFN 2' INTERCEPT 9 PER 3 3 3 TRT
0 0 9 PRIORTRT 2 2 2 3/DIVISOR9
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Covariance Parameter Estimates Covariance Parameter Estimates
Cov Parm Estimate
SUBJ(SEQ) 3.0855
Residual 1.0028
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Type 3 Tests of Fixed Effects Type 3 Tests of Fixed Effects Type 3 Tests of Fixed Effects Type 3 Tests of Fixed Effects Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr gt F
SEQ 5 30.6 1.05 0.4072
TRT 2 66 124.51 lt.0001
PER 1 66 15.08 0.0002
PRIORTRT 2 66 2.22 0.1168
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Estimates Estimates Estimates Estimates Estimates Estimates
Label Estimate Standard Error DF t Value Pr gt t
A LSM DFN 2 23.6769 0.3438 45.2 68.87 lt.0001
B LSM DFN 2 22.8484 0.3438 45.2 66.46 lt.0001
C LSM DFN 2 26.7970 0.3438 45.2 77.94 lt.0001
A LSM DFN 3 22.3750 0.3698 57.2 60.50 lt.0001
B LSM DFN 3 21.5465 0.3698 57.2 58.26 lt.0001
C LSM DFN 3 25.4951 0.3698 57.2 68.94 lt.0001
A-B 0.8285 0.2639 66 3.14 0.0025
A-C -3.1201 0.2639 66 -11.82 lt.0001
B-C -3.9486 0.2639 66 -14.96 lt.0001
PER 1 LSM 23.1389 0.3370 42.1 68.66 lt.0001
PER 2 LSM 24.6333 0.3370 42.1 73.10 lt.0001
PER 3 LSM 25.5500 0.3370 42.1 75.82 lt.0001
C_A 25.1556 0.3761 60 66.89 lt.0001
C_B 25.4285 0.3761 60 67.62 lt.0001
C_C 24.6910 0.3761 60 65.66 lt.0001
C_A - C_B -0.2729 0.3541 66 -0.77 0.4436
C_A - C_C 0.4646 0.3541 66 1.31 0.1940
C_B - C_C 0.7375 0.3541 66 2.08 0.0411
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Estimates Estimates Estimates Estimates Estimates Estimates
Label Estimate Standard Error DF t Value Pr gt t
A LSM DFN 2 23.6769 0.3438 45.2 68.87 lt.0001
B LSM DFN 2 22.8484 0.3438 45.2 66.46 lt.0001
C LSM DFN 2 26.7970 0.3438 45.2 77.94 lt.0001
A LSM DFN 3 22.3750 0.3698 57.2 60.50 lt.0001
B LSM DFN 3 21.5465 0.3698 57.2 58.26 lt.0001
C LSM DFN 3 25.4951 0.3698 57.2 68.94 lt.0001
A-B 0.8285 0.2639 66 3.14 0.0025
A-C -3.1201 0.2639 66 -11.82 lt.0001
B-C -3.9486 0.2639 66 -14.96 lt.0001
PER 1 LSM 23.1389 0.3370 42.1 68.66 lt.0001
PER 2 LSM 24.6333 0.3370 42.1 73.10 lt.0001
PER 3 LSM 25.5500 0.3370 42.1 75.82 lt.0001
C_A 25.1556 0.3761 60 66.89 lt.0001
C_B 25.4285 0.3761 60 67.62 lt.0001
C_C 24.6910 0.3761 60 65.66 lt.0001
C_A - C_B -0.2729 0.3541 66 -0.77 0.4436
C_A - C_C 0.4646 0.3541 66 1.31 0.1940
C_B - C_C 0.7375 0.3541 66 2.08 0.0411
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Contrasts Contrasts Contrasts Contrasts Contrasts
Label Num DF Den DF F Value Pr gt F
TRT 2 66 124.51 lt.0001
PERIOD 2 66 53.17 lt.0001
CARRYOVER 2 66 2.22 0.1168
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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.
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(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
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(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.
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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) Wh
en 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.
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Goad and Johnson (2000) provide some alternative
analyses for crossover experiments.
Consider again, the three period/three treatment
crossover design in six sequences.
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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
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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).
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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).
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TITLE1 'CRSOVR EXAMPLE - A THREE PERIOD/THREE TRT
DESIGN' TITLE2 'ASSUMES CARRYOVER AND UNEQUAL
VARIANCES' PROC MIXED CLASSES SEQ PER TRT
PRIORTRT SUBJ MODEL Y SEQ TRT PER
PRIORTRT/DDFMKR LSMEANS TRT PER
PRIORTRT/PDIFF REPEATED TRT/SUBJECTSUBJ
TYPECSH ESTIMATE 'A LSM DFN 2' INTERCEPT 9 PER
3 3 3 TRT 9 0 0 PRIORTRT 2 2 2
3/DIVISOR9 ESTIMATE 'B LSM DFN 2' INTERCEPT 9
PER 3 3 3 TRT 0 9 0 PRIORTRT 2 2 2
3/DIVISOR9 ESTIMATE 'C LSM DFN 2' INTERCEPT 9
PER 3 3 3 TRT 0 0 9 PRIORTRT 2 2 2
3/DIVISOR9
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Tests for equal treatment effects.
N 6 ?.5 ?B2 ?C4 Analysis Assumptions Analysis Assumptions Analysis Assumptions Analysis Assumptions
Simulation C0V0 C0V1 C1V0 C1V1
C0V0 ?.053 ?(1) 1.0 ?.057 ?(1) 1.0 ? .057 ?(1) 1.0 ?.051 ?(1)1.0
C0V1 ?.066 ?(1) .50 ?.057 ?(1) .88 ? .049 ?(1) .42 ?.049 ?(1).67
C1V0 ?.138 ?(1) 1.0 ?.149 ?(1) 1.0 ? .057 ?(1) 1.0 ?.051 ?(1)1.0
C1V1 ?.070 ?(1) .32 ?.069 ?(1) .73 ? .049 ?(1) .42 ?.049 ?(1).67
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Tests for equal treatment effects.
N 12 ?.5 ?B2 ?C4 Analysis Assumptions Analysis Assumptions Analysis Assumptions Analysis Assumptions
Simulation C0V0 C0V1 C1V0 C1V1
C0V0 ?.049 ?(1) 1.0 ?.052 ?(1) 1.0 ? .054 ?(1) 1.0 ?.055 ?(1)1.0
C0V1 ?.070 ?(1) .89 ?.053 ?(1) .99 ? .055 ?(1) .77 ?.046 ?(1).94
C1V0 ?.227 ?(1) 1.0 ?.232 ?(1) 1.0 ? .054 ?(1) 1.0 ?.055 ?(1)1.0
C1V1 ?.081 ?(1) .70 ?.100 ?(1) .97 ? .055 ?(1) .77 ?.046 ?(1).94
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Tests for equal treatment effects.
N 18 ?.5 ?B2 ?C4 Analysis Assumptions Analysis Assumptions Analysis Assumptions Analysis Assumptions
Simulation C0V0 C0V1 C1V0 C1V1
C0V0 ?.054 ?(1) 1.0 ?.056 ?(1) 1.0 ? .048 ?(1) 1.0 ?.053 ?(1)1.0
C0V1 ?.071 ?(1) .99 ?.051 ?(1) 1.0 ? .054 ?(1) .91 ?.051 ?(1).99
C1V0 ?.370 ?(1) 1.0 ?.378 ?(1) 1.0 ? .048 ?(1) 1.0 ?.053 ?(1)1.0
C1V1 ?.094 ?(1) .90 ?.125 ?(1) 1.0 ? .054 ?(1) .91 ?.051 ?(1).99
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Tests for Carryover
Simulation C1V0 C1V1
N 6 ?A1 ?B1 ?C1 ? .043 ?(.5) .52 ?(1) .99 ?.040 ?(.5).53 ?(1) .99
N 6 ?A1 ?B.5 ?C.25 ? .054 ?(.5) .86 ?(1) 1.0 ?.044 ?(.5).99 ?(1) 1.0
N 6 ?A1 ?B2 ?C4 ? .048 ?(.5) .10 ?(1) .25 ?.044 ?(.5).13 ?(1) .35
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Tests for Carryover
Simulation C1V0 C1V1
N 12 ?A1 ?B1 ?C1 ? .040 ?(.5) .85 ?(1) 1.0 ?.042 ?(.5).85 ?(1) 1.0
N 12 ?A1 ?B.5 ?C.25 ? .047 ?(.5) 1.0 ?(1) 1.0 ?.046 ?(.5)1.0 ?(1) 1.0
N 12 ?A1 ?B2 ?C4 ? .064 ?(.5) .15 ?(1) .45 ?.056 ?(.5).20 ?(1) .62
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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.