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
CROSSOVER DESIGNS
Two Period - Two Treatment Crossover Design
Period 1
SEQ 1
A
SEQ 2
B
Period 2
B
A
We have n1 subjects randomly assigned to
sequence 1, and n2 subjects randomly
assigned to sequence 2.
Suppose nj subjects are assigned to the
jth sequence of treatments. Let yijk denote
the observed response from the kth subject
in the jth sequence during the ith period;
i=1,2; j=1,2; k = 1,2,...,nj.
Means Model:
yijk = ij + jk + ijk for all i,j,k
Ideal Conditions:
jk ~ iid N(0,2),
ijk ~iid N(0,2), and
all jk’s and ijk’s are independent.
Main Advantage:
Each subject serves as his/her own
control.
Most appropriate for conditions that
reoccur.
Disadvantages:
The designs cannot be used for treatment
comparisons when the condition being
treated is cured during the first
period.
The two period/two treatment crossover
design should not be used when carryover
effects exist unless one can include a
“washout period” prior to administering
a treatment in the second period.
IDEAL TREATMENT STRUCTURE MODEL
Model
Parameters
Period 1
Period 2
SEQ 1
SEQ 2
CARRYOVER MODEL:
Model
Parameters
Period 1
Period 2
SEQ 1
SEQ 2
Model
Parameters
Period 1
Period 2
NOTE:
SEQ 1
SEQ 2
Therefore
,
,
.
If carry-over exists, the difference in
treatment effects can only be obtained from
Period 1 data. Then
.
Usual Analysis
SV
Seq
df
1
MS
F
S12 S12/S22
Sub(Seq) n1+n2-2 S22
Trt
1
S32
S32/S52
Per
1
S42
S42/S52
Error
n1+n2-2 S52
Hyp Tstd
Question: What if the treatments have different
variances?
Let yjk be the vector of responses for the kth
subject in the jth sequence of treatments, and
suppose we assume that
yjk ~ N( j,  )
where
 1 j 
 j   
 2j
 *11  *12   2   *11  2   *12   11  12 
1
1


  
 *
*    2
*
2
*  
1
1

  21  22      21     22   21  22 
2
ij     i  S j     

*
and where and 
are determined by i and j.
Two period/two treatment crossover designs
where the treatments have unequal
variances.
Here we assume that
  A2
 
  A B
 A B 

 B2 
for the AB sequence
and that
  B2
 
  A B
 A B 

 A2 
for the BA sequence.
Note that
y11k  y21k ~ N ( A   B  1   2 ,  A2  2  A B   B2 ),
k  1, 2,..., n1
and
y22 k  y12 k ~ N ( A   B   2  1 ,  A2  2  A B   B2 ),
k  1, 2,..., n2
Shanga (2003) showed that in the two
period/two treatment crossover design
without carryover effects, the estimated
standard errors for testing both between
subject and within subject contrasts of the
ij’s when the treatments have unequal
variances are exactly the same as they are
when the treatments have equal variances.
Thus if one is only interested in the
difference between the two treatments, then
there is no need to worry about unequal
variances in this design.
Other questions:
Are there any consequences to ignoring
carryover and/or unequal variances in the
two period/two treatment crossover design?
In particular - does unequal variances show
up as carryover or does carryover show up
as unequal variances?
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).
Each of 1000 sets of data under each of these
conditions was analyzed four different ways
assuming:
(1)
(2)
(3)
(4)
no carryover and equal variances (C0V0),
no carryover and unequal variances(C0V1),
carryover and equal variances (C1V0), and
carryover and unequal variances (C1V1).
PROC MIXED;
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;
PROC MIXED;
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;
Tests for equal treatment effects.
Analysis Assumptions
N= 6,
=.5,
B=2
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
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
N= 12,
=.5,
B=2
Tests for equal treatment effects.
Analysis Assumptions
N= 18,
=.5,
B=2
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
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
N= 30,
=.5,
B=2
NOTE: Failing to assume carryover
when carryover exists invalidates
the tests for equal treatment
effects and the invalidation
generally gets worse as the
THREE TREATMENT - THREE PERIOD
CROSSOVER DESIGN
Period 1
1 2
A A
SEQUENCE
3 4 5
B B C
6
C
Period 2
B C
A
C
A
B
Period 3
C B
C
A
B
A
Carryover Case:
Table 1. Model parameters for the 3 period/3 treatment
crossover design with carryover.
Sequence
Per
1
2
3
+ 1+ A
+ 1+ B
4
5
+1+B
+1+ C
+2+A+C
1
+ 1+A
2
+2+ B+A
+ 2+C+A
+ 2+A+B
+ 2+C+B
3
+ 3+C+B
+ 3+B+C
+ 3+C+A
+ 3+A+C
+ 3+ B+A
6
+ 1+C
+ 2+B+C
+ 3+A+B
Let
represent the expected
response in Period i of Sequence j.
that
Note
Suppose nj subjects are assigned to the jth
sequence of treatments.
Let yijk denote the observed response from the
kth subject in the jth sequence during the ith
period;
i=1,2,3; j=1,2,...,6; k=1,2,...,nj.
Means Model:
yijk = ij + jk + ijk for all i,j,k
where
represents the expected response
in Period i of Sequence j.
Ideal Conditions:
jk ~ iid N(0,2),
ijk ~iid N(0,2), and
all jk’s and ijk’s are independent.
Remarks:
A crossover experiment is really
a special type of a repeated measures
experiment where the treatment is
changing over time.
We know that traditional ANOVA
analyses of repeated measures
experiments are only valid when the
repeated measures satisfy compound
symmetry and tests for differences
across time points are valid if and only
if the repeated measures satisfy the
Hyuhn-Feldt Conditions.
Question:
What do the above remarks have to do
with the validity of our analyses of
crossover experiments described previously?
The analyses that we have performed up
to this point in time using our ideal
conditions are valid if the vector of
measurements on a subject within each
sequence satisfies compound symmetry. That
is, the variance of the measured response
is the same for each treatment*period
combination and the correlation between
measurements in different periods is the
same for all pairs of periods.
Again, let yjk be the vector of
responses for the kth subject in the jth
sequence of treatments, and suppose we
assume that
yjk ~ N( j,  )
where
 1 j 
 
2j
j   
  
 
  pj 
,
  11  12

 22
21



 

  p1  p 2
  1p 
 2p 
 , and
  

  pp 
ij     i  S j     

*
and where and 
are determined by i and
j.
Remark:
The covariance matrix  is said to
possess compound symmetry if
 = 2[(1- )I +  J]for some 2 and  .
Remark:
The covariance matrix 
satisfy the H-F conditions if
is
said
 = I +  j’ + j ’ for some  and  .
to
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.
(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
(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.
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  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  satisfies the H-F conditions.
Goad and Johnson (2000) provide some
alternative analyses for crossover experiments.
Consider again, the three period/three
treatment crossover design in six sequences.
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:
  A2
Σ1    B A
  
 C A
 A B
 B2
 C B
 A C 
 B C 
 C2 
Shanga simulated three period/three treatment
crossover experiments satisfying four different
conditions:
(1)
(2)
(3)
(4)
no carryover and equal variances (C0V0),
no carryover and unequal variances(C0V1),
carryover and equal variances (C1V0), and
carryover and unequal variances (C1V1).
Each of 1000 sets of data under each of these
conditions was analyzed four different ways
assuming:
(1)
(2)
(3)
(4)
no carryover and equal variances (C0V0),
no carryover and unequal variances(C0V1),
carryover and equal variances (C1V0), and
carryover and unequal variances (C1V1).
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/DDFM=KR;
LSMEANS TRT PER PRIORTRT/PDIFF;
REPEATED TRT/SUBJECT=SUBJ TYPE=CSH;
RUN;
Tests for equal treatment effects.
N= 6
=.5
B=2
C=4
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
Tests for equal treatment effects.
N= 12
=.5
B=2
C=4
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
Tests for equal treatment effects.
N= 18
=.5
B=2
C=4
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
Tests for Carryover
Simulation
C1V0
C1V1
N= 6
A=1
B=1
C=1
= .043
(.5) =.52
(1) =.99
=.040
(.5)=.53
(1) =.99
N= 6
A=1
B=.5
C=.25
= .054
(.5) =.86
(1) =1.0
=.044
(.5)=.99
(1) =1.0
N= 6
A=1
B=2
C=4
= .048
(.5) =.10
(1) =.25
=.044
(.5)=.13
(1) =.35
Tests for Carryover
Simulation
C1V0
C1V1
N= 12
A=1
B=1
C=1
= .040
(.5) =.85
(1) =1.0
=.042
(.5)=.85
(1) =1.0
N= 12
A=1
B=.5
C=.25
= .047
(.5) =1.0
(1) =1.0
=.046
(.5)=1.0
(1) =1.0
N= 12
A=1
B=2
C=4
= .064
(.5) =.15
(1) =.45
=.056
(.5)=.20
(1) =.62
In the three treatment/three period/six sequence
crossover design, Shanga also considered testing
H 0 :  A2   B2   C2
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.
That’s All For Now!