Analysis ofAnalysis of
2 x 2 Crossover Designs2 x 2 Crossover Designswith Continuous Datawith Continuous Data
OrawanOrawan sAETANsAETANBiostatistician
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OverviewOverview Crossover designs Common Crossover design Possible effects Dealing with aliasing
- Methodology- Statistical Analysis
Example
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Crossover DesignsCrossover DesignsCrossover Designs“Each treatment is administered to each
patient at different times in the study”
Chronic & stable disease (asthma, arthritis, diabetes, hypertension, migraine…)
Chronic & stable disease Chronic & stable disease (asthma, arthritis, diabetes, (asthma, arthritis, diabetes, hypertension, migrainehypertension, migraine……))
subjects may undergo an active drug for 6 weeks and then “cross over” to the
placebo for 6 weeks
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Common Crossover DesignCommon Crossover Design
Figure 1: 2 x 2 crossover design
Sequence 1
Sequen
ce 2
Run-in(Baseline)Run-in
(Baseline)Washout(Baseline)Washout(Baseline)
Treatment A
Treatment A
Treatment B
Treatment B
Treatment A
Treatment A
Treatment B
Treatment B
Period 1 Period 2
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A comparative study of heat effect between hot pack and Thai herbal ball on pain and physiological changes
Run-in(Baseline)Run-in
(Baseline)Washout
(1 wk)Washout
(1 wk)
Hot packHot pack
Thai herbal ball
Thai herbal ball
Hot packHot pack
Thai herbal ball
Thai herbal ball
Example StudyExample Study (My Research)(My Research)
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Crossover DesignsCrossover Designs
Advantages:Advantages: Own control Within-subject comparison Removal of intersubject variability Reducing of the costs Increasing of Precision & power Small sample size
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Crossover DesignsCrossover DesignsCrossover Designs
Disadvantages:Disadvantages:
Carryover effects
Drop out
The analysis is more complex than
in a parallel groups design
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Possible EffectsPossible Effects
Direct treatment effect ( )
Period effect ( )
Carryover effect ( )
Treatment-by-period interaction ( )
Sequence (Group) effect ( )
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Direct treatment effectDirect treatment effect Period effectPeriod effect
Treatment A better
Treatment B
Treatment A better
Treatment B
2B 1B
1A 2A
0
0,5
1
1,5
2
2,5
1 2
period
me
an
1B
2B
2A1A
0
0.5
1
1.5
2
2.5
1 2period
mea
n
(a) (b)
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Carryover effectCarryover effect TreatmentTreatment--byby--period interactionperiod interaction
1B
2B
2A
1A
0
0.5
1
1.5
2
2.5
3
1 2p e r iod
me
an
1B
2B
2A
1A
0
0.5
1
1.5
2
2.5
3
1 2pe r iod
mea
n
1B
2B
2A
1A
0
0.5
1
1.5
2
2.5
3
1 2pe r iod
mea
n
Sequence effect
Treatment A better
Treatment B
Treatment A better
Treatment B
(c)
(e)
(d)
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Dealing with aliasingDealing with aliasing MethodologyMethodology
Latin square for
crossover
designs
Washout period
Statistical Statistical AnalysisAnalysis
Preliminary test
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Latin square for crossover designsLatin square for crossover designs
√√√√ABBA/BAAB/AABB/BBAA
×√√√AB/BA
√√√×ABB/BAA, AB/BA/AA/BB
√√×√AABBA/ABBAA
×√√×ABA/BAB
×√×√AABA/ABAA
√√××AABBA/BAABB
×√××ABAA/BAAB
×××√ABB/BAB
××××AAB/ABB
StronglyBalanced
BalancedUniform within Periods
Uniform within Sequences
Examples
Table 1Table 1:: comparison of twocomparison of two--treatmenttreatment crossovercrossover designs (designs (PiantadosiPiantadosi, 2003))
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Model: Continuous dataModel: Continuous data
where,overall mean effect of jth patient with ith sequence &
is ~N (0, ) effect of kth period treatment effect of mth treatmentcarryover effect of mth treatment random error and is ~N (0, )
ijkmmkijijk bY
ijb
k
m
m
ijk
2b
2b
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Statistical AnalysisStatistical Analysis1. Graph Subject profiles plot Group by period plot
2. Preliminary test Equal of carryover effect
3. Estimation of treatment effect 2 periods 1stperiod
2B 1B
1A 2A
0
0,5
1
1,5
2
2,5
1 2
period
mea
n
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TwoTwo--stage procedurestage procedurePreliminary test for
carryover effect 10% 2-side level10% 2-side level
5% 2-side level5% 2-side level
two-sample t-test or ANOVA
two-sample t-test or ANOVA
)( BA
Estimate the treatment effect
of 2 periods
Estimate the treatment effect of the 1stperiod
SigNon-S
ig
By…Grizzle’s procedure (1965)
)( BA )( BA
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22
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Preliminary test forPreliminary test forcarryover effectcarryover effect
A 1 AB 2
B 1 BA 22 (Sequence BA)
1 (Sequence AB)
Period 2Period 1Group
sequence AB = sequence BAA + B = B + A
=H0:
A 1 AB 2 B 1 BA 2
BA
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Estimation of treatment effectEstimation of treatment effect1: Estimate the treatment effect of 2 periods
½ (A – B) = ½ (B – A)
=
=
H0:
A 1 )2 AB B 1)2 BA ½( ½(A A B- ½ B - ½
BA BA
2: Estimate the treatment effect of the1stperiodA = B
=
H0:
A 1 B 1
BA
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Statistical AnalysisStatistical Analysis
Assumptions
The repeated measurements on each subject are independent
Normally distributed random variables with equal variances
Residual Analysis Residual Analysis
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Physical Carryover Effects
Psychological Carryover Effects
Treatment-by-Period Interaction
Group Difference
BA Cause ofCause of
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ExampleExample
Table 2: Grizzle’s data (Grizzle, J.E. The two-period changeover design and its use in clinical trials. Biometrics, 1965; 21: 467-80.)
0.91.00.6-0.3-1.01.7-0.30.9
1.3-2.30.0-0.8-0.4-2.9-1.9-2.9
12345678
1.0-0.70.21.10.41.2
0.20.0-0.80.60.31.5
123456
Period 2Period 1SubjectPeriod 2Period 1Subject
Group 2 (BA)Group 1 (AB)
2 x 2 crossover design2 x 2 crossover design
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ExampleExample
2A
1A
2B
1B
-1,4
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
period
mea
n r
esp
on
ds
Figure 2: Group-by-period plot for Grizzle’s data
• Strongly carryover
effect
•Treatment-by period
interaction
•Sequence effect
• Strongly carryover
effect
•Treatment-by period
interaction
•Sequence effect1 2
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ExampleExample
,1.0 05.0
Table 3: Two-sample t-test for 2 x 2 crossover design from Grizzle’s data
0.09-0.13 to 1.570.380.72**Direct treatment(two period )
0.040.06 to 3.010.681.54**Direct treatment(first period )
0.05-0.03 to 3.30.761.63*Carryover
p-value95% CISEdiffvariable
* **
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ExampleExample (My Research)(My Research)
A comparative study of heat effect between hot pack and
Thai herbal ball on pain and
physiological changes
ReportData Analysis
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