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

11

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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