Analysis Of 2x2 Cross Over

Analysis of 2 x 2 Crossover Designs with Continuous Data Orawan sAETAN Biostatistician

Overview Crossover designs Common Crossover design Possible effects Dealing with aliasing - Methodology - Statistical Analysis  Example    

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Crossover Designs “Each treatment is administered to each patient at different times in the study” subjects may undergo an active drug for 6 weeks and then “cross over” to the placebo for 6 weeks Chronic & stable disease (asthma, arthritis, diabetes, hypertension, migraine …) migraine…) 3

Common Crossover Design Se e qu e nc 1

Treatment Treatment A A Washout Washout (Baseline) (Baseline)

Run-in Run-in (Baseline) (Baseline)

qu e S

ce n e

Treatment Treatment B B

2

Treatment Treatment B B

Treatment Treatment A A

Period 1

Period 2

Figure 1: 2 x 2 crossover design 4

Example Study (My Research) A comparative study of heat effect between hot pack and Thai herbal ball on pain and physiological changes Hot Hot pack pack

Thai Thai herbal herbal ball ball Washout Washout (1 (1 wk) wk)

Run-in Run-in (Baseline) (Baseline) Thai Thai herbal herbal ball ball

Hot Hot pack pack

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Crossover Designs Advantages:      

Own control Within-subject comparison Removal of intersubject variability Reducing of the costs Increasing of Precision & power Small sample size 6

Crossover Designs Disadvantages:  Carryover effects  Drop out  The analysis is more complex than in a parallel groups design 7

Possible Effects  Direct treatment effect ( )  Period effect ()  Carryover effect ( )  Treatment-by-period interaction ( )  Sequence (Group) effect () 8

 Direct treatment effect

 Period effect

2,5 1A

2A

2

1

2A 1A

1,5 2B

1B

mean

mean

2

2.5

1.5

1B

1

0,5

2B 0.5

0 1

0

2

1

period

(a)

Treatment A better Treatment B

period

2

(b)

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 Carryover effect  Treatment-by-period interaction 3

Treatment A better Treatment B

2A

1 .5

1A 1B

1 2B 0 .5 0 1

p e r io d

2.5

2A

2 1.5

1B 1A

1 0.5

2B

0

2

1

(c)

2

1B

2.5 2

(e)

p e r io d

(d)

3

mean

mean

2

3

mean

2 .5

1A 1.5 2A

1 2B

0.5



Sequence effect

0 1

p e r io d

2

10

Dealing with aliasing  Methodology  Latin square for

crossover designs

 Statistical Analysis  Preliminary

test

 Washout period 11

Latin square for crossover designs Examples

Uniform within Sequences

Uniform within Periods

Balanced

Strongly Balanced

AAB/ABB

×

×

×

×

ABB/BAB

√

×

×

×

ABAA/BAAB

×

×

√

×

AABBA/BAABB

×

×

√

√

AABA/ABAA

√

×

√

×

ABA/BAB

×

√

√

×

AABBA/ABBAA

√

×

√

√

ABB/BAA, AB/BA/AA/BB

×

√

√

√

AB/BA

√

√

√

×

ABBA/BAAB/AABB/BBAA

√

√

√

√

Table 1: comparison of two-treatment crossover designs (Piantadosi, Piantadosi 2003) 12

Model: Continuous data Yijk    bij   k  m  m   ijk

where,  overall mean bij effect of jth patient with ith sequence & 2 is ~N (0,  b )  k effect of kth period  m treatment effect of mth treatment m carryover effect of mth treatment 2  ijk random error and is ~N (0, b ) 13

Statistical Analysis

2,5

1. Graph  Subject profiles plot  Group by period plot 2. Preliminary test  Equal of carryover effect 3. Estimation of treatment effect  2 periods  1stperiod

mean

2

1A

2A

1,5 1

2B

1B

0,5 0 1

2

period

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Two-stage procedure Preliminary test for carryover effect

1

( A   B )

two-sample two-sample t-test t-test or or ANOVA ANOVA

2

ig S n No

Estimate the treatment effect of 2 periods

( A   B )

10% 10% 2-side 2-side level level

Si g

Estimate the treatment effect of the 1stperiod

5% 5% 2-side 2-side level level

( A   B )

By…Grizzle’s procedure (1965)

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Preliminary test for carryover effect Group

Period 1

Period 2

1 (Sequence AB)

  1   A

   2  B  A

2 (Sequence BA)

  1  B

   2   A  B

sequence AB = sequence BA A+B=B+A   1   A     2   B   A =   1   B     2   A  B  A  B H0: 16

Estimation of treatment effect 1: Estimate the treatment effect of 2 periods ½ (A – B) = ½ (B – A) ½(   1   A     2   A  B ) = ½(   1   B     2   B   A )  A- ½  A =  B - ½  B

 A  B

H0:

A B

2: Estimate the treatment effect of the1stperiod A=B   1   A =   1  B H0:  A B 17

Statistical Analysis  Assumptions

Residual Analysis

The repeated measurements on each subject are independent Normally distributed random variables with equal variances 18

Cause of  A  B  Physical Carryover Effects  Psychological Carryover Effects  Treatment-by-Period Interaction  Group Difference 19

Example 2 x 2 crossover design Group 1 (AB)

Group 2 (BA)

Subject

Period 1

Period 2

Subject

Period 1

Period 2

1 2 3 4 5 6

0.2 0.0 -0.8 0.6 0.3 1.5

1.0 -0.7 0.2 1.1 0.4 1.2

1 2 3 4 5 6 7 8

1.3 -2.3 0.0 -0.8 -0.4 -2.9 -1.9 -2.9

0.9 1.0 0.6 -0.3 -1.0 1.7 -0.3 0.9

Table 2: Grizzle’s data (Grizzle, J.E. The two-period changeover design and its use in clinical trials. Biometrics, 1965; 21: 467-80.) 20

Example 0,8 1B

0,6 0,4

1A 2A

mean responds

0,2

• • Strongly Strongly carryover carryover

0

effect effect

-0,2

•Treatment-by •Treatment-by period period

-0,4 -0,6

interaction interaction

-0,8 -1 -1,2 -1,4

•Sequence •Sequence effect effect

2B

1

period

2

Figure 2: Group-by-period plot for Grizzle’s data 21

Example variable

diff

SE

95% CI

p-value

1.63

0.76

-0.03 to 3.3

0.05

**Direct treatment (first period )

1.54

0.68

0.06 to 3.01

0.04



**Direct treatment (two period )

0.72

0.38

-0.13 to 1.57

0.09



*Carryover

*

 0.1,

** 

 0.05

Table 3: Two-sample t-test for 2 x 2 crossover design from Grizzle’s data 22

Example (My Research) A comparative study of heat effect between hot pack and Thai herbal ball on pain and physiological changes

Data

Analysis

Report 23

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