Simple Cross – over Design(แผนการทดลองแบบเปล��ยนสล�บ

อย�างง�าย)

ByDr.Wuttigrai Boonkum

Dept.Animal Science, Fac. AgricultureKKU

Simple Cross-Over Design

• Other name “Simple Change-over Design” or “Reversal design”

• Look like Repeated Measurement Exp.• About 3 factors are treatments, Animal

and time.• Researcher must change – over all

treatments in each animal.• Response measured of treatment effect

in each animal and each time.

Objective

• To compare between cross-over design and switch-back design.

• Can calculated statistic parameters in cross-over design and switch-back design.

• Can interpretation and conclusion of results from SAS program.

• Tell differentiate of Type of Replicated Latin Square.

Step by Step of Cross-over Step by Step of Cross-over Design Design

Classify FactorsClassify Factors

Consideration of number of Animal, Treatment and Time Consideration of number of Animal, Treatment and Time

Statistical model, Hypothesis setting, Lay outStatistical model, Hypothesis setting, Lay out

ANOVA analysis using SAS programANOVA analysis using SAS program

Interpretation and ConclusionInterpretation and Conclusion

Statistical model

ijkkjiijky

error

effecttreatment

effectcolumn

effectrow

meanoverall

treatmentofnobservatioy

ijk

k

j

i

ijk

Hypothesis setting

• Look like Latin Square Design such as:• Trt = 2, hypothesis is:

21

210

:

:

AH

H

Lay outLay out

A A B A B B

B B A B A A

Period1Period1

Period2Period2

A A B A B -

B B A B A A

Period1Period1

Period2Period2

Transition periodTransition period

Resting periodResting period

12 EU.; A = Animal12 EU.; A = Animal

A1A1 A2A2 A3A3 A4A4 A5A5 A6A6

SAS code

Data……; input row col trt y;Cards; x x x x x x x x x x x x;Proc anova data =………….; class row col trt; model y = row col trt; means trt /duncan;Run;

Like Latin square design

SAS outputSAS output

ANOVA Table

SOV df SS MS F P-value

Period p-1

Animal a-1

Treatment

t-1

Error (t-1)*(t-2)

Total n-1

Interpretation is likely LSD

P-value > 0.05 non-significant; nsP-value < 0.05 significant; *P-value < 0.01 highly significant; **

AdvantagesAdvantages

1. Have efficiency more than CRD

2. Good for budget limitation

3. Increase precision for Experimental design

Switch-back Design

• Look like cross-over design.

• But turn around 1st treatment when cross-over each treatments.

• This design is appropriate for high effect of time on treatment

• The example this design such as: lactation trait, growth trait, traits about time period etc.

ExampleExampleA

A

B

B

B

ASequence A B A

Sequence B A B

Lay out

AA AA BB AA BB BB

BB BB AA BB AA AA

AA AA BB AA BB BB

Period1

Period2

Period3

Sequence A B A

This lay out have 2 sequence:

Sequence B A B

Animal 1 Animal 2 Animal 3 Animal 4 Animal 5 Animal 6

18 EU.

Statistical model

ijkijjikiijky )(

error

errorsequencewithinanimal

effectperiodandeffectsequenceoferactionint

effectperiod

effectsequence

meanoverall

treatmentofnobservatioy

ijk

ik

ij

j

i

ijk

)(

Hypothesis setting• Look like Cross-over Design such as:• Trt = 2, hypothesis is:

0)(2:

0)(2:

02/)(:

02/)(:

0

0

BAH

BAH

or

BAAH

BAAH

A

A

0)(2:

0)(2:

02/)(:

02/)(:

0

0

ABH

ABH

or

ABBH

ABBH

A

A

Sequence B A B Sequence A B A

ANOVA

Note: Animal(sq) = Animal within sequence error; P = Period (is regression)

SOV df SS MS F P-value

Sequence s-1

Animal(sq) s(a-1)

Period p-1

P*Sequence 1*(s-1)

P*Animal(sq) 1*s(a-1)

Treatment t-1

Error dftot-dfother

Total n-1

SAS codeData……; input row col trt observ; If cow = 1 or cow = 2 or cow = 3 THEN seq = 1 ELSE seq = 2;P = period;Cards; x x x x x x x x x x x x;Proc GLM data =………….; class seq cow period trt ; model observ = seq cow(seq) period p*seq p*cow(seq) trt /SS1; Test H = period p*seq E = p*cow(seq); Test H = seq E = cow(seq); Lsmeans trt ;Run;

SAS outputSAS output

Interpretation

Check P-value of adjusted p * sequence interaction

Check P-value of adjusted period and sequence respectively

Check P-value of treatment effect

ns * , **

conclusion Treatment mean analysis

AdvantagesAdvantages

• Precision morn than cross-over design

• Appropriate for time period traits

Replicated Latin Square Design

• Use case more than 2 treatment

• Researcher want to change-over trt.

• To decrease error of sequence so must have a square.

• Each square must difference of sequence so may be called “balanced square” or “orthogonal square”.

Replicated Latin Square Design

3 type of Replicated Latin Square

1. Type I: originally animal set, time difference.

Square1 Square2

Period Anim 1 Anim 2 Anim 3 Period Anim 1 Anim 2 Anim 3

1 C B A 4 A C B

2 A C B 5 C B A

3 B A C 6 B A C

2. Type II: new animal set, same time.

Square1 Square2

Period Anim 1 Anim 2 Anim 3 Period Anim 4 Anim 5 Anim 6

1 C B A 1 A C B

2 A C B 2 C B A

3 B A C 3 B A C

3. Type III: new animal set, time difference.

Square1 Square2

Period Anim 1 Anim 2 Anim 3 Period Anim 4 Anim 5 Anim 6

1 C B A 4 A C B

2 A C B 5 C B A

3 B A C 6 B A C

Orthogonal or balanced squareOrthogonal or balanced square

Example : A, B, C and D are treatmentsExample : A, B, C and D are treatments

CCBBAA

AA

AA

AA

DD

CCBB DD

CCBBDD

CC BBDD

Orthogonal or balanced squareOrthogonal or balanced square

Example : A, B, C, D and E are treatmentsExample : A, B, C, D and E are treatments

AA

AA

AA

AA

AA

Statistical model and ANOVAStatistical model and ANOVA

SOVSOV DfDf

SqSq s-1s-1

AnimAnim p-1p-1

P(Sq)P(Sq) s(p-1)s(p-1)

TT p-1p-1

ErrorError spsp22-p(s+2)+2-p(s+2)+2

TotalTotal spsp22-1-1

ijklkljilijkl TPASyAType )(:

Statistical model and ANOVA

SOVSOV DfDf

SqSq s-1s-1

Anim(Sq)Anim(Sq) s(p-1)s(p-1)

PP p-1p-1

TT p-1p-1

ErrorError spsp22-p(s+2)+2-p(s+2)+2

TotalTotal spsp22-1-1

ijklkjlilijkl TPASyBType )(:

Statistical model and ANOVA

SOVSOV DfDf

SqSq s-1s-1

Anim(Sq)Anim(Sq) s(p-1)s(p-1)

P(Sq)P(Sq) s(p-1)s(p-1)

TT p-1p-1

ErrorError spsp22-p(2s+1)+s+1-p(2s+1)+s+1

TotalTotal spsp22-1-1

ijklkljlilijkl TPASyCType )()(:

SAS code• Type A:

Proc anova data = ……….;

class sq anim period trt;

model Y = sq anim period(sq) trt;

means trt /Duncan;

Run;

• Type B:

Proc anova data = ……….;

class sq anim period trt;

model Y = sq anim(sq) period trt;

means trt /Duncan;

Run;

SAS code• Type C:

Proc anova data = ……….;

class sq anim period trt;

model Y = sq anim(sq) period(sq) trt;

means trt /BON;

Run;

SAS outputSAS output Type AType A

SAS outputSAS output Type BType B

SAS outputSAS output Type CType C

Latin square Design to Estimate Residual Effects

• Transition period limited.

• Some treatments may have residual effects.

• Sometime Researcher interested in residual effects.

• Example residual effects such as antibiotic, hormones etc.

SAS data setData; input sq anim period trt $ milk Resid;Cards;1 1 1 A 38 X1 1 2 B 25 A1 1 3 C 15 B1 2 1 B 109 X1 2 2 C 86 B1 2 3 A 39 C1 3 1 C 124 X1 3 2 A 72 C1 3 3 B 27 A2 4 4 A 86 X2 4 5 C 76 A2 4 6 B 46 C2 5 4 B 75 X2 5 5 A 35 B2 5 6 C 34 A2 6 4 C 101 X2 6 5 B 63 C2 6 6 A 1 B;

X

A

B

SAS code

Proc GLM data =……….;

class sq anim period trt resid;

model milk = sq anim(sq) period(sq) trt resid;

Run;

Graeco Latin Square Design

• Researcher can separate a variable later (greek letter)

• Level of effects equal row effect, column effect and treatment effect.

Statistical model

effecttreatment

effectgreek

effectcolumn

effectrow

meanoverall

nobservatioy

when

y

l

k

j

i

ijkl

ijkllkjiijkl

Lay outLay outrow Col 1 Col 2 Col 3 row Col 1 Col 2 Col 3

1 A B C 1 α β γ

2 B C A 2 γ α β

3 C A B 3 β γ α

row Col 1 Col 2 Col 3

1A

αB

βC

γ

2B

γC

αA

β

3C

βA

γB

α

SAS codeSAS codeData…………; input row col trt $ greek $ observe;Cards;x x x x xx x x x xx x x x xx x x x x;Proc anova data =…..; class row col greek trt; model observe = row col greek trt; means trt / duncan;Run;

ANOVA of Graeco Latin Square Design

SOVSOV DfDf

RowRow r-1r-1

ColumnColumn c-1c-1

TreatmentTreatment t-1t-1

GreekGreek g-1g-1

ErrorError ResidualsResiduals

TotalTotal n-1n-1

The EndNext time I will lecture about …

Incomplete block design