Simple Cross – over Design (แผนการทดลองแบบเปลี่ยนสลับอย่างง่าย) By Dr.Wuttigrai Boonkum Dept.Animal Science, Fac. Agriculture KKU

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 Design Classify Factors Consideration of number of Animal, Treatment and Time Statistical model, Hypothesis setting, Lay out ANOVA analysis using SAS program Interpretation and Conclusion

Statistical model

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

Lay out A B A B - A1 A2 A3 A4 A5 A6 Transition period Resting period 12 EU.; A = Animal A B - Period1 Period2

SAS code Data……; input row col trt y; Cards; 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 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; ns P-value < 0.05 significant; * P-value < 0.01 highly significant; **

Advantages Have efficiency more than CRD Good for budget limitation 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.

Example A B B A Sequence A  B  A Sequence B  A  B

Lay out Animal 1 Animal 2 Animal 3 Animal 4 Animal 5 Animal 6 A B Period1 Period2 Period3 18 EU. Sequence A  B  A This lay out have 2 sequence: Sequence B  A  B

Statistical model

Hypothesis setting ) ( 2 : / ¹ - = + A B H or + - = ) ( 2 : / ¹ - = + Look like Cross-over Design such as: Trt = 2, hypothesis is: Sequence B  A  B Sequence A  B  A ) ( 2 : / ¹ - = + A B H or + - = ) ( 2 : / ¹ - = + B A H or

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

SAS code Data……; 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 ; 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 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

Advantages 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 1 C B A 4 2 5 3 6

2. Type II: new animal set, same time. Square1 Square2 Period Anim 1 Anim 2 Anim 3 Anim 4 Anim 5 Anim 6 1 C B A 2 3

3. Type III: new animal set, time difference. Square1 Square2 Period Anim 1 Anim 2 Anim 3 Anim 4 Anim 5 Anim 6 1 C B A 4 2 5 3 6

Orthogonal or balanced square Example : A, B, C and D are treatments A B C D B C D A D A B C C D A B

Orthogonal or balanced square Example : A, B, C, D and E are treatments A A A A A

Statistical model and ANOVA SOV Df Sq s-1 Anim p-1 P(Sq) s(p-1) T Error sp2-p(s+2)+2 Total sp2-1

Statistical model and ANOVA SOV Df Sq s-1 Anim(Sq) s(p-1) P p-1 T Error sp2-p(s+2)+2 Total sp2-1

Statistical model and ANOVA SOV Df Sq s-1 Anim(Sq) s(p-1) P(Sq) T p-1 Error sp2-p(2s+1)+s+1 Total sp2-1

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 output Type A

SAS output Type B

SAS output Type 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 set X A B Data; input sq anim period trt $ milk Resid; Cards; 1 1 1 A 38 X 1 1 2 B 25 A 1 1 3 C 15 B 1 2 1 B 109 X 1 2 2 C 86 B 1 2 3 A 39 C 1 3 1 C 124 X 1 3 2 A 72 C 1 3 3 B 27 A 2 4 4 A 86 X 2 4 5 C 76 A 2 4 6 B 46 C 2 5 4 B 75 X 2 5 5 A 35 B 2 5 6 C 34 A 2 6 4 C 101 X 2 6 5 B 63 C 2 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

Lay out row Col 1 Col 2 Col 3 1 A B C α β γ 2 3 row Col 1 Col 2 Col 3

SAS code Data…………; input row col trt $ greek $ observe; Cards; x 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 SOV Df Row r-1 Column c-1 Treatment t-1 Greek g-1 Error Residuals Total n-1

The End Next time I will lecture about … Incomplete block design