Simple Cross over Design By Dr Wuttigrai Boonkum
Simple Cross – over Design (แผนการทดลองแบบเปลยนสล บอยางงาย ) 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 1 A 2 A 3 A 4 A 5 A 6 Period 1 A A B B Transition period Period 2 B B A A Resting period 12 EU. ; A = Animal Period 1 A A B - Period 2 B B A A
SAS code Data……; input row col trt y; Cards; 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 Period p-1 Animal a-1 Treatment t-1 Error (t-1)*(t-2) Total n-1 SS Interpretation is likely LSD P-value > 0. 05 P-value < 0. 01 non-significant; ns significant; * highly significant; ** MS F P-value
Advantages 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 1 st 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 Sequence A B A A B Sequence B A B
Lay out Animal 1 Animal 2 Animal 3 Animal 4 Period 1 A A B B Period 2 B B A A Period 3 A A B B Animal 5 Animal 6 18 EU. Sequence A B A This lay out have 2 sequence: Sequence B A B
Statistical model
Hypothesis setting • Look like Cross-over Design such as: • Trt = 2, hypothesis is: Sequence B A B Sequence A B A H 0 : ) B + B( / 2 - A = 0 H A : ) B + B( / 2 - A ¹ 0 H 0 : ) A + A( / 2 - B = 0 H A : ) A + A( / 2 - B ¹ 0 or or H 0 : B - 2) A ( = 0 H A : B - 2) A ( ¹ 0 H 0 : A - 2) B ( = 0 H A : A - 2) B ( ¹ 0
ANOVA SOV Sequence Animal(sq) Period P*Sequence P*Animal(sq) Treatment df SS MS F s-1 s(a-1) p-1 1*(s-1) 1*s(a-1) t-1 Error dftot-dfother Total n-1 Note: Animal(sq) = Animal within sequence error; P = Period (is regression) P-value
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 ; Proc GLM data =…………. ; class seq cow period trt ; model observ = seq cow(seq) period p*seq p*cow(seq) trt /SS 1; 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. Square 1 Square 2 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. Square 1 Square 2 Period Anim 1 Anim 2 Anim 3 Period Anim 4 Anim 5 Anim 6 1 C B A 1 A C B 2 C B A 3 B A C
3. Type III: new animal set, time difference. Square 1 Square 2 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 square Example : A, B, C and D are treatments A 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
Statistical model and ANOVA SOV Df Sq s-1 Anim p-1 P(Sq) s(p-1) T p-1 Error sp 2 -p(s+2)+2 Total sp 2 -1
Statistical model and ANOVA SOV Df Sq s-1 Anim(Sq) s(p-1) P p-1 T p-1 Error sp 2 -p(s+2)+2 Total sp 2 -1
Statistical model and ANOVA SOV Df Sq s-1 Anim(Sq) s(p-1) P(Sq) s(p-1) T p-1 Error sp 2 -p(2 s+1)+s+1 Total sp 2 -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 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 1 α β γ 2 B C A 2 γ α β 3 C A B 3 β γ α row 1 2 3 Col 1 Col 2 A Col 3 B α B C β C γ C A α A β γ β B γ α
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
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