Sa 601 The Design of Experimens Yuqing Xu Deparmen of Saisics Universiy of Wisconsin Madison, WI 53706, USA December 1, 2016 Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 1 / 17
Lain Squares Definiion of Lain Squares A Lain Square design for reamens uses 2 unis. The 2 unis are represened as a square. Each reamen occurs once in each row and once in each column. A Lain Square blocks on boh rows and columns simulaneously. Noe: If you ignore he row blocking facor, he LS design is an RCB for he column blocking facor. If you ignore he column blocking facor, he LS design is an RCB for he row blocking facor. Figure: Examples of Lain squares Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 2 / 17
Lain Squares Pros and Cons of Lain Squares Con: A resricion on : can no be oo large. Pro: Allows wo blocking effecs (e.g spaial and emporal ogeher) Example: Blood concenraion of drug Treamens: 3 mehods for delivering a drug: a soluion, a able, and a capsule. Response: concenraion Design: There are 3 paiens, and each paiens will be given he drug hree imes, once wih each of he hree mehods. Each drug will be used once in he firs period, once in he second period, and once in he hird period. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 3 / 17
Lain Squares Pros and Cons of Lain Squares Con: A resricion on : can no be oo large. Pro: Allows wo blocking effecs (e.g spaial and emporal ogeher) Example: Blood concenraion of drug Treamens: 3 mehods for delivering a drug: a soluion, a able, and a capsule. Response: concenraion Design: There are 3 paiens, and each paiens will be given he drug hree imes, once wih each of he hree mehods. Each drug will be used once in he firs period, once in he second period, and once in he hird period. Example: Blood concenraion of drug 3 3 Lain square, wih row block: paiens, column block: ime periods Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 3 / 17
Lain Squares Example: Blood concenraion of drug 3 3 Lain square, wih row block: paiens, column block: ime periods: Table: Design of example Period1 Period 2 Period 3 Paien1 S T C Paien2 T C S Paien3 C S T Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 4 / 17
Lain Squares Analysis of Lain squares Model: Y ijk = µ + r i + c j + k + ɛ ijk. where r i for row blocking, c j for column blocking, k for reamens. y ijk = y... +(y i.. y... )+(y.j. y... )+(y..k y... )+(y ijk y i.. y.j. y..k +2y...) Source DF SS MS F Row -1 i=1 (y i.. y...) 2 SSR ( 1) Column -1 j=1 (y.j. y...) 2 SSC ( 1) Treamen -1 j=1 (y..k y...) 2 SSTr 1 SSE Error (-2)(-1) By subracion ( 2)( 1) Toal 2 1 ijk (y ijk y...) 2 MSTr MSE Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 5 / 17
Lain Squares Figure: Examples of Lain squares Anova Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 6 / 17
Replicaed Lain Squares Same row blocks and same column blocks Model: Y ijkl = µ + r i + c j + k + δ l + ɛ ijkl where r i for row blocking, c j for column blocking, k for reamens, δ l for squares (l = 1...., m) Source DF SS MS F Row -1 i=1 m(y i... y...) 2 SSR ( 1) Column -1 j=1 m(y.j.. y...) 2 SSC ( 1) Treamen -1 k=1 m(y..k. y...) 2 SSTr 1 Square(bach) m-1 m l=1 2 (y...l y...) 2 SSS (m 1) SSE Error (m+m-3)(-1) By subracion (m+m 3)( 1) Toal m 2 1 ijkl (y ijkl y...) 2 MSTr MSE Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 7 / 17
Replicaed Lain Squares Differen row blocks and same column blocks Model: Y ijkl = µ + r i(l) + c j + k + δ l + ɛ ijkl where r i(l) for row blocking (relaed o square l), c j for column blocking, k for reamens, δ l for squares (l = 1...., m) ˆr i(l) = y i..l y...l Source DF SS MS F Row m(-1) m i=1 l=1 (y i..l y...l ) 2 SSR ( 1) Column -1 j=1 m(y.j.. y...) 2 SSC ( 1) Treamen -1 k=1 m(y..k. y...) 2 SSTr 1 Square(bach) m-1 m l=1 2 (y...l y...) 2 Error By subracion By subracion Toal m 2 1 ijkl (y ijkl y...) 2 SSS (m 1) MSTr MSE Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 8 / 17
Replicaed Lain Squares Differen row blocks and Differen column blocks Model: Y ijkl = µ + r i(l) + c j(l) + k + δ l + ɛ ijkl where r i(l) for row blocking (relaed o square l), c j(l) for column blocking (relaed o square l), k for reamens, δ l for squares (l = 1...., m) ˆr i(l) = y i..l y...l, ĉ j(l) = y.j.l y...l Source DF SS MS F Row m(-1) m i=1 l=1 (y i..l y...l ) 2 SSR ( 1) Column m(-1) m j=1 l=1 (y.j.l y...l ) 2 SSC ( 1) Treamen -1 k=1 m(y..k. y...) 2 SSTr 1 Square(bach) m-1 m l=1 2 (y...l y...) 2 Error By subracion By subracion Toal m 2 1 ijkl (y ijkl y...) 2 SSS (m 1) MSTr MSE Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 9 / 17
Two-Levels(Series) Facorials Definiion of 2 k design A wo-levels(series) facorial design is one in which all he facors have jus wo levels. For k facors, we call his a 2 k design, because here are 2 k differen facor-level combinaions. The wo levels for each facor are generally called low and high. Denoe a level combinaion by a sring of lower-case leers. e.g. bcd Denoe a facor by capial leers. e.g. A,B,C, BC The model can be wrien as y = µ + x 1 α + x 2 β + x 1 x 2 (αβ) + ɛ where parameers α is defined as he average of high and low in facor A. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 10 / 17
Two-Levels(Series) Facorials Example: 2 3 wihou ineracion If we consider 2 3 wihou ineracions, he model is y = µ + x 1 α + x 2 β + x 3 γ + ɛ Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 11 / 17
Two-Levels(Series) Facorials Example: 2 3 wih ineracions If we consider 2 3 wihineracions, he model is y = µ + x 1 α + x 2 β + x 1 x 2 (αβ) + x 3 γ + x 1 x 3 (αγ) + x 2 x 3 (βγ) + x 1 x 2 x 3 (αβγ) + ɛ Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 12 / 17
Two-Levels(Series) Facorials Analysis for 2 3 The main effec parameer α can be esimaed by: ˆα = 1 8 [(ȳ a + ȳ ab + ȳ ac + ȳ abc ) (ȳ (1) + ȳ b + ȳ c + ȳ bc )] Suppose we have r replicaes for each reamen level, he sum of squares for A can be wrien SSA = 4r(ˆα) 2 + 4r( ˆα) 2 = r 8 [(ȳ a + ȳ ab + ȳ ac + ȳ abc ) (ȳ (1) + ȳ b + ȳ c + ȳ bc )] 2 Similarly you can work ou his for any effec. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 13 / 17
Daniel s Mehod for Tesing in 2 k Model Some Facs by Normal Assumpion Our original daa are independen and normally disribued wih consan variance. Effecs conrass in Table 10.3 gives us resuls ha are also independen and normally disribued wih consan variance. The expeced value of any of hese conrass is zero if he corresponding null hypohesis is rue. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 14 / 17
Daniel s Mehod for Tesing in 2 k Model Technique of Daniel s Plo The main effec α can be esimaed by: ˆα = 1 8 [(y a + y ab + y ac + y abc ) (y (1) + y b + y c + y bc )] Le  = 2ˆα. Conrass corresponding o null effecs should look like a sample from a normal disribuion wih mean zero and fixed variance. Conrass corresponding o non-null effecs will have differen means and should look like ouliers Key Assumpion: We assume ha we will have mosly null resuls, wih a few non-null resuls ha should look like ouliers. Noe: These echniques will work poorly if here are many non-null effecs. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 15 / 17
Daniel s Mehod for Tesing in 2 k Model Example of Normal Plo Numbers indicaes sandard order Effec 1 and 2 he main effec of A, B, appears as a clear oulier, and he res appear o follow a line. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 16 / 17
Daniel s Mehod for Tesing in 2 k Model Con. Normal Plo Wha abou changing he posiive A, B, C... o negaive ones? I is supposed o be working oo... The whole plo changes! No uniqueness. Yuqing Xu (UW-Madison) Sa 601 Week 12 December 1, 2016 17 / 17