ex 2. Questionnaires: main treatments. Interviewers/questionnaire: subtreatments

Transcription

1 Split Plot Deign -ha type of treatment: main (A) and u (B). -each plot of a main plot i plit into a numer of uplot to accomodate an equal numer of uplot treatment.suplot yield within a main plot are correlated ince they receive the ame main treatment. -aign utreatment to main plot at random independently. ex 1. Field: Main plot Varietie of wheat: main treatment (i.e. whole plot treatment). EAch field i divided into uplot (and harveted each week). Harveting time: u-treatment (or u-plot treatment). The comparion on varietie ha field-to-field variation o it i le precie than the comparion of harveting time (the uplot trt.). ex. Quetionnaire: main treatment. Interviewer/quetionnaire: utreatment ex 3. Different machine for milking cow: machine are main plot treatment. A method of paturizing the milkk i a uplot treatment. ex 4. Compare different furnace for preparation of alloy. Compare different type of mold into which alloy may e poured (comparing furnace require much greater amount of alloy than comparing mold). Advantage: -can etimate uplot treatment contrat -etimation of uplot trt (B) and interaction (AB) effect i more precie than main plot treatment effect (A). -accomodate the need of different amount of experimental material y variou treatment. Can comine factor needing large and mall amount of information. Factor A need a large ample whera factor B need only a mall one. -accomodate the need to inert additional factor into an experiment already in progre. Factor B can alo e exaimined (& AB interaction) for only a minimal cot. Randomized lock with plit-plot: lock, t treatment per lock, uplot per treatment trt 1 uplot 1 uplot uplot trt trt t 1

2 Randomization: treatment are applied to main plot at random (ay RB deign). Suplot treatment are applied to uunit at random within each main plot independently. Linear Model: y ijk i e ijk ; i 1,...,; j 1,...,t; k 1,..., e ijk e ij f ijk, where e ij ~N 0, e, f ijk ~N 0, f ; V y ijk e f Aume E(e ijk ) 0; E(e ijk ) V(e ijk ) Since all uplot within main plot get a common treatment, E(e ijk x e ijk ), k. For all other comination, e ijk are uncorrelated. So COV(y ijk, y i j k ), k,i i, j 0, if i i,j i.e. y ijk and y ijk are correlated (k o y ij1,...,y ij are correlated for each pair (i,j). Since y ijk are correlated within pair (i,j), we want to make an orthogonal tranformation on the y to get new uncorrelated variale o that we can ue linear model theory. Conider y ij y ij1 y ij. Define c ij y ij where i an orthogonal matrix uch that 1 1 for all pair (i,j), where kp i a typical element. kp Since i an orthogonal matrix, p 1 p 1 for each k,..., (i.e. each row ha length 1) 1 p 1 p p 0,k,..., p 1 p 0,k,..., p 0 p 1 Conider the new random variale c ijk :

3 1) c ij1 1 k 1 y ijk y ij. ; V c ij1 1 V 1 1 y ijk 1 V y ijk Cov y ijk, y ijk ( 1 w 1,ay ) c ijk p 1 p y ijp (k 1 ; V c ijk V p 1 p 1 p p y ijp, (k 1 V y ijp p p Cov y ijp,y ijp 1 p p 1 ( 1 w, ay 3) Cov (c ijk,c ijk ) Cov p 1 p y ijp, p 1 py ijp (k ; k, k 1; i.e. k,k,..., p 1 p 1 q 1 4) Cov(c ij1,c ijk Cov 1 l 1 p qcov y ijp, y ijq p p Cov y ijp, y ijp p p 1 p y ijl, p 1 p y ijp 1 p Cov y ijp, y ijp 1 p 1 l 1 q 1,q p p qcov y ijp,y ijq l 1 p 1,p l p 1 p 1 p Cov y ijl,y ijp p 1,p l p Cov y ijl,y ijp 5) Further, for all other pair c ijk, c i j k, etc. (i.e. different i or j ), the covariance i 0, ince y are uncorrelated. Thu the orthogonal tranformation preerve independence. So we have uncorrelated random variale c ijk, i 1,...,; j 1,...,t; k 1,..., where V(c ij1 ) 1 w 1 1 and V(c ijk ) 1, k,...,. w Now we can apply weighted leat quare or tranform variale to get a common variance and ue the Gau-Markov theorem. Tranforming variale, define c ij1 w 1 c ij1, c ijk w c ijk, k,...,. Now V(c ij1 w 1 V c ij1 w 1 w w V c ijk w w 1 1, k,..., (i.e. a common variance of 1) V(c ijk Leat quare: Minimize 3

4 i c ijk E c ijk i c ij1 E c ij1 (w 1,w 0) i.e. we want to minimize i c ij1 E c ij1 w w 1 c ijk E c ijk i w 1 i c ij1 E c ij1 w i c ijk E c ijk i c ijk E c ijk E(c ij1 ) E y ij. i ( 0 k (model retriction)) For k,...,: E(c ijk p p E y ijp p p i p p p p p p Minimize i c ij1 i w w 1 i c ijk p p p p Normal equation: (1) c i j ij1 i () c ij1 i i (3) i c ij1 i (4) p w w 1 i c ijk p p p p p (5) p w w 1 i c ijk p p p p p On the aove equation, impoe i i j 0, and et all partial derivative equal to 0. Reult are numered (1) through (5) elow: (1) i c ij1 ij i.e. t i c ij1 or t i c ij1 i 1 t 1 t 1 y ijk i y... 1 y ijk i y ijk, o () c ij1 t t i 0 i c ij1 i.e. i 1 t 1 t y ijk y ijk j k t y i... So i i y i.. y i.. (3) c ij1 1 i y ijk y.j. y..., o y.j. y.j. Var i i Var y i.. y i.. Var y i.. Var y i.. lock) (covariance 0 ince different 4

5 k But Var y i.. Var y ijk j Var c ij1 1 Var i i 1 t t t 1 t t j 1 V c ij1 t t 1 1 t Var Var y.j. y.j. V y.j. V y.j. (ince covariance i 0) But Var y.j. Var i y ijk (4) i Var 1 RHS i i i t Var i c ij1 1 1 i 1 V c ij1 1 1 p c ijk i p p p p p p p p p p kp p kp p kp p p p kpp p p p p p p p p p p p In the aove, note that p 1 1 ince column alo form an orthonormal et; alo note that ince p 1 1, it follow that p p 1. Thu, RHS may e continued a follow: i p p p p p p p ( p p p p 0) t p p Impoe p 0 RHS t p LHS i p p p y ijp i kp y ijp p p p p y ijp i 1 1 y ijp 1 p p y ijp i y ijp 1 i p y ijp Equating RHS LHS: p i y ijp t 1 t i p y ijp y..p y... y..k y..k Var Var y..k y..k Var y..k Cov y..k, y..k Cov t t i y ijk, i y ijk t t i Cov y ijk, y ijk (the term inide the ummation i t alanced deign) t t t 1 t (i.e. a contant for all treatment; a 5

6 (5) i p c ijk i RHS i p i p p p p p p p p p p p p p p p p i p p p p p p p p p p p i 1 p i 1 jp 1 i p p p p p (due to model retriction p p p p 0 LHS i p p p y ijp i kp y ijp p p p p y ijp i 1 1 y ijp 1 i p p y ijp i y ijp 1 i p y ijp LHS RHS i y ijp 1 i p p y ijp p p o p y.jp y.j. y..p y... So p p y.jp y.jp y..p y..p ; V p p 1 1 t p p y.jp y.j p y.j. y.j.; V p p 1 1 V p p V y.jp y.jp y..p y..p V y.jp V y..p Cov y.jp,y.jp Cov y.jp,y..p Cov y.jp, y..p Cov y.jp,y..p Cov y.jp,y..p Cov y..p, y..p Note that y.jp i y ijp V p p t 1 1 t y ijp i j,y..p, y t.jp i yijp t t t,y i j y ijp. Thu,..p t t t 1 t t 1 1 t V p p V y.jp y.j p y.j. y.j. V y.jp V y.j. Cov y.jp, y.j p Cov y.jp, y.j. Note that V y.jp,v y.j. 1 i y ijk Cov y.jp, y.j. Cov y.j p, y.j. Cov y.j p,y.j. Cov y.j.,y.j. 1 1 i V y ijk i k Cov y ijk, y ijk 1 1 Cov y.jp,y.j. 1 i 1 Cov y ijp, y ijk 1 Cov y.j p,y.j. All other covariance are 0 ince j j. 6

7 V p p Find F for teting i (auming w /w 1 i known). ANOVA For Split-Plot: Source df SS Block -1 t i 1 y i.. y... t Main trt t-1 1 y.j. y... Error(a) [lock x trt] (-1)(t-1) i 1 Between wholeplot t-1 um utreatment -1 t y..k y... t 1 y ij. y i.. y.j. y... MT x ST (t-1)(-1) y.jk y.j. y..k y... Error () uplot error t(-1)(-1) y utraction Sutotal t(-1) i c ijk i y ijk y ij. Total t-1 utotal um i y ijk y... Note with repect to the aove that MS MT MS error a ~F t 1, 1 t 1, MS ST MS error ~F 1,t 1 1, MS MTxST MS error ~F t 1 1,t 1 1. To jutify thee F tet, we need to how that (i) y.j. y... t 1 c.j1 c..1 ~ t 1 central under H 0 : 0 (ii) t p y..p y... t c..k ~ 1 central under H 0 : 0. (iii) y.jk y.j. y..k y... t 1 1 c.jk ~ 1 t 1 central under H 0 :k 0. Firt jutify the equalitie and then ue RHS to how E(RHS) 0 to how centrality. NOTE: Sutreatment and interaction are etimated more preciely than main treatment. Optimal R 1 -R 0 tet depend on / 1 or. If unknown, optimal tet are not availale. However, uing Cochran theorem, exact tet can e otained, from c ijk * and c ijk repectively where V(c ijk *),k,..., and V(c ij1 *) 1. Motivation for Model (randomization theory): y iuvjk denote the yield of the v th uplot in the u th whole plot when (jk) i applied in lock i, and x iuv i the yield of the v th uplot in the u th main plot of lock i. 7

8 y iuvjk x iuv t jk y... y i... y... y...jk y... y i..jk y i... y...jk y... y iuvjk y i..jk x... t.. x i.. x... t jk t.. x iuv x i.. Note in the aove expreion that the firt term i, the econd term i i, the third term i (, and the final term i (e ij f ijk. t jk t.. t.. t j. t.. t.k t.. t jk t j. t.k t.. Note in the aove expreion that the econd term i, the third term i and the final term i k. x iuv x i.. x iu. x i.. x iuv x iu. Note in the aove expreion that the firt term i e ij, and the econd term i f ijk. oerved yield of (jk) treatment in lock i i: y ijk i e ij f ijk ; V y ijk e f i i j k j k k k 0; i 1,...,; j 1,...,t; k 1,..., j j e ij u iu x iu. x i.. where iu f ijk u,v k iuv x iuv x iu. 1 if MT j applied to MP u, and equal 0 otherwie. jk ( v iuv 1 or 0) y ijk u v k iuv y iuvjk Make an orthogonal tranformation c ij y ij where firt row of i( 1,,..., 1,. Now c ij1 y ij. i e ij ecaue k f ijk 0 i.e. uual RB analyi for c ij1 i e ij Conider now all other c ijk : Since x iu. -x i.. i the ame for all plit plot of ame whole plot a regard error we have orthogonal contrat of quantitie x iuv -x iu.. We can otain comparion of thee y taking correponding comparion of oerved mean. 8

9 Source df MS E(MS) Block -1 T (whole plot trt.) t-1 T t 1 BxT (-1)(t-1) W f e S (-1) S t 1 SxT (-1)(t-1) I t 1 1 k jk Error (-1)t(-1) E Compare T / W for H 0 : 0 ~F t 1, 1 t 1. Compare S / E for H 0 : 0 ~F 1, 1 t 1. Compare I / E for H 0 : k 0 ~F t 1 1, 1 t 1. Split Plot in a Latin Square: y ijkl yield of (j,k) th plot having l th utreatment ij i Split-plit plot (e.g. 3-tage ampling) : -further udiviion of u-plot k i l i ij il l y ijkl i l l l l ijkl ANOVA 9

10 Block (rep) MT Error (1) (rep x MT) Sutreatment (ST) ST x MT Error () SS(lock x ST) SS(lock x (STxMT)) Su-u treatment (SST) MT x SST ST x SST (MT x ST) x SST Error (3) (y utraction) Total Strip plot: -do not ue independent randomization in different main plot, i.e. utreatment are arranged in trip in each lock. y ijk i k ijk Cov ijk, ijk 1 Cov ijk, ij Make orthogonal tranformation y ij. and y i.k. Need orthogonal matrice, one with firt row ( 1,..., 1 and one with firt row ( 1 1,...,. t t We get c ij1, c ijp a efore; c ij1,c ij 1 are correlated; alo c ijp, c ij p. Uing ( 1 t,..., 1 t to tranform, we get independence ut with 3 different variance. RB Analyi 10

11 Block -1 MT t-1 Error (1) (-1)(t-1) ST -1 Error () (-1)(-1) Mt x ST (-1)(t-1) Error (3) (-1)(t-1)(-1) Yield of 3 varietie of alfalfa (ton per acre) in 1944 following 4 date of final cutting in 1943: Variety Date A B Ladak C D Total Variety Date A B Caack C D Total Variety Date A B Ranger C D Total Grand Total over all oervation:

12 Variety A B C D Ladak Caack Ranger Total Mean (ton/acre) Source of variation df SS MS Main Plot Varietie (t) t Block () Main plot error (-1)(t-1) Suplot Date of cutting () ** Date x variety (-1)(t-1) Suplot erroe Total C G /N / TSS C Main plot : C Varietie : C Block : C Main plot error : Su clae in variety datetale : C.3511 Date : C Date x variety : Suplot error : The ize difference among date of cutting were not unexpected, nor were the maller yield following B and C. The lat harvet hould e either early enough to allow renewed growth and retoration of the conequent depletion of root reerve, or o late that no growth and depletion will enue. The urpriing feature: (i) yield following C greater than B, 1

13 ince late Septemer i uually conidered a poor time to cut alfalfa in Iowa, and (ii) the aence of interaction etween date and variety Ladak i low to renew growth after cutting and might have reacted differently from the other varietie. In ome experiment, may ue either plit-plot deign or ordinary randomized lock in which the main treatment/treatment comination are randomized within each lock. On the average, the arrangement have the ame overall accuracy. Relative to randomized lock, the plit-plot deign give reduced accuracy on the mmain plot treatment and increaed accuracy on uplot treatment and interaction. In ome indutrial experiment conducted a plit-plot deign, erroneouly analyzed a RCB too low error to main plot treatment and too high error to uplot treatment. example/ aignment A plit-plit plot experiment on corn wa conducted to try 3 rate of planting (tand) with 3 level of fertilizer in irrigated and non-irrigated plot. The deign wa randomized lock with 4 replication. The main plot carried the irrigation treatment. On each there were uplot with 3 tand (10000, and plant/acre). Finally, each uplot wa divided into 3 part repectively fertilized with 60, 10 and 180 pound of nitrogen. The yield are in uhel/acre. Reult for non-irrigated: Fertilizer Stand Block Reult for irrigated: Block 13

14 Fertilizer Stand Source df MS Main plot lock 3 irrigation (I) error (A) Suplot Stand (S) I x S * Error (B) Su-u-plot Fertilizer (F) I x F 476.7** S x F I x S x F Error (C) Total 71 NOTES: -IS and IF interaction ignificant -ISF error IS interaction much the ame at all F level OR IF interaction much the ame at all S level. Hence, each -way tale give information: Fertilizer Stand 14

15 F 1 F F 3 S S S 3 not irrigated irrigated Neither F or S affected yield materially on non-irrigated plot. With irrigation, the effect of each wa pronounced. Examine eparately the plit-plot experiment on the irrigated plot. df MS tand linear ** deviation 1 96 Error(A) Fertilizer linear 1 688** deviation SF 4 9 error (B) NOTE: Planting and fertilizer rate were well choen for the unirrigated plot, ut on the irrigated plot they were too low to allow any evaluation of the optima. Thi ugget that irrigation hould not e a factor in uch experiment. But in order to compare cot and return over a numer of year, experiment (one with and one without irrigation) hould e randomly interplanted to control fertility difference. 15