Analysis of Variance and Design of Experiments-II
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1 Anlyi of Vrince nd Deign of Experiment-II MODULE VI LECTURE - 7 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shlbh Deprtment of Mthemtic & Sttitic Indin Intitute of Technology Knpur
2 Anlyi of covrince ith one plit-plot covrite There re vriou poibilitie for the nlyi of covrince in plit-plot experiment, e.g., there cn be covrite for the hole-plot nd not for the plit-plot, covrite for the plit-plot nd not for the hole-plot, or to hve different covrite for the hole nd plit-plot. The djutment to the tretment men cn be mey nd o chooe the model crefully. There i no imple unique y to ue nd djut for covrite in plit-plot experiment. Aume tht the covrite re prt of the experimentl unit rther thn repone to the tretment pplied. Thi men tht tretment do not ffect the covrite nd o the covrite re vilble to the experimenter t plnning or execution tge. The covrite re obervble contnt in the model. Development of model for one covrite t the plit-plot level Conider very bic model for plit-plot experiment ith hole-plot rrnged in n RBD nd one covrite tht i ocited ith the plit-plot experimentl unit, y µ + r + + ε(1 + + ( + βx + ε(, hij h i hi j ij hij hij here h 1,..., r, i 1,..., t nd j 1,..., nd the covrite i xhij. Aume tht the hole-nd plit-plot tretment re fixed effect, implying tht ( ( 0. The x ' re oberved contnt. We further ume o o oj io tht ε (1 hi nd ε re identiclly nd independently normlly ditributed, ech ith men 0 nd vrince 1 nd ( hij repectively. Moreover, they re mutully independent lo. Rerite the model to iolte the covrite contribution to bi nd vrince. Since there re to type of experimentl x unit, to ource of error, firt to plit into x, (correponding to the hole-plot nd ( x x (for the plit-plot. hij hio hij hij hio σ σ,
3 The model i generlized to llo different regreion coefficient by introducing nd β for the plit-plot prt. The model then become y µ + r + + β x + ε(1 + + ( + β ( x x + ε(. hij h i hio hi j ij hij hio hij for the hole-plot prt of the nlyi No rite the model in form tht explicitly ho ho the covrite contribute to the bi of the etimted hole-nd plit-plot fctor effect nd the vrince component. Uing the identitie β 3 nd x x + ( x x + ( x x + ( x x x + x hio ooo hoo ooo oio ooo hio hoo oio ooo ( x x ( x x + ( x x x + x + ( x x x + x, hij hio ooj ooo oij oio ooj ooo hij hio oij oio the model i reritten y µ + β x + r + β ( x x + + β ( x x + β ( x x x + x + ε(1 + hij ooo h hoo ooo i oio ooo hio hoo oio ooo hi j + β ( x x + ( + β ( x x x + x + β ( x x x + x + ε( ( ooj ooo ij oij oio ooj ooo hij hio oij oio hij µ + r + + β ( x x h i hio hoo xoio + xooo + ε(1 hi + j + ( ij + β ( xhij xhio xoij + xoio + ε( hij, here µ µ + β x ooo r r + β ( x x h h hoo ooo + β ( x x i i oio ooo + β ( x x j j ooj ooo ( jk ( ij + β ( xoij xoio xooj + xooo.
4 4 The extr term in µ, rh, i, j nd ( ij repreent the contribution to the bi from the experimentl unit vi the covrite nd β ( x x x + x nd β ( x x x + x the contribution to vrince. The nlyi of hio hoo oio ooo covrince provide djutment to remove ll of thee. Anlyi of covrince tble hij hio oij oio Once model h been contructed, the hole-plot prt of the deign mtrix i orthogonl to the plit-plot prt nd it i poible to do the nlyi of covrince nd etimte ll the prmeter. The firt tep i to contruct compct nlyi of covrince tble follo: Anlyi of covrince tble Source y vrible Whole-Plot Covrite Split-plot Covrite Men M M x y M xx Block B B x y B xx W W W x y W xx Error (1 E (1 E (1 x y E (1 x x S S W S W S S xy W S xy S xx W S xx Error ( E ( E ( xy E ( xx Totl T
5 5 The quntitie in the column lbeled y-vrible re the uul nlyi of vrince um of qure. The column under the heding Whole-plot covrite contin the um of qure computed uing the hole-plot covrite. The other column contin the um of cro-product involving the y-vrible nd then hole-plot covrite. Similrly, the to column under the Split-Plot Covrite heding contin the um of qure nd the cro-product involving the plit-plot covrite. The x nd x ubcript identify term computed uing the hole-plot nd plit-plot covrite, repectively. For exmple nd E(1 ( x x x + x xx hio oi hoo ooo h i ( x (. x hij hio oij + oio h i j E x x x x The expected vlue re follo: E E(1 ( r1( t 1( σ + σ + ( β ( x x x + x 1 hio oio hoo ooo h i E E(1 xy β ( xhio xoio xhoo + xooo h i E ( E(1 x ( β ( x x x + x + ( σ + σ ( x x x + x y hio oio hoo ooo 1 hio oio hoo ooo h i h i ( ( E E( ( r1 t ( 1 σ + ( β x x x + x hij hio oij oio h i j E E( β x x x + x x y hij hio oij oio h i j x ( ( ( y β hij hio oij oio σ hij hio oij oio h i j h i j E ( E( x x x + x + x x x + x.
6 6 Then ˆ β ˆ β E(1 E(1 E( E( x y x x xy xx ( ˆ E β β ( ˆ E β β ˆ ( σ + σ1 Vr[ β] ( x x x + x Vr[ ˆ β ] h ( σ + σ1 E(1 i x x h i j xx σ E( σ ( x x x + x. hio oio hoo ooo hij hio hoj hoo Note tht there i no ted degree of freedom if there relly i one covrite in the hole-plot trtum nd nother in the plit-plot trtum nd the to effect re dditive in the hole-plot trtum.
7 7 Adjuting the hole-plot error for the covrite give 1 E(1 x y MSE(1 E(1 [( r1( t1 1 ] E(1 xx ith ( r1( t1 1 degree of freedom. Notice the upercript indicte men qure djuted for the covrite. E MSE(1 σ + σ 1. Adjuting the plit-plot error give 1 E( xy MSE( E( [( r1 t ( 1 1 ] E(1 xx ith ( r1 t ( 1 1 degree of freedom. E MSE( σ The tretment nd interction um of qure mut lo be djuted for the covrite to provide proper tet of hypothee. In the hole-plot trtum, the djuted hole-plot tretment um of qure ith (t 1 degree of freedom i MSE Similrly, in the plit-plot trtum, MSS 1 Wx (1 y+ E xy W. ( t 1 Wx (1 y E + xy 1 Sx ( y+ E xy S ( 1 Sxx E( + xx 1 ( W S ( ( xy + E xy E xy MS( W S ( W S +. ( t1( 1 ( W Sxx + E( ( xx E xx
8 8 Tet of hypothee re performed uing the djuted men qure. For hiole-plot tretment, ue MSW F MSE (1 ith t -1 nd ( r1( t1 1 degree of freedom. In the plit-plot trtum, tet the plit-plot tretment uing F MSS MSE ( ith -1 nd ( r1 t ( 1 1 degree of freedom nd the interction of hole-plot nd plit-plot tretment i teted uing MS( W S F MSE( ith ( t1( 1 nd ( r1 t ( 1 1 degree of freedom.
Analysis of Variance and Design of Experiments-II
Anly of Vrne Degn of Experment-II MODULE VI LECTURE - 8 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr Shlbh Deprtment of Mthemt & Sttt Indn Inttute of Tehnology Knpur Tretment ontrt: Mn effet The uefulne of hvng
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