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 ovrte n the model provde more urte more pree etmte of tretment ontrt Now dut for the ovrte to remove be ttrbutble to dfferene mong expermentl unt Now redue the b by dutng tretment men ontrt Conder the whole-plot prt of the degn Conder the model gven n equton ( whh follow: w w w w y = µ + β x + r + β ( x x + w + β ( x x + β ( x x x + x + ε(1 + h ooo h hoo ooo oo ooo ho hoo oo ooo h + β ( x x + ( w + β ( x x x + x + β ( x x x + x + ε( ( oo ooo o oo oo ooo h ho o oo h = µ + r + w + β ( x x w h ho hoo x + x + + + w + x x x + x + oo ooo ε(1 h ( β ( h ho o oo ε( h, where w µ = µ + β x ooo r = r + β ( x x w h h hoo ooo w = w + β ( x x w oo ooo = + β ( x x oo ooo ( w k = ( w + β ( xo xoo xoo + xooo
3 µ w The unbed etmte of re obtned from th model follow: y oo = ˆ µ + wˆ y = wˆ oo wth = 0 But the unbed etmte of ontrt of the form w re needed nted of thee ontrt Ung the defnton mpled n equton (, the unbed etmte re gven ˆw ˆ ˆ ˆw y = µ + β x + w + β ( x x oo ooo oo ooo ˆ ˆ w y = w + β x oo ooo Sne xh ' re the oberved ontnt, o ˆ ˆ ˆ w µ + w = y β x oo oo ˆ ˆ w w = y β x oo oo Thee re the requred ontrt the duted mn effet ontrt
4 Further, ( x oo ( 1 Vr ˆ w σ + σ = + r E(1 x w x w E MSE(1 = σ + σ1 MSE (1 h ( r 1( t 1 1 degree of freedom Compron mong the plt-plot tretment level re mlr Then The unbed etmte of the ontrt of the form unbed etmte re gven by y oo = ˆ µ + ˆ ˆ oo = y ˆw ˆ ˆ ˆ y = µ + β x + + β ( x x oo ooo oo ooo ˆ ˆ y = + β x oo oo Thee expreon n be re-expreed re needed Ung the defnton n equton (, the ˆ ˆ ˆ ˆw µ + = y ( β x x β xˆ oo oo ooo ooo ˆ ˆ = y β x oo oo Alo x oo Vr ˆ = + σ rt E( E MSE( = σ MSE ( h ( r 1 t ( 1 1 degree of freedom
5 Interton ontrt Interton ontrt re of the form y o wth = 0 The trd error of uh ontrt depend on the nture of the ontrt Conder the ontrt mong plt-plot level for fxed whole-plot level Thee re of the form y o wth = 0 wth pef vlue Then The duted ontrt t vrne E y = + w o ( ( ( ( β ( o oo = + w + x x ( ˆ ( ˆ + w = y β ( x x o o oo ( x x 1 o oo σ r E ( Next onder the ontrt mong whole-plot tretment level, ether t the me plt-plot tretment level or t the dfferent plt-plot tretment level Thee ontrt tke the generl form y o, where 0 Then E y = w + + w o ( ( w = ( w + β ( x x + + ( w + β ( x x oo ooo o oo =
6 The duted ontrt ˆw ( ˆ ˆ ( ( ( ˆ w + + w = y β x x β ( x x o oo ooo o oo h vrne ( σ1 + σ ( x ( oo x x o x ooo oo + ( σ1 + σ + σ r E(1 x ( wx E w There no ne etmte of th vrne A moderte mplfton obtned by plttng th n two e Conder frt the whole-plot ontrt t one plt-plot level Th mple tht the ' re zero for ll ' the generl duted ontrt then beome t vrne beome ˆ β ( An unbed etmte of th vrne gven by ( ( wˆ + ˆ + ( w = y x x, ' ' ' ' o ' o ' oo ' o ' oo ( 1 ( x o ' xoo ( 1 σ + σ ' σ + + σ = σ + σ r E( r E( ( x x 1 o ' oo 1 ' + MSE + ' MSE r E r ( ( (1 It dffult to fnd t ext degree of freedom The Stterwte formul ued to pproxmte the degree of freedom whh ued for the tet of hypothe onfdene ntervl etmton For the more generl e, the unbed etmte of the vrne gven by The ext degree of freedom re dffult to obtn The pproxmte degree of freedom n be obtned ung Stterthwte pproxmton ( x x ( x x ( (1 w w w w 1 1( xoo xooo o oo 1 ( xoo xooo + + MSE + + MSE r E(1 x x E( x x r E(1 x x
7 Anly of ovrne wth one whole-plot ovrte n RBD We onder llutrte the nly of ovrne for the plt-plot experment wth whole-plot n n RBD one ovrte oted wth the whole-plot Developng the model We begn wth the model y = µ + r + w + βx + ε(1 + + ( w + ε(, h= 1,, r, = 1,, t, = 1,,, h h h h h where h = 1,,,r; = 1,,,t = 1,,,, the ovrte Aume tht both the whole- plt-plot tretment re fxed effet, mplyng tht wo = o = ( w o = ( w o = 0 The xh ' re oberved ontnt The ε (1 h ε ( h re dentlly ndependently normlly dtrbuted eh wth men 0 vrne σ σ repetvely Moreover they re mutully ndependent Rewrte the model to olte the ovrte ontrbuton to b vrne Ung x = x + ( x x + ( x x + ( x x x + x, ho oo ho oo o oo h ho o oo the model rewrtten x h 1 y = µ + βx + r + β( x x + w + β( x x + β( x x x + x + ε(1 + + ( w + ε( h oo h ho oo o oo h ho o oo h h = µ + r + w + β( x x x + x + ε(1 + + ( w + ε( h h ho o oo h h where µ = µ + βx, oo r = r + β ( x x, h h ho oo w = w + β ( x x o oo
8 The term n β ( x x x + x h ho o remove ll of thee µ, r, w h repreent the ontrbuton to b from the expermentl unt v the ovrte, repreent the ontrbuton to the vrne The nly of ovrne provde dutment to Computton n the nly of ovrne One the model ontruted, the next tep to ontrut the ompt nly of ovrne tble follow: Anly of ovrne tble for the plt-plot wth only whole-plot ovrte Soure y vrble Covrte Men M M M Blok B B B W W W W Error (1 E (1 E (1 E (1 S S W S W S Error ( E ( Totl T T T
9 There re no ovrne dutment n the plt-plot trtum In the whole-plot trtum, we hve ˆ β = E(1 E(1 ˆ ( σ + σ1 Vr( β = E(1 1 E (1 MSE(1 = E(1 [( r 1( t 1 1] E(1 MSW 1 [ W + E(1 ] E(1 = W + ( t 1 W + E(1 E(1 Contrt Sne there only whole-plot ovrte, o only the whole-plot tretment ontrt re duted Conder the ontrt wth = 0 Rewrte th Th ontrt h vrne The etmte of y = wˆ + ˆ β ( x x oo o oo wˆ = y ˆ β x oo oo x o 1 + + r E(1 ( σ σ1 ( σ + σ wth ( r 1( t 1 1 degree of freedom gven by MSE(1 1