Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

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1 Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur

2 Tukey s rocedure for multle comarso (T- method) The T-method uses the dstrbuto of studetzed rage statstc (The S-method (dscussed ext) utlzes the F- dstrbuto) The T-method ca be used to make the smultaeous cofdece statemets about cotrasts amog a set of arameters { β, β,, β} ( β β ) ad a estmate s of error varace f certa restrctos are satsfed These restrctos have to be vewed accordg to the gve codtos For examle, oe of the restrctos s that all ˆ ' s have equal varaces I the setu of oe way classfcato, ˆβ σ β β has ts mea Y ad ts varace s Ths reduces to a smle codto that all ' s are same, e, = for all so that all the varaces are same Aother assumto s to assume that ˆ β, ˆ β ˆ,, β ( ) ' the dffereces { β β, =,,, } are statstcally deedet ad the oly cotrasts cosdered are We make the followg assumtos: The ˆ β are statstcally deedet, ˆ β,, ˆ β ˆ β s a kow costat t ~ N( β, a σ ), =,,,, a > 0 s s s a deedet estmate of σ wth degrees of freedom e, ~ χ ( ) (Here = - ) ad σ v s s statstcally deedet of ˆ β ˆ ˆ, β,, β

3 3 The statemet t t of T-method d s as follows: Uder the assumtos ()-(v), the robablty s L = Cβ ( C = 0) = = smultaeously satsfy ˆ ˆ L Ts C L L + Ts C = = ( α) that the values of cotrasts where Lˆ = ˆ ˆ Cβ, β s the maxmum lkelhood (or least squares) estmate of β, T = aq α,,, wth qα,, beg the = uer 00 α % ot of the dstrbuto of Studetzed rage Note that f L s a cotrast lke β ( ) the ad the varace s so that ad the terval β C = σ a = = smlfes to ( ˆ β ˆ β ) Ts β β ( ˆ β ˆ β ) + Ts where T = q Thus the maxmum lkelhood (or least squares) estmate of s sad to be α,, Lˆ = ˆ β ˆ β L = β β sgfcatly dfferet from zero accordg to T-crtero f the terval ( ˆ β ˆ β Ts, ˆ β ˆ β + Ts) does ot cover β β = 0, e e, f ˆ β ˆ β > Ts Ts or more geeral f ˆ L > Ts C =

4 4 The stes volved the testg ow volve followg stes: Comute ( ˆ ˆ ) Lˆ β β or Comute all ossble arwse dffereces Comare all the dffereces wth s qα,, C = ˆ ( ˆ ˆ ) If L or β β > Ts C = the ˆ ad ˆ are sgfcatly dfferet where T β Tables for T are avalable q = α,, β Whe samle szes are ot equal, the Tukey-Kramer Procedure suggests to comare ˆL wth qα,, s + C = or T + C =

5 5 The Scheffe s method (S-method) of multle comarso S-method geerally gves shorter cofdece tervals the T-method It ca be used a umber of stuatos where T-method s ot alcable, eg, whe the samle szes are ot equal { ψ } A set L of estmable fuctos s called a -dmesoal sace of estmable fuctos f there exsts learly deedet estmable fuctos ( ψ, ψ,, ψ ) such that every ψ L s of the form ψ = Cy where C, C,, C = are kow costats I other words, L s the set of all lear combatos of,,, ψ ψ ψ Uder the assumto that the arametrc sace s Y ~ N( Xβ, σ I) wth rak( X ) =, β = ( β,, β ), X s matrx, cosder a -dmesoal sace L of estmable fuctos geerated by a set of learly deedet estmable fuctos { ψ, ψ,, ψ } Ω For ay ψ L, Let = Cy be ts least squares (or maxmum lkelhood) estmator, = C = Var( ) = σ = σ ψ ( say) ad ψ s C = ˆ σ = s ( ) where s the mea square due to error wth degrees of freedom

6 6 The statemet of S-method s as follows: Uder the arametrc sace Ω, the robablty s ( α) that smultaeously for all ψ L, S ˆ σ ψ + S ˆ σ where the costat S = F (, ) α Method For a gve sace L of estmable fuctos ad cofdece coeffcet ( α), the least square (or maxmum lkelhood) estmate of ψ L wll be sad to be sgfcatly dfferet from zero accordg to S-crtero f the cofdece terval ( S ˆ σ ψ + S ˆ σ ) does ot cover ψ = 0, e, f > ˆ σ S ψ ˆ The S-method s less sestve to the volato of assumtos of ormalty ad homogeety of varaces

7 7 Comarso of Tukey s ad Scheffe s methods Tukey s method ca be used oly wth equal samle sze for all factor level but S-method s alcable whether the samle szes are equal or ot Although, Tukey s method s alcable for ay geeral cotrast, the rocedure s more owerful whe comarg smle arwse dffereces ad ot makg more comlex comarsos 3 It oly arwse comarsos are of terest, ad all factor levels have equal samle szes, Tukey s method gves shorter cofdece terval ad thus s more owerful 4 I the case of comarsos volvg geeral cotrasts, Scheffe s method teds to gve arrower cofdece terval ad rovdes a more owerful test 5 Scheffe s method s less sestve to the volatos of assumtos of ormal dstrbuto ad homogeety of varaces