Chapter 3 Experimental Design Models

Transcription

1 Chater 3 Exermetal Desg Models We cosder the models whch are used desgg a exermet. The exermetal codtos, exermetal setu ad the obectve of the study essetally determe that what tye of desg s to be used ad hece whch tye of desg model ca be used for the further statstcal aalyss to coclude about the decsos. These models are based o oe-way classfcato, two way classfcatos (wth or wthout teractos), etc. We dscuss them ow detal few setus whch ca be exteded further to ay order of classfcato. We dscuss them ow uder the set u of oe-way ad two-way classfcatos. t may be oted that t has already bee descrbed how to develo the lkelhood rato tests for the testg the hyothess of equalty of more tha two meas from ormal dstrbutos ad ow we wll cocetrate more o dervg the same tests through the least squares rcle uder the setu of lear regresso model.the desg matrx s assumed to be ot ecessarly of full rak ad cossts of 0 s ad s oly. Oe way classfcato: Let radom samles from ormal oulatos wth same varaces but dfferet meas ad dfferet samle szes have bee deedetly draw. Let the observatos Y follow the lear regresso model setu ad Y deotes the th observato of deedet varable Y whe effect of th level of factor s reset. The Y are deedetly ormally dstrbuted wth EY ( ) µ + α,,,...,,,,..., VY ( ) where µ s the geeral mea effect. - s fxed. - gves a dea about the geeral codtos of the exermetal uts ad treatmets. α s the effect of th level of the factor. - ca be fxed or radom. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

2 Examle: Cosder a medce exermet whch there are three dfferet dosages of medces - mg., 5 mg., 0 mg. whch are gve to atets for cotrollg the fever. These are the 3 levels of medces, ad so deote α mg., α 5 mg., α 3 0 mg. Let Y deotes the tme take by the medce to reduce the body temerature from hgh to ormal. Suose two atets have bee gve mg. of dosage, so Y ad Y wll deote ther resoses. So we ca wrte that whe α mg s gve to the two atets, the EY ( ) µ + α ;,. Smlarly, f α 5 mg. ad α 3 0 mg. of dosages are gve to 4 ad 7 atets resectvely the the resoses follow the model EY ( ) µ + α;,,3,4 EY ( ) µ + α ;,,3, 4,5,6, Here µ deotes the geeral mea effect whch may be thought as follows: The huma body has tedecy to fght agast the fever, so the tme take by the medce to brg dow the temerature deeds o may factors lke body weght, heght, geeral health codto etc. of the atet. So µ deotes the geeral effect of all these factors whch s reset all the observatos. the termology of lear regresso model, µ deotes the tercet term whch s the value of the resose varable whe all the deedet varables are set to take value zero. exermetal desgs, the models wth tercet term are more commoly used ad so geerally we cosder these tyes of models. Also, we ca exress Y µ + α + ε;,,...,,,,..., where ε s the radom error comoet Y. t dcates the varatos due to ucotrolled causes whch ca fluece the observatos. We assume that ε s are detcally ad deedetly dstrbuted as N(0, ) wth E( ε ) 0, ( ε ) Var. Note that the geeral lear model cosdered s EY ( ) Xβ Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

3 for whch Y ca be wrtte as EY ( ) β. Whe all the etres X are 0 s or s, the ths model ca also be re-exressed the form of EY ( ) µ + α. Ths gves rse to some more ssues. Cosder ad rewrte EY ( ) β where β + ( β β) µ + α µ β β α β β. Now let us see the chages the structure of desg matrx ad the vector of regresso coeffcets. The model EY ( ) β µ + α ca ow be rewrtte as EY ( ) X* β * Cov( Y ) where β * ( µ, α, α,..., α ) s a vector ad X X* s a ( + ) matrx, ad X deotes the earler defed desg matrx whch - frst rows as (,0,0,,0), - secod rows as (0,,0,,0) -, ad - last rows as (0,0,0,,). Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 3

4 We earler assumed that rak ( X ) but ca we also say that rak ( X *) s also the reset case? Sce the frst colum of X* s the vector sum of all ts remag colums, so rak ( X *). t s thus aaret that all the lear arametrc fuctos of α, α,..., α are ot estmable. The questo ow arses that what kd of lear arametrc fuctos are estmable? Cosder ay lear estmator L wth ay C a Now EL ( ) a EY ( ) a ( µ + α ) µ a + a α µ ( C ) + Cα. Thus C C α s estmable f ad oly f 0,.e., C α s a cotrast. Thus, geeral ether estmable. α or ay µα,, α,..., α s estmable. f t s a cotrast, the t s Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 4

5 Ths effect ad outcome ca also be see from the followg exlaato based o the estmato of arameters µα,, α,..., α. Cosder the least squares estmato ˆ, µαˆ ˆ ˆ, α,..., α of µ, α, α,..., α resectvely. Mmze the sum of squares due to ε ( y µ α ) S to obta ˆ, µαˆ ˆ,..., α. ε ' s S ( a) 0 ( y µ α ) 0 µ S ( b) 0 ( y µ α ) 0,,,...,. α Note that (a) ca be obtaed from (b) or vce versa. So (a) ad (b) are learly deedet the sese that there are ( + ) ukows ad learly deedet equatos. Cosequetly ˆ, µαˆ,..., ˆ α do ot have a uque soluto. Same ales to the maxmum lkelhood estmato of µα,,... α.. f a sde codto that ˆ α 0 or α 0 s mosed the (a) ad (b) have a uque soluto as ˆ µ y yoo, ˆ α ˆ y µ o y y oo where. case, all the samle szes are same, the the codto αˆ 0 or α 0 reduces to ˆ α 0 or α 0. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 5

6 So the model y µ + α + ε eeds to be rewrtte so that all the arameters ca be uquely estmated. Thus Y µ + α + ε ( µ + α) + ( α α) + ε * * µ + α + ε where * µ µ + α * α α α α α ad α 0 * Ths s a rearameterzed form of the lear model. Thus a lear model whe X s ot of full rak, the the arameters do ot have uque estmates. such codtos, a restrcto α 0 (or equvaletly α 0 case all s are ot same) ca be added ad the the least squares (or maxmum lkelhood) estmators obtaed are uque. The model * * EY ( ) µ * + α ; α 0 s called a rearametrzato of the orgal lear model. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 6

7 Let us ow cosder the aalyss of varace wth addtoal costrat. Let Y β + ε,,,..., ;,,..., β + ( β β) + ε µ + α + ε wth µ β β, α β β, α 0, ad ε s are detcally ad deedetly dstrbuted wth mea 0 ad varace. The ull hyothess s H : α α... α 0 0 ad the alteratve hyothess s H : atleast oe α α for all,. Ths model s a oe-way layout the sese that the observatos y ' s are assumed to be affected by oly oe treatmet effect α. So the ull hyothess s equvalet to testg the equalty of oulato meas or equvaletly the equalty of treatmet effects. We use the rcal of least squares to estmate the arameters µα,, α,... α. Mmze the error sum of squares ε ( y µ α ) E wth resect to µα,, α,..., α. The ormal equatos are obtaed as Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 7

8 E 0 ( y µ α ) 0 µ or E 0 ( y µ α ) 0 α or µ + α y () µ + α y (,,..., ). () Usg α 0 () gves G ˆ µ y y oo where G y s the grad total of all the observatos. Substtutg ˆµ () gves ˆ α ˆ y µ T ˆ µ y y o oo wheret y s the treatmet total due to th effect α,.e., total of all the observatos recevg the th treatmet ad y o y. Now the ftted model s y ˆ µ + ˆ α ad the error sum of squares after substtutg ˆµ ad ˆ α E becomes Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 8

9 E ( y ˆ µ ˆ α ) ( y yoo) ( yo yoo) ( y yoo) ( yo yoo) G T G y where the total sum of squares ( TSS ) TSS ( y y ) G y oo, ad G s called as correcto factor ( CF ). To obta a measure of varato due to treatmets, let H0 α α... α 0 be true. The the model becomes Y µ + ε,,,..., ;,,...,. Mmzg the error sum of squares E ( y µ ) wth resect to µ, the ormal equato s obtaed as E 0 ( y µ ) 0 µ or G ˆ µ y oo. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 9

10 Substtutg ˆ µ E, the error sum of squares becomes E ( y ˆ µ ) ( y y ) G y. oo Note that E : Cotas varato due to treatmet ad error both E: Cotas varato due to treatmet oly So E E: cota varato due to treatmet oly. The sum of quares due to treatmet ( SSTr ) s gve by SSTr E E SSTr ( y y ) o oo T G. The followg quatty s called the error sum of squares or sum of squares due to error (SSE) ( o). SSE y y These sum of squares forms the bass for the develomet of tools the aalyss of varace. We ca wrte TSS SSTr + SSE. The dstrbuto of degrees of freedom amog these sum of squares s as follows: The total sum of squares s based o quattes subect to the costrat that ( y yoo) 0 so TSS carres ( ) degrees of freedom. The sum of squares due to the treatmets s based o quattes subect to the costrat Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 0

11 ( yo yoo) 0so SSTr has ( ) degrees of freedom. The sum of squares due to errors s based o quattes subect to costrats ( y y ) 0,,,..., o so SSE carres ( ) degrees of freedom. Also ote that TSS SSTr + SSE, the TSS has bee dvded to two orthogoal comoets - SSTr ad SSE. Moreover, all TSS, SSTr ad SSE ca be exressed a quadratc form. Sce ε are assumed to be detcally ad deedetly dstrbuted followg N(0, ), so y are also deedetly dstrbuted followg N µ α ( +, ). Now usg the theorems 7 ad 8 wth q SSTr, q SSE, we have uder H, 0 SSTr ad SSE ~ χ ( ) χ ~ ( ). Moreover, SSTr ad SSE are deedetly dstrbuted. The mea squares s defed as the sum of squares dvded by the degrees of freedom. So the mea square due to treatmet s SSTr MSTr ad the mea square due to error s SSE MSE. Thus, uder H, 0 Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

12 MSTr F ~ F(, ). MSE The decso rule s that reect H 0 f F > F α,, atα % level of sgfcace. f H 0 does ot hold true, the MSTr ~ ocetral F(,, δ ) MSE α whereδ s the ocetralty arameter. Note that the test statstc MSTr MSE ca also be obtaed from the lkelhood rato test. f H 0 s reected, the we go for multle comarso tests ad try to dvde the oulato to several grous havg the same effects. The aalyss of varace table s as follows: Source Degrees Sum of Mea sum F-value of varato of freedom squares of squares Treatmet SSTr MSTr MSTr MSE Error SSE MSE Total TSS Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

13 Now we fd the exectatos of SSTr ad SSE. E( SSTr) E ( yo yoo) where E {( µ α ε ) ( µ ε )} o oo α ε ε, ε ε ad 0. o oo E( SSTr) E + ( o oo) Sce { α ε ε } E ( α ) E ( εo εoo) E( ε ) Var( ε ) Var ε o o E( εoo) Var( εoo) Var ε E( ε ε ) Cov( ε, ε ) o oo o oo Cov εε. E( SSTr) α + or α + ( ) α SSTr + E α or ( ). E MSTr + Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 3

14 Next E( SSE) E ( y yo ) E { ( µ α ε ) ( µ α εo) } E ( ε εo) E + ( ε εo εε o) + or ( ). ( ) ( ) ( ) SSE or E E MSE Thus MSE s a ubased estmator of. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 4

15 Two way classfcato uder fxed effects model Suose the resose of a outcome s affected by the two factors A ad B. For examle, suose vartes of magoes are grow o dfferet lots of same sze each of the dfferet locatos. All lots are gve same treatmet lke equal amot of water, equal amout of fertlzer etc. So there are two factors the exermet whch affect the yeld of magoes. - Locato (A) - Varety of magoes (B) Such a exermet s called two factor exermet. The dfferet locatos corresod to the dfferet levels of A ad the dfferet vartes corresod to the dfferet levels of factor B. The observatos are collected o the bass of er lot. The combed effect of the two factors (A ad B our case) s called the teracto effect (of A ad B). Mathematcally, let a ad b be the levels of factors A ad B resectvely the a fucto f( ab, ) s called a fucto of o teracto f ad oly f there exsts fuctos gaad ( ) hb ( ) such that f( ab, ) ga ( ) + hb ( ). Otherwse the factors are sad to teract. For a fucto f( ab, ) of o teracto, f( a, b) ga ( ) + hb ( ) f( a, b) ga ( ) + hb ( ) f( a, b) f( a, b) ga ( ) ga ( ) ad so t s deedet of b. Such o teracto fuctos are called addtve fuctos. Now there are two otos: - Oly oe observato er lot s collected. - More tha oe observatos er lot are collected. f there s oly oe observato er lot the there caot be ay teracto effect amog the observtos ad we assume t to be zero. f there are more tha oe observatos er lot the teracto effect amog the observatos ca be cosdered. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 5

16 We cosder here two cases. Oe observato er lot whch the teracto effect s zero.. More tha oe observatos er lot whch the teracto effect s reset. Two way classfcato wthout teracto Let y be the resose of observato from th level of frst factor, say A ad th level of secod factor, say B. So assume Y are deedetly dstrbuted as Ths ca be rereseted the form of a lear model as EY ( ) µ where µ µ wth µ + ( µ µ ) + ( µ µ ) + ( µ µ µ + µ ) oo o oo o oo o o oo µ + α + β + γ oo α µ µ o oo β µ µ o oo γ µ µ µ + µ o o oo α ( µ µ ) 0 o oo β ( µ µ ) 0 o oo Here α : effect of th level of factor A or excess of mea of th level of A over the geeral mea. β : effect of th level of B or excess of mea of th level of B over the geeral mea. γ : teracto effect of th level of A ad th level of B. Here we assume γ 0 as we have oly oe observato er lot. N µ (, ),,...,,,,...,. We also assume that the model EY ( ) µ s a full rak model so that µ ad all lear arametrc fuctos of µ are estmable. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 6

17 The total umber of observatos are whch ca be arraged a two way calssfed table where the rows corresods to the dfferet levels of A ad the colum corresods to the dfferet levels of B. The observatos o Y ad the desg matrx X ths case are Y µ α α α β β β y y y y y y f the desg matrx s ot of full rak, the the model ca be rearameterzed. such a case, we ca start the aalyss by assumg that the model EY ( ) µ + α + β s obtaed after rearameterzato. There are two ull hyothess of terest: H H : α α... α 0 0α : β β... β 0 0β agast H α : at least oe α (,,..., ) s dfferet from others H β at least oe β (,,..., ) s dfferet from others. : Now we derve the least squares estmators (or equvaletly the maxmum lkelhood estmator) of µα, ad β,,,...,,,,..., by mmzg the error sum of squares ( ). E y µ α β The ormal equatos are obtaed as Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 7

18 E 0 ( y µ α β ) 0 µ E 0 ( y µ α β ) 0,,,..., α E 0 ( y µ α β ) 0,,,...,. β, Solvg the ormal equatos ad usg α 0 ad β 0, the least squares estmator are obtaaed as G ˆ µ y y T ˆ α y y y y y,,..., oo oo o oo ˆ B β y y y y y,,,..., oo oo o oo where T oo : treatmet totals due to th α effect,.e., sum of all the observatos recevg the th treatmet effect. B : block totals due to th β effect,.e., sum of all the observatos the th block. Thus the error sum of squares s SSE M E µα,, β ˆ ( y ˆ ˆ µ α β ) y yoo yo yoo yo y oo ( y yo yo yoo) y yoo yo yoo yo yoo ( ) ( ) ( ) + ( ) ( ) ( ) whch carres ( ) ( ) ( )( ) degrees of freedom. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 8

19 Next we cosder the estmato of µ ad β uder the ull hyothess H 0 α : α α... α 0 by mmzg the error sum of squares E ( y µ β ). The ormal equato are obtaed by E µ 0 E ad 0,,,..., β whch o solvg gves the least square estmates ˆ µ yoo ˆ β y y. o oo The sum of squares due to H0 α s M E M( y µ β ) µβ, µβ, ˆ ( y ˆ µ β ) yo yoo + y yo yo + yoo ( ) ( ). Sum of squares due to factor A Error sum of squares Thus the sum of squares due to devato from H 0α (or sum of squares due to rows or sum of squares are to factor A) ( o oo) o oo SSA y y y y ad carres ( ) ( )( ). degrees of freedom. Now we fd the estmates of µ ad α uder H0 β : β β... β 0 by mmzg E ( y µ α ). Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 9

20 The ormal equatos are E µ 0 E ad 0,,,..., α whch o solvg gve the estmators as ˆ µ y oo ˆ α y y. o oo The mmum value of the error sum of squares s obtaed by M E µα, ( y ˆ µ ˆ α ) ( y y ) o ( o oo) ( o o oo) y y + y y y + y Sum of squares due to factor B Error sum of squares The sum of squares due to devato from H0 β (or the sum of squares due to colums or sum of squares due to factor B) s SSB ( y y ) y y o oo o oo ad ts degrees of freedom are ( ) ( )( ). Note that the total sum of squares s ( oo) ( yo yoo) ( yo yoo) ( y yo yo y ) oo ( yo yoo) ( yo yoo) ( y yo yo yoo) TSS y y SSA + SSB + SSE. The arttog of degrees of freedom to the corresodg grous s ( ) + ( ) + ( )( ). Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 0

21 Note that SSA, SSB ad SSE are mutually orthogoal ad that s why the degrees of freedom ca be dvded lke ths. Now usg the theory exlaed whle dscussg the lkelhood rato test or assumg y ' s to be deedetly dstrbuted as N µ + α + β afd usg the Theorems (, ),,,..., ;,,...,, 6 ad 7, we ca wrte SSA SSB SSE ~ χ ( ) χ ~ ( ) χ ~ (( )( )). So the test statstc for H0 α s obtaed as SSA / F SSE / ( )( ) ( )( ) SSA. ( ) SSE MSA ~ F (( ), ( ) ( )) uder H MSE where SSA MSA SSE MSE. ( )( ) 0α Same statstc s also obtaed usg the lkelhood rato test for H0 α. The decso rule s 0α α [ ] Reect H f F > F ( ), ( ) ( ). Uder H α, F α follows a ocetral F dstrbuto F ( δ, ( ), ( )( )) where δ the assocated ocetralty arameter. s Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

22 Smlarly, the test statstc for H0 β s obtaed as SSB / F SSE / ( )( ) ( )( ) SSB ( ) SSE MSB ~ F (( ),( )( )) uder H MSE SSB where MSB. The decso rule s Reect H f F > F (( ),( )( )). 0β α 0β The same test statstc ca also be obtaed from the lkelhood rato test. The aalyss of varace table s as follows: Source of Degrees Sum of Mea sum F-value varato of freedom squares of squares Factor A (or rows) ( ) SSA MSA F MSA MSE Factor B (or colum) ( ) SSB MSB F Error ( )( ) SSE MSE (by subtracto) Total TSS MSB MSE t ca be foud o smlar les as the case of oe way classfcato that E( MSA) + ( ). α E( MSB) + E MSE β Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur

23 f the ull hyothess s reected, the we use the multle comarso tests to dvde the α ' s (or β ' s) to grous such that α ' s (or β ' s) belogg to the same grou are equal ad those belogg to dfferet grous are dfferet. Geerally, ractce, the terest of exermeter s more usg the multle comarso test for treatmet effects rather o the block effects. So the multle comarso test are used geerally for the treatmet effects oly. Two way classfcato wth teractos: Cosder the two way classfcato wth a equal umber, say K observatos er cell. Let yk : k th observato (, ) th cell,.e., recevg the treatmets th level of factor A ad th level of factor B,,,..., ;,,..., k ;,,..., Kad y are deedetly draw from k y k µ + εk N µ so that the lear model uder cosderato s (, ) whereε k are detcally ad deedetly dstrbuted followg N(0, ). Thus E( y ) µ where wth k µ µ µ + ( µ µ ) + ( µ µ ) + ( µ µ µ + µ ) oo α µ µ o oo o oo oo o oo o oo o o oo µ + α + β + γ β µ µ γ µ µ µ + µ o o oo α 0, β 0, γ 0, γ 0. Assume that the desg matrx X s of full rak so that all the arametrc fuctos of µ are estmable. The ull hyothess are H H H 0α 0β 0γ : α α... α 0 : β β... β 0 :All γ 0 for all,. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 3

24 The corresodg alteratve hyothess s H : At least oe α α, for α H : At least oe β β, for β H : At least oe γ γ, for k. γ Mmzg the error sum of squares K ( k ), k E y µ α β γ The ormal equatos are obtaed as k E E E E 0, 0 for all, 0 for all ad 0 for all ad µ α β γ The least squares estmates are obtaed as ˆ µ yooo K k ˆ α y y y y oo ooo k ooo K ˆ β y y y y ˆ γ oo ooo k ooo K y y y + y K K y y y + y o oo oo ooo y k k oo oo ooo. The error sum of square s SSE M ( y µ α β γ ) ˆ µ, ˆ α, ˆ β, ˆ γ k K ( y ˆ µ ˆ α ˆ β ˆ γ ) k K ( y y ) k SSE K k wth ~ ( ( )). k k o χ K Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 4

25 Now mmzg the error sum of squares uder H 0α α α... α 0,.e., mmzg K k E ( y µ β γ ) k wth resect to µβ, ad γ ad solvg the ormal equatos E E E 0, 0 for all ad 0 for all ad µ β γ gves the least squares estmates as ˆ µ yˆ ooo ˆ β y y oo ooo ˆ γ y y y + y. o ooo oo ooo The sum of squares due to H0 α, s M µβ,, γ K k K k ( y µ β γ ) k ( y ˆ µ ˆ β ˆ γ ) k K ( yk yo ) K( yoo yooo) k + ( oo ooo). SSE + K y y Thus the sum of squares due to devato from H0 α or the sum of squares due to effect A s SSA Sum of squares due to H SSE K ( y y ) SSA χ wth ~ ( ). 0α Mmzg the error sum of squares uder H0 β : β β... β 0.e., mmzg K E ( y k µ α γ ), k ad solvg the ormal equatos E E E 0, 0 for all ad 0 for all ad µ α γ yelds the least squares estmators as oo ooo Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 5

26 ˆ µ y ooo ˆ α y y ooo ooo ˆ γ y y y + y. o oo oo ooo The mmum error sum of squares s K k ( y ˆ µ ˆ α ˆ γ ) k SSE + K ( y y ) oo ooo ad the sum of squares due to devato from wth ~ ( ). SSB Sum of squares due to H SSE K ( y y ) SSB χ 0β Ho β or the sum of squares due to effect B s oo ooo Next, mmzg the error sum of squares uder H0 γ : all γ 0 for all,,.e., mmzg E ( y µ α β ) 3 K k k wth resect to µα, ad β ad solvg the ormal equatos E3 E3 E3 0, 0 for all ad 0 for all µ α β yelds the least squares estmators ˆ µ y ooo ˆ α y y oo ooo ˆ β y y. oo ooo The sum of squarers due to H0 γ s K M ( yk µ α β ) µα,, β k K ˆ ( y ˆ ˆ k µ α β ) k SSE + K( yo yoo yoo + yooo). Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 6

27 Thus the sum of squares due to devato from H0 γ or the sum of squares due to teracto effect AB s SSAB Sum of squares due to H SSE K ( y y y + y ) SSAB χ wth ~ (( ) )). 0γ o oo oo ooo The total sum of squares ca be arttoed as TSS SSA + SSB + SSAB + SSE where SSA, SSB, SSAB ad SSE are mutually orthogoal. So ether usg the deedece of SSA, SSB, SSAB ad SSE as well as ther resectve χ dstrbutos or usg the lkelhood rato test aroach, the decso rules for the ull hyothess at α level of sgfcace are based o F- statstc as follows ( K ) SSA F. ~ F [ (, ( K ) ] uder H0 α, SSE ( K ) SSB F. ~ F [ (, ( K ) ] uder H0β, SSE ad ( K ) SSAB F3. ~ F [( )( ), ( K ) ] uder H0γ. ( )( ) SSE So Reect H0 α f F > F α[ ( ), ( K ) ] Reect H0β f F > F α[ ( ), ( K ) ] Reect H f F > F ( )( ), ( K ). 0γ 3 α [ ] f H0 α or H0 β s reected, oe ca use t -test or multle comarso test to fd whch ars of α ' s or β ' s are sgfcatly dfferet. f H0 γ s reected, oe would ot usually exlore t further but theoretcally t- test or multle comarso tests ca be used. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 7

28 t ca also be show that K E( SSA) + ( ). α K E( SSB) + E( SSAB) + E SSE β K γ ( )( ) The aalyss of varace table s as follows: Source of Degrees Sum of Mea sum F-value varato of freedom squares of squares Factor A ( ) SSA Factor B ( ) SSB teracto AB ( )( ) SSAB Error ( K ) SSE MSA SSA F MSA MSE SSB MSB MSB F MSE SSAB MSAB MSAB F3 ( )( ) MSE SSE MSE ( K ) Total ( K ) TSS Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 8

29 Tukey s test for oaddtvty: Cosder the set u of two way classfcato wth oe observato er cell ad teracto as y µ + α + β + γ + ε,,...,,,,..., wth α 0, β 0. The dstrbuto of degrees of freedom ths case s as follows: Source Degrees of freedom A B AB(teracto) ( )( ) Error 0 Total There s o degree of freedom for error. The roblem s that the two factor teracto effect ad radom error comoet are subsumed together ad caot be searated out. There s o estmate for. f o teracto exsts, the H : γ 0 for all, s acceted ad the addtve model 0 y µ + α + β + ε s well eough to test the hyothess H : α 0 ad H : β wth error havg ( )( ) degrees of freedom. f teracto exsts, the H : γ 0 s reected. such a case, f we assume that the structure of 0 teracto effect s such that t s roortoal to the roduct of dvdual effects,.e., γ λα β the a test for testg H : 0 0 λ ca be costructed. Such a test wll serve as a test for oaddtvty. t wll hel kowg the effect of resece of teract effect ad whether the teracto eters to the model addtvely. Such a test s gve by Tukey s test for oaddtvty whch requres oe degree of freedom leavg ( -)( -) - degrees of freedom for error. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 9

30 Let us assume that dearture from addtvty ca be secfed by troducg a roduct term ad wrtg the model as E( y ) µ + α + β + λα β ;,,...,,,,..., wth α 0, β 0. Whe λ 0, the model becomes olear model ad the least squares theory for lear models s ot alcable. Note that usg α 0, β 0, we have y y µ + α + β + λα β + ε oo λ µ + α + β + ( α )( β ) + ε µ + ε oo E( y ) µ oo ˆ µ y. oo oo Next y y µ + α + β + λα β + ε o µ + α + β + λα β + ε µ + α + ε E( y ) µ + α o o o ˆ α y ˆ µ y y. o o oo Smlarly y o µ + β ˆ β y ˆ µ y y o o oo Thus ˆ, µαˆ ad ˆβ rema the ubased estmators of µα, ad β, resectvely rresectve of whether λ 0 or ot. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 30

31 Also E y yo yo + y oo λαβ or E ( y yoo) ( yo yoo) ( yo yoo) λαβ. Cosder the estmato of µα,, β ad λ based o the mmzato of S ( y µ α β λα β ) S. The ormal equatos are solved as S 0 S 0 µ ˆ µ y S 0 ( + λβ ) S 0 α S 0 ( + λα ) S 0 β oo S 0 αβ S 0 λ or αβ ( y µ α β λαβ ) 0 αβ y or λ λ (say α β ) whch ca be estmated rovded α ad β are assumed to be kow. Sce α ad β ca be estmated by ˆα y y ad ˆβ y y rresectve of whether λ 0 o oo o oo or ot, so we ca substtute them lace of α ad β λ whch gves Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 3

32 ˆ αβˆ ˆ ˆ y ( ) αβ y ˆ λ ˆ ˆ ˆ α ˆ β α β ( y y )( y y ) y o oo o oo SS ˆ A α o oo where S ( y y ) A S y y B ˆ B β ( o oo Assumg α ad β to be kow ). Var ( λ) ( ) 0 αβ Var y + αβ α β α β α β usg Var( y ), ( Cov y, y ) 0 for all k. k Whe α ad β are estmated by ˆα ad ˆ β, the substtute them back the exresso of Var( λ) ad treatg t as Var( ˆ λ ) gves ˆ Var( λ) ˆ ˆ α β SS A for gve ˆα ad ˆβ. B Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 3

33 Note that f λ 0, the αβ y ˆ ˆ E λ/ ˆ α, β for all, E αβ αβ ( µ + α + β ε ) E ( α )( β ) 0 0. ( α ) ( β ) As ˆα ad ˆβ remas vald rresectve of λ 0 or ot, ths sese ˆλ s a fucto of y ad hece ormally dstrbuted as ˆ λ ~ N 0,. SS A B Thus the statstc ˆ ( λ) Var( ˆ λ) N A B A B ( yo yoo)( yo yoo)( y yo yo + yoo) SS S follows a S N ˆ ˆ αβ y SS ( yo yoo)( yo yoo) y SS A χ - dstrbuto wth oe degree of freedom where ( yo yoo)( yo yoo)( y yo yo + yoo) SS s the sum of squares due to oaddtvty. A B B Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 33

34 Note that S ( y y y + y ) o o oo AB follows χ (( )( )). so SN SAB χ. s oegatve ad follows [ ( )( ) ] The reaso for ths s as follows: y µ + α + β + o addtvty + ε ad so TSS SSA + SSB + SN + SSE SSE TSS SSA SSB S has degrees of freedom ( ) ( ) ( ) ( )( ) We eed to esure that SSE > 0. So usg the result f QQ, ad Q are quadratc forms such that N Q Q+ Q wth Q~ χ ( a), Q ~ χ ( b) ad Q s o-egatve, the Q ~ χ ( a b)" esures that the dfferece SN SAB s oegatve. Moreover S N (SS due to oaddtvty) ad SSE are orthogoal. Thus the F-test for oaddtvty s S N / F SSE / ( )( ) SSN SSE ~ F,( )( ) uder H. [( )( ) ] [ ] 0 Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 34

35 So the decso rule s Reect H : λ 0 wheever 0 [ ] F> F α, ( )( ) The aalyss of varace table for the model cludg a term for oaddtvty s as follows: Source of Degrees Sum of Mea sum F-value varato of freedom squares of squares A B S A S B Noaddtvty N MS MS A B S A SB S MSN SN MS N MSE Error ( )( ) SSE (By substracto) SSE MSE ( )( ) Total TSS Comarso of Varaces Oe of the basc assumtos the aalyss of varace s that the samles are draw from dfferet ormal oulatos wth dfferet meas but same varaces. So before gog for aalyss of varace, the test of hyothess about the equalty of varace s eeded to be doe. We dscuss the test of equalty of two varaces ad more tha two varaces. Case : Equalty of two varaces H :. 0 Suose there are two deedet radom samles A: x, x,..., x ; x ~ N( µ, ) A A B: y, y,..., y ; y ~ N( µ, ) B B The samle varace corresodg to the two samles are Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 35

36 s x x x ( ) s ( y y). y Uder H0 : A B, ( ) s ( ) s x χ y χ ~ ( ) ~ ( ). Moreover, the samle varaces s ad s ad are deedet. So x y ( ) s x ( ) s y s s x y ~ F., So for testg H0 : versus H:, the ull hyothess H 0 s reected f F > F or F < F where F α α ;, ; ;, α ;, F α. ;, f the ull hyothess H : s reected, the the roblem s termed as the Fster-Behre s 0 roblem. The solutos are avalable for ths roblem. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 36

37 Case : Equalty of more tha two varaces: Bartlett s test H :... k ad 0 H k : for atleast oe,,...,. Let there be k deedet ormal oulato N µ each of sze,,,..., k. Let (, ) s, s,..., s k be k deedet ubased estmators of oulato varaces,,..., k resectvely wth ν, ν,..., ν k degrees of freedom. Uder H 0, all the varaces are same as, say ad a ubased estmate of s s ν s where,. k k ν ν ν ν Bartlett has show that uder H 0 k s ν l s k + 3( ) k ν ν s dstrbuted as χ ( k ) based o whch H 0 ca be tested. Aalyss of Varace Chater 3 Exermetal Desg Models Shalabh, T Kaur 37