Chapter 4 Experimental Design and Their Analysis

Transcription

1 Chapter 4 Expermental Desgn and her Analyss Desgn of experment means how to desgn an experment n the sense that how the obseratons or measurements should be obtaned to answer a query n a ald, effcent and economcal way he desgnng of experment and the analyss of obtaned data are nseparable If the experment s desgned properly keepng n mnd the queston, then the data generated s ald and proper analyss of data prodes the ald statstcal nferences If the experment s not well desgned, the aldty of the statstcal nferences s questonable and may be nald It s mportant to understand frst the basc termnologes used n the expermental desgn Expermental unt: For conductng an experment, the expermental materal s dded nto smaller parts and each part s referred to as expermental unt he expermental unt s randomly assgned to a treatment s the expermental unt he phrase randomly assgned s ery mportant n ths defnton Experment: A way of gettng an answer to a queston whch the expermenter wants to know reatment Dfferent objects or procedures whch are to be compared n an experment are called treatments Samplng unt: he object that s measured n an experment s called the samplng unt hs may be dfferent from the expermental unt Factor: A factor s a arable defnng a categorzaton A factor can be fxed or random n nature A factor s termed as fxed factor f all the leels of nterest are ncluded n the experment A factor s termed as random factor f all the leels of nterest are not ncluded n the experment and those that are can be consdered to be randomly chosen from all the leels of nterest Replcaton: It s the repetton of the expermental stuaton by replcatng the expermental unt Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

2 Expermental error: he unexplaned random part of araton n any experment s termed as expermental error An estmate of expermental error can be obtaned by replcaton reatment desgn: A treatment desgn s the manner n whch the leels of treatments are arranged n an experment Example: (Ref: Statstcal Desgn, G Casella, Chapman and Hall, 008) Suppose some aretes of fsh food s to be nestgated on some speces of fshes he food s placed n the water tanks contanng the fshes he response s the ncrease n the weght of fsh he expermental unt s the tank, as the treatment s appled to the tank, not to the fsh Note that f the expermenter had taken the fsh n hand and placed the food n the mouth of fsh, then the fsh would hae been the expermental unt as long as each of the fsh got an ndependent scoop of food Desgn of experment: One of the man objectes of desgnng an experment s how to erfy the hypothess n an effcent and economcal way In the contest of the null hypothess of equalty of seeral means of normal populatons hang same arances, the analyss of arance technque can be used Note that such technques are based on certan statstcal assumptons If these assumptons are olated, the outcome of the test of hypothess then may also be faulty and the analyss of data may be meanngless So the man queston s how to obtan the data such that the assumptons are met and the data s readly aalable for the applcaton of tools lke analyss of arance he desgnng of such mechansm to obtan such data s acheed by the desgn of experment After obtanng the suffcent expermental unt, the treatments are allocated to the expermental unts n a random fashon Desgn of experment prodes a method by whch the treatments are placed at random on the expermental unts n such a way that the responses are estmated wth the utmost precson possble Prncples of expermental desgn: here are three basc prncples of desgn whch were deeloped by Sr Ronald A Fsher () Randomzaton () Replcaton () Local control Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

3 () Randomzaton he prncple of randomzaton noles the allocaton of treatment to expermental unts at random to aod any bas n the experment resultng from the nfluence of some extraneous unknown factor that may affect the experment In the deelopment of analyss of arance, we assume that the errors are random and ndependent In turn, the obseratons also become random he prncple of randomzaton ensures ths he random assgnment of expermental unts to treatments results n the followng outcomes a) It elmnates the systematc bas b) It s needed to obtan a representate sample from the populaton c) It helps n dstrbutng the unknown araton due to confounded arables throughout the experment and breaks the confoundng nfluence Randomzaton forms a bass of ald experment but replcaton s also needed for the aldty of the experment If the randomzaton process s such that eery expermental unt has an equal chance of receng each treatment, t s called a complete randomzaton () Replcaton: In the replcaton prncple, any treatment s repeated a number of tmes to obtan a ald and more relable estmate than whch s possble wth one obseraton only Replcaton prodes an effcent way of ncreasng the precson of an experment he precson ncreases wth the ncrease n the number of obseratons Replcaton prodes more obseratons when the same treatment s used, so t ncreases precson For example, f arance of x s σ than arance of sample mean x based on n obseraton s σ n So as n ncreases, Var( x ) decreases () Local control (error control) he replcaton s used wth local control to reduce the expermental error For example, f the expermental unts are dded nto dfferent groups such that they are homogeneous wthn the blocks, than the araton among the blocks s elmnated and deally the error component wll contan the araton due to the treatments only hs wll n turn ncrease the effcency Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 3

4 Complete and ncomplete block desgns: In most of the experments, the aalable expermental unts are grouped nto blocks hang more or less dentcal characterstcs to remoe the blockng effect from the expermental error Such desgn are termed as block desgns he number of expermental unts n a block s called the block sze If sze of block = number of treatments and each treatment n each block s randomly allocated, then t s a full replcaton and the desgn s called as complete block desgn In case, the number of treatments s so large that a full replcaton n each block makes t too heterogeneous wth respect to the characterstc under study, then smaller but homogeneous blocks can be used In such a case, the blocks do not contan a full replcate of the treatments Expermental desgns wth blocks contanng an ncomplete replcaton of the treatments are called ncomplete block desgns Completely randomzed desgn (CRD) he CRD s the smplest desgn Suppose there are treatments to be compared All expermental unts are consdered the same and no dson or groupng among them exst In CRD, the treatments are allocated randomly to the whole set of expermental unts, wthout makng any effort to group the expermental unts n any way for more homogenety Desgn s entrely flexble n the sense that any number of treatments or replcatons may be used Number of replcatons for dfferent treatments need not be equal and may ary from treatment to treatment dependng on the knowledge (f any) on the arablty of the obseratons on nddual treatments as well as on the accuracy requred for the estmate of nddual treatment effect Example: Suppose there are 4 treatments and 0 expermental unts, then - the treatment s replcated, say 3 tmes and s gen to 3 expermental unts, - the treatment s replcated, say 5 tmes and s gen to 5 expermental unts, - the treatment 3 s replcated, say 6 tmes and s gen to 6 expermental unts and - fnally, the treatment 4 s replcated [0-(6+5+3)=]6 tmes and s gen to the remanng 6 expermental unts Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 4

5 All the arablty among the expermental unts goes nto expermented error CRD s used when the expermental materal s homogeneous CRD s often neffcent CRD s more useful when the experments are conducted nsde the lab CRD s well suted for the small number of treatments and for the homogeneous expermental materal Layout of CRD Followng steps are needed to desgn a CRD: Dde the entre expermental materal or area nto a number of expermental unts, say n Fx the number of replcatons for dfferent treatments n adance (for gen total number of aalable expermental unts) No local control measure s proded as such except that the error arance can be reduced by choosng a homogeneous set of expermental unts Procedure Let the treatments are numbered from,,, and n be the number of replcatons requred for th treatment such that = n = n Select n unts out of n unts randomly and apply treatment to these n unts (Note: hs s how the randomzaton prncple s utlzed s CRD) Select n unts out of ( n n ) unts randomly and apply treatment to these n unts Contnue wth ths procedure untl all the treatments hae been utlzed Generally equal number of treatments are allocated to all the expermental unts unless no practcal lmtaton dctates or some treatments are more arable or/and of more nterest Analyss here s only one factor whch s affectng the outcome treatment effect So the set-up of one-way analyss of arance s to be used y : Inddual measurement of j th expermental unts for th treatment =,,,, j =,,, n y : Independently dstrbuted followng µ : oerall mean N µ + wth ( α, σ ) nα = 0 = Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 5

6 α : th treatment effect H0 α α α : : = = = = 0 H All ' α are not equal s he data set s arranged as follows: reatments y y y y y y y y y n n n where n = y s the treatment total due to th effect, j= n s the grand total of all the obseratons = = j= G = = y In order to dere the test for H 0, we can use ether the lkelhood rato test or the prncple of least squares Snce the lkelhood rato test has already been dered earler, so we choose to demonstrate the use of least squares prncple he lnear model under consderaton s y = µ + α + ε, =,,,, j =,,, n where ε ' s are dentcally and ndependently dstrbuted random errors wth mean 0 and arance σ he normalty assumpton of ε s s not needed for the estmaton of parameters but wll be needed for derng the dstrbuton of arous noled statstcs and n derng the test statstcs Let n n ε µ α = j= = j= S = = ( y ) Mnmzng S wth respect to µ and α, the normal equatons are obtaned as Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 6

7 S = 0 nµ + nα = 0 µ = n S = 0 nµ + nα = y =,,, α j= Solng them usng ˆ µ = y oo ˆ α = y y o oo = nα = 0, we get where y o n = y s the mean of obseraton receng the th treatment and n j = of all the obseratons y oo n n = j = = y s the mean he ftted model s obtaned after substtutng the estmate ˆµ and ˆ α n the lnear model, we get y ˆ ˆ ˆ = µ + α + ε or y = yoo + ( yo yoo) + ( y yo ) or ( y y ) = ( y y ) + ( y y) oo o oo Squarng both sdes and summng oer all the obseraton, we hae n n ( y yoo) = n ( yo yoo) + ( y yoo) = j= = = j= Sum of squares otal sum Sum of squares or = due to treatment + of squares due to error effects or SS = SSr + SSE Snce n ( y yoo) = 0, so SS s based on the sum of ( n ) squared quanttes he SS = j= carryes only ( n ) degrees of freedom Snce o oo = n ( y y ) = 0, so SSr s based only on the sum of ( -) squared quanttes he SSr carres only ( -) degrees of freedom Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 7

8 Snce n n ( y yo ) = 0 for all =,,,, so SSE s based on the sum of squarng n quanttes lke = ( y y ) wth constrants o n j= ( y y ) = 0, So SSE carres (n ) degrees of freedom o Usng the Fsher-Cochram theorem, SS = SSr + SSE wth degrees of freedom parttoned as (n ) = ( - ) + (n ) Moreoer, the equalty n SS = SSr + SSE has to hold exactly In order to ensure that the equalty holds exactly, we fnd one of the sum of squares through subtracton Generally, t s recommended to fnd SSE by subtracton as SSE = SS - SSr n SS = ( y y ) where G = j= = j= n G = y n n = = j= n j= y SSr = n ( y y ) o o oo G = = n n where = G n n j= y : correcton factor Now under H 0 : α = α = = α = 0, the model become Y = µ + ε, and mnmzng S n = = j= ε wth respect to µ ges Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 8

9 S 0 ˆ G = µ = = yoo µ n he SSE under H 0 becomes n SSE = ( y y ) = j= and thus SS = SSE oo hs SS under H 0 contans the araton only due to the random error whereas the earler SS = SSr + SSE contans the araton due to treatments and errors both he dfference between the two wll prodes the effect of treatments n terms of sum of squares as ( oo) = SSr = n y y Expectatons n o = j= n ( ε εo) = j= n E ε ne εo = j= = σ nσ n = n E( SSE) = E( y y ) = = ( ) ( ) = = ( n ) σ SSE E( MSE) = E = σ n ( ) = ( o oo) = = ne ( α + εo εoo) = nα nεo nεoo = = σ σ = nα + n n = = n n = nα + ( ) σ = SStr = = α + σ = E SSr n E y y = + E( MSr) E n Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 9

10 In general ( ) E( MSr) E MSr = σ σ but under H, 0 all α = 0 and so Dstrbutons and decson rules: Usng the normal dstrbuton property of ε ' s, we fnd that y ' s are also normal as they are the lnear combnaton of ε ' s SSr ~ χ ( ) under H 0 σ SSE ~ χ ( n ) under H 0 σ SSr and SSE are ndependently dstrbuted MStr ~ F (, n ) under H0 MSE Reject H at * leel of sgnfcance f F > F α n 0 α *;, [Note: We denote the leel of sgnfcance here by α * because α has been used for denotng the factor] he analyss of arance table s as follows Source of Degrees Sum of Mean sum F araton of freedom squares of squares Between treatments - SSr MSr MSr MSE Errors n - SSE MSE otal n - SS Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 0

11 Randomzed Block Desgn If large number of treatments are to be compared, then large number of expermental unts are requred hs wll ncrease the araton among the responses and CRD may not be approprate to use In such a case when the expermental materal s not homogeneous and there are treatments to be compared, then t may be possble to group the expermental materal nto blocks of szes unts Blocks are constructed such that the expermental unts wthn a block are relately homogeneous and resemble to each other more closely than the unts n the dfferent blocks If there are b such blocks, we say that the blocks are at b leels Smlarly f there are treatments, we say that the treatments are at leels he responses from the b leels of blocks and leels of treatments can be arranged n a two-way layout he obsered data set s arranged as follows: Blocks Block otals b y y y y b B = y o y y y y b B = y o reatments j y j y j y y bj B j = y oj y y y y b B b = y ob reatment otals = y o =y o =y o y o Grand otal G= y oo Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

12 Layout: A two-way layout s called a randomzed block desgn (RBD) or a randomzed complete block desgn (RCB) f wthn each block, the treatments are randomly assgned to expermental unts such that each of the! ways of assgnng the treatments to the unts has the same probablty of beng adopted n the experment and the assgnment n dfferent blocks are statstcally ndependent he RBD utlzes the prncples of desgn - randomzaton, replcaton and local control - n the followng way: Randomzaton: - Number the treatments,,, - Number the unts n each block as,,, - Randomly allocate the treatments to expermental unts n each block Replcaton Snce each treatment s appearng n the each block, so eery treatment wll appear n all the blocks So each treatment can be consdered as f replcated the number of tmes as the number of blocks hus n RBD, the number of blocks and the number of replcatons are same 3 Local control Local control s adopted n RBD n followng way: - Frst form the homogeneous blocks of the xpermental unts - hen allocate each treatment randomly n each block he error arance now wll be smaller because of homogeneous blocks and some arance wll be parted away from the error arance due to the dfference among the blocks Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

13 Example: Suppose there are 7 treatment denoted as,,, 7 correspondng to 7 leels of a factor to be ncluded n 4 blocks So one possble layout of the assgnment of 7 treatments to 4 dfferent blocks n a RBD s as follows Block Block Block 3 Block Analyss Let y : Inddual measurements of j th treatment n th block, =,,,b, j =,,, y s are ndependently dstrbuted followng where µ : oerall mean effect β : th block effect τ j : j th treatment effect N µ β τ σ ( + + j, ) such that b β = 0, τ j = 0 = j= here are two null hypotheses to be tested - related to the block effects H : β = β = = β = 0 0B - related to the treatment effects H : τ = τ = = τ = 0 0 he lnear model n ths case s a two-way model as y = µ + β + τ + ε, =,,, b; j =,,, j b where ε are dentcally and ndependently dstrbuted random errors followng a normal dstrbuton wth mean 0 and arance σ Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 3

14 he tests of hypothess can be dered usng the lkelhood rato test or the prncple of least squares he use of lkelhood rato test has already been demonstrated earler, so we now use the prncple of least squares b b ε y µ β τ j = j= = j= Mnmzng S = = ( ) and solng the normal equaton S S S = 0, = 0, = 0 for all =,,, b, j =,,, µ β τ j the least squares estmators are obtaned as ˆ µ = y, oo ˆ β = y y, o oo ˆ τ = y y j oj oo he ftted model s y = ˆ µ + ˆ β + ˆ τ + ˆ ε j = y + ( y y ) + ( y y ) + ( y y y + y ) oo o oo oj oo o oj oo Squarng both sdes and summng oer and j ges b b b ( y yoo) = ( yo yoo) + b ( yoj yoo) + ( y yo yoj + yoo) = j= = j= = j= or SS = SSBl + SSr + SSE wth degrees of freedom parttoned as b = ( b ) + ( ) + ( b )( ) he reason for the number of degrees of freedom for dfferent sums of squares s the same as n the case of CRD b Here SS = ( y y ) = j= b G = y b = j= oo G : b correcton factor b G = y : Grand total of all the obseraton = j= Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 4

15 SSBl = ( y y ) j= = n = th B = y : block total o b B G = b b j= j= j G oo SSr = b ( y y ) oj = b b th = y : j treatment total j = oo b ( o oj oo) = j= SSE = y y y + y he expectatons of mean squares are SSBl E( MSBl) = E = σ + b b SSr b E( MSr) = E = σ + SSE E MSE = E = σ ( b )( ) b β = τ j j= ( ) Moreoer, SSBl σ SSr χ σ SSE σ ( b ) ~ χ ( b ) ( ) ~ ( ) ( b )( ) ~ ( b )( ) χ Under H : β = β = = β = 0, 0B E( MSBl) = E( MSE) and SSBl and SSE are ndependent, so MSBl Fbl = ~ F(( b,( b )( )) MSE Smlarly, under H0 : τ = τ = = τ = 0, so E( MSr) = E( MSE) b Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 5

16 and SSr and SSE are ndependent, so MSr Fr = ~ F ( ),( b )( )) MSE Reject H0 f F > F (( b ), ( b )( ) B be Reject H0 f F > F (( ),( b )( )) r α α If H 0B s accepted, then t ndcates that the blockng s not necessary for future expermentaton If H 0 s rejected then t ndcates that the treatments are dfferent hen the multple comparson tests are used to dde the entre set of treatments nto dfferent subgroup such that the treatments n the same subgroup hae the same treatment effect and those n the dfferent subgroups hae dfferent treatment effects he analyss of arance table s as follows Source of Degrees Sum of Mean F araton of freedom squares squares Blocks b - SSBl MSBl F Bl reatments - SSr MSr F r Errors (b - )( - ) SSE MSE otal b - SS Latn Square Desgn he treatments n the RBD are randomly assgned to b blocks such that each treatment must occur n each block rather than assgnng them at random oer the entre set of expermental unts as n the CRD here are only two factors block and treatment effects whch are taken nto account and the total number of expermental unts needed for complete replcaton are b where b and are the numbers of blocks and treatments respectely Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 6

17 If there are three factors and suppose there are b, and k leels of each factor, then the total number of expermental unts needed for a complete replcaton are bk hs ncreases the cost of expermentaton and the requred number of expermental unts oer RBD In Latn square desgn (LSD), the expermental materal s dded nto rows and columns, each hang the same number of expermental unts whch s equal to the number of treatments he treatments are allocated to the rows and the columns such that each treatment occurs once and only once n the each row and n the each column In order to allocate the treatment to the expermental unts n rows and columns, we take the help from Latn squares Latn Square: A Latn square of order p s an arrangement of p symbols n p cells arranged n p rows and p columns such that each symbol occurs once and only once n each row and n each column For example, to wrte a Latn square of order 4, choose four symbols A, B, C and D hese letters are Latn letters whch are used as symbols Wrte them n a way such that each of the letters out of A, B, C and D occurs once and only once s each row and each column For example, as A B C D B C D A C D A B D A B C hs s a Latn square We consder frst the followng example to llustrate how a Latn square s used to allocate the treatments and n gettng the response Example: Suppose dfferent brands of petrol are to be compared wth respect to the mleage per lter acheed n motor cars Important factors responsble for the araton n the mleage are - dfference between nddual cars - dfference n the drng habts of drers We hae three factors cars, drers and petrol brands Suppose we hae Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 7

18 - 4 types of cars denoted as,, 3, 4-4 drers that are represented by as a, b, c, d - 4 brands of petrol are ndcated by as A, B, C, D Now the complete replcaton wll requre 4 4 4= 64 number of experments We choose only 6 experments o choose such 6 experments, we take the help of Latn square Suppose we choose the followng Latn square: A B C D B C D A C D A B D A B C Wrte them n rows and columns and choose rows for cars, columns for drers and letter for petrol brands hus 6 obseratons are recorded as per ths plan of treatment combnaton (as shown n the next fgure) and further analyss s carred out Snce such desgn s based on Latn square, so t s called as a Latn square desgn Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 8

19 Another choce of a Latn square of order 4 s C B A D B C D A A D C B D A B C hs wll agan ge a desgn dfferent from the preous one he 6 obseratons wll be recorded agan but based on dfferent treatment combnatons Snce we use only 6 out of 64 possble obseratons, so t s an ncomplete 3 way layout n whch each of the 3 factors cars, drers and petrol brands are at 4 leels and the obseratons are recorded only on 6 of the 64 possble treatment combnatons hus n a LSD, the treatments are grouped nto replcaton n two-ways once n rows and and n columns, rows and columns aratons are elmnated from the wthn treatment araton In RBD, the expermental unts are dded nto homogeneous blocks accordng to the blockng factor Hence t elmnates the dfference among blocks from the expermental error In LSD, the expermental unts are grouped accordng to two factors Hence two effects (lke as two block effects) are remoed from the expermental error So the error arance can be consderably reduced n LSD he LSD s an ncomplete three-way layout n whch each of the the three factors, z, rows, columns and treatments, s at leels each and obseratons only on taken Each treatment combnaton contans one leel of each factor of the 3 possble treatment combnatons are he analyss of data n a LSD s condtonal n the sense t depends on whch Latn square s used for allocatng the treatments If the Latn square changes, the conclusons may also change We note that Latn squares play an mportant role s a LSD, so frst we study more about these Latn squares before descrbng the analyss of arance Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 9

20 Standard form of Latn square A Latn square s n the standard form f the symbols n the frst row and frst columns are n the natural order (Natural order means the order of alphabets lke A, B, C, D, ) Gen a Latn square, t s possble to rearrange the columns so that the frst row and frst column reman n natural order Example: Four standard forms of 4 4 Latn square are as follows A B C D A B C D A B C D A B C D B A D C B C D A B D A C B A D C C D B A C D A B C A D B C D A B D C A B D A B C D C B A D C B A For each standard Latn square of order p, the p rows can be permuted n p! ways Keepng a row fxed, ary and permute (p - ) columns n (p - )! ways So there are p!(p - )! dfferent Latn squares For llustraton Sze of square Number of Standard squares Value of p!( - p)! otal number of dfferent squares 3 x 3 4 x x x Conjugate: wo standard Latn squares are called conjugate f the rows of one are the columns of other For example A B C D A B C D B C D A and B C D A C D A B C D A B D A B C D A B C are conjugate In fact, they are self conjugate A Latn square s called self conjugate f ts arrangement n rows and columns are the same Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 0

21 ransformaton set: A set of all Latn squares obtaned from a sngle Latn square by permutng ts rows, columns and symbols s called a transformaton set From a Latn square of order p, p!(p - )! dfferent Latn squares can be obtaned by makng p! permutatons of columns and (p - )! permutatons of rows whch leaes the frst row n place hus Number of dfferent p!(p - )! X number of standard Latn Latn squares of order = squares n the set p n a transformaton set Orthogonal Latn squares If two Latn squares of the same order but wth dfferent symbols are such that when they are supermposed on each other, eery ordered par of symbols (dfferent) occurs exactly once n the Latn square, then they are called orthogonal Greco-Latn square: A par of orthogonal Latn squares, one wth Latn symbols and the other wth Greek symbols forms a Greco-Latn square For example A B C D B A D C C D A B D C B A α β γ δ δ γ β α β α δ γ γ δ α β s a Greco-Latn square of order 4 Greco Latn squares desgn enables to consder one more factor than the factors n Latn square desgn For example, n the earler example, f there are four drers, four cars, four petrol and each petrol has four aretes, as αβγ,, and δ, then Greco-Latn square helps n decdng the treatment combnaton as follows: Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

22 Cars 3 4 a Aα Bβ Cγ Dδ b Bδ Aγ Dβ Cα Drers c Cβ Dα Aδ Bγ d Dγ Cδ Bα Aβ Now Aα means: Drer a wll use the α arant of petrol A n Car Bγ means: Drer c wll use the γ arant of petrol B n Car 4 and so on Mutually orthogonal Latn square A set of Latn squares of the same order s called a set of mutually orthogonal Latn square (or a hyper Greco-Latn square) f eery par n the set s orthogonal he total number of mutually orthogonal Latn squares of order p s at most (p - ) Analyss of LSD (one obseraton per cell) In desgnng a LSD of order p, choose one Latn square at random from the set of all possble Latn squares of order p Select a standard latn square from the set of all standard Latn squares wth equal probablty Randomze all the rows and columns as follows: - Choose a random number, less than p, say n and then nd row s the n th row - Choose another random number less than p, say n and then 3 rd th row s the n row and so on - hen do the same for column For Latn squares of order less than 5, fx frst row and then randomze rows and then randomze columns In Latn squares of order 5 or more, need not to fx een the frst row Just randomze all rows and columns Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur

23 Example: Suppose followng Latn square s chosen A B C D E B C D E A D E A B C E A B C D C D E A B Now randomze rows, eg, 3 rd row becomes 5 th row and 5 th row becomes 3 rd row he Latn square becomes A B C D E B C D E A C D E A B E A B C D D E A B C Now randomze columns, say 5 th column becomes st column, st column becomes 5 th column E B C A D A C D B E D A B E C C E A D B B D E C A column becomes 4 th column and 4 th Now use ths Latn square for the assgnment of treatments y : Obseraton on k th treatment n th row and j th block, =,,,, j =,,,, k =,,, k rplets (I,j, k) take on only the experment y k s are ndependently dstrbuted as alues ndcated by the chosen partcular Latn square selected for the N µ + α + β + τ σ ( j k, ) Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 3

24 Lnear model s y = µ + α + β + τ + ε, =,,, ; j=,,, k ; =,,, k j k k where ε k are random errors whch are dentcally and ndependently dstrbuted followng N(0, σ ) wth α = 0, β = 0, τ = 0, k = j= k= α : man effect of rows β : man effect of columns γ k j : man effect of treatments he null hypothess under consderaton are H H H : α = α = = α = 0 0R : β = β = = β = 0 0C : τ = τ = = τ = 0 0 he analyss of arance can be deeloped on the same lnes as earler Mnmzng S = ε wth respect to µα,, β and τ k gen the least squares estmate as = j= k= k j ˆ µ = y ooo ˆ α = y y =,,, oo ooo ˆ β = y y j =,,, j ojo ooo ˆ τ = y y k =,,, k ook ooo Usng the ftted model based on these estmators, the total sum of squares can be parttoned nto mutually orthogonal sum of squares SSR, SSC, SSr and SSE as SS = SSR + SSC + SSr + SSE Where SS: otal sum of squares = G = ( yk yooo) yk = j= k= = j= k= SSR: Sum of squares due to rows = R = G ( oo ooo) = ; = k = j= k= y y R y Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 4

25 SSC: Sum of squares due to column = C j = G ( ojo ooo) = ; j = k j= = k= y y C y SSr : Sum of squares due to treatment = k = G ( ook ooo) = ; k = k k= = j= y y y Degrees of freedom carred by SSR, SSC and SSr are ( - ) each Degrees of freedom carred by SS s Degree of freedom carred by SSE s ( - ) ( - ) he expectatons of mean squares are obtaned as SSR E( MSR) = E = σ + α = SSC E( MSC) = E = σ + β j j= SSr E( MSr) = E = σ + SSE E MSE = E = σ ( )( ) τk k = ( ) hus MSR - under H0R, FR = ~ F(( ),( )( )) MSE MSC - under H0C, FC = ~ F(( ),( )( )) MSE MSr - under H0, F = ~ F(( ),( )( )) MSE Decson rules: Reject H at leel α f F > F 0R R α ; ( ),( )( ) Reject H at leel α f F > F 0C C α ;( ),( )( ) Reject H at leel α f F > F 0 α ;( ),( )( ) If any null hypothess s rejected, then use multple comparson test Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 5

26 he analyss of arance table s as follows Source of Degrees Sum of Mean sum F araton of freedom squares of squares Rows - SSR MSR F R Columns - SSC MSC F C reatments - SSr MSr F Error ( - )( - ) SSE MSE otal SS Mssng plot technques: It happens many tme n conductng the experments that some obseraton are mssed hs may happen due to seeral reasons For example, n a clncal tral, suppose the readngs of blood pressure are to be recorded after three days of gng the medcne to the patents Suppose the medcne s gen to 0 patents and one of the patent doesn t turn up for prodng the blood pressure readng Smlarly, n an agrcultural experment, the seeds are sown and yelds are to be recorded after few months Suppose some cattle destroys the crop of any plot or the crop of any plot s destroyed due to storm, nsects etc In such cases, one opton s to - somehow estmate the mssng alue on the bass of aalable data, - replace t back n the data and make the data set complete Now conduct the statstcal analyss on the bass of completed data set as f no alue was mssng by makng necessary adjustments n the statstcal tools to be appled Such an area comes under the purew of mssng data models and lot of deelopment has taken place Seeral books on ths ssue hae appeared, eg Lttle, RJA and Rubn, DB (00) Statstcal Analyss wth Mssng Data, nd edton, New York: John Wley Schafer, JL (997) Analyss of Incomplete Multarate Data Chapman & Hall, London etc We dscuss here the classcal mssng plot technque proposed by Yates whch nole the followng steps: Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 6

27 Estmate the mssng obseratons by the alues whch makes the error sum of squares to be mnmum Substtute the unknown alues by the mssng obseratons Express the error sum of squares as a functon of these unknown alues Mnmze the error sum of squares usng prncple of maxma/mnma, e, dfferentatng t wth respect to the mssng alue and put t to zero and form a lnear equaton Form as many lnear equaton as the number of unknown alues (e, dfferentate error sum of squares wth respect to each unknown alue) Sole all the lnear equatons smultaneously and solutons wll prode the mssng alues Impute the mssng alues wth the estmated alues and complete the data Apply analyss of arance tools he error sum of squares thus obtaned s corrected but treatment sum of squares are not corrected he number of degrees of freedom assocated wth the total sum of squares are subtracted by the number of mssng alues and adjusted n the error sum of squares No change n the degrees of freedom of sum of squares due to treatment s needed Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 7

28 Mssng obseratons n RBD One mssng obseraton: Suppose one obseraton n (, j) th cell s mssng and let ths be x he arrangement of obseratons n RBD then wll look lke as follows: Blocks Block otal b y y y y b B = y o y y y y b B = y o reatments j y j y j y = x y bj B j = x ' y oj + y y y y b B b = y ob reatment otals = y o = y o = y ' o + x y o Grand otal G = y + x ' oo where ' y oo : total of known obseratons y : total of known obseratons n j th block ' oj ' y o : total of known obseratons n th treatment ' ( G ') ( yoo + x) Correcton factor ( CF) = = n b b = j= ( x terms whch are constant wth respect to x) CF [( ' ) o terms whch are constant wth respect to ] SS = y CF = + SSBl = y + x + x CF b SSr [( ' = yoj + x) ++ terms whch are constant wth respect to x] CF SSE = SS SSBl SSr ' ' ' ( yoo + x) = x ( yo + x) ( yoj + x) + + (terms whch are constant wth respect to x) CF b b Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 8

29 Fnd x such that SSE s mnmum ' ' ' ( SSE) ( y ) ( ) o + x yoj + x ( yoo + x) = 0 x + = 0 x b b ' ' ' yo + byoj yoo or x = ( b )( ) wo mssng obseratons: If there are two mssng obseraton, then let they be x and y - Let the correspondng row sums (block totals) are ( R+ x) and ( R + y) - Column sums (treatment totals) are ( C+ x) and ( C + y) - otal of known obseratons s S hen SSE = x + y [( R + x) + ( R + y) ] [( C + x) + ( C + y) ] + ( S + x + y) b b + terms ndependent of x and y Now dfferentate SSE wth respect to x and y, as ( SSE) R+ x C+ x S + x + y = 0 x + = 0 x b b b ( SSE) R + y C + y S + x + y = 0 y + = 0 y b hus solng the followng two lnear equatons n x and y, we obtan the estmated mssng alues ( b )( ) x = br + C S y ( b )( ) y = br + C S x Adjustments to be done n analyss of arance () Obtan the wthn block sum of squares from ncomplete data () Subtract correct error sum of squares from () hs gen the correct treatment sum of squares () Reduce the degrees of freedom of error sum of squares by the number of mssng obseratons () No adjustments n other sum of squares are requred Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 9

30 Mssng obseratons n LSD Let - x be the mssng obseraton n (, j, k) th cell, e, y, =,,,, j=,,, k, =,,, k Now - R: otal of known obseratons n th row - C: otal of known obseratons n j th column - : otal of known obseraton receng the k th treatment - S: otal of known obseratons ( S+ x) Correcton factor ( CF) = otal sum of squares ( SS) = x + term whch are constant wth respect to x - CF ( R+ x) Row sum of squares ( SSR) = + term whch are constant wth respect to x - CF ( C+ x) Column sum of squares ( SSC) = + term whch are constant wth respect to x - CF ( + x) reatment sum of squares( SSr) = + term whch are constant wth respect to x - CF Sum of squares due to error ( SSE) = SS - SSR - SSC - SSr Choose x such that SSE s mnmum So d( SSE) = 0 dx 4( S+ x) x ( R+ C+ + 3 x) ) + V( R+ C+ ) S or x = ( )( ) ( S + x) = x ( R x) ( C x) ( x) Adjustment to be done n analyss of arance: Do all the steps as n the case of RBD o get the correct treatment sum of squares, proceed as follows: - Ignore the treatment classfcaton and consder only row and column classfcaton - Substtute the estmated alues at the place of mssng obseraton Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 30

31 - Obtan the error sum of squares from complete data, say SSE - Let SSE be the error sum of squares based on LSD obtaned earler - Fnd corrected treatment sum of squares = SSE SSE - Reduce of degrees of freedom of error sum of squares by the number of mssng alues Analyss of Varance Chapter 4 Extal Desgn and her Analyss Shalabh, II Kanpur 3