Chapter 6 Balanced Incomplete Block Design (BIBD)

Transcription

1 Chapte 6 Balanced Incomplete Bloc Design (BIBD) The designs lie CRD and RBD ae the complete bloc designs We now discuss the balanced incomplete bloc design (BIBD) and the patially balanced incomplete bloc design (PBIBD) which ae the incomplete bloc designs A balanced incomplete bloc design (BIBD) is an incomplete bloc design in which - b blocs hae the same numbe of plots each and - eey teatment is eplicated times in the design - Each teatment occus at most once in a bloc, ie, n i 0 o whee n is the numbe of i times the th teatment occus in i th bloc, i,,, b;,,, - Eey pai of teatments occus togethe is λ of the b blocs Such design is denoted by 5 paametes Dbλ (,,,; ) The paametes b,,, and λ ae not chosen abitaily They satisfy the following elations: ( I) b ( II) λ( ) ( ) ( III) b (and hence > ) Hence n fo all i i n fo all i and n n + n n + + n n λ fo all,,, Obiously i i b b So the design is not othogonal n i cannot be a constant fo all

2 Example of BIBD In the design D( b, ;, ; λ) : conside b 0 ( say, B,, B ), 6 ( say, T,, T ), 3, 5, λ Blocs Teatments B T T T 3 B T T T 4 B3 T T T 3 4 B4 T T T 4 6 B5 T T T 5 6 B6 T T T 3 6 B7 T T T 4 5 B8 T T T 5 6 B9 T T T B0 T T T Now we see how the conditions of BIBD ae satisfied ( i) b and b ( ii) λ( ) 5 0 and ( ) 5 0 λ( ) ( ) ( iii) b 0 6 Een if the paametes satisfy the elations, it is not always possible to aange the teatments in blocs to get the coesponding design The necessay and sufficient conditions to be satisfied by the paametes fo the existence of a BIBD ae not nown The conditions (I)-(III) ae some necessay condition only The constuction of such design depends on the actual aangement of the teatments into blocs and this poblem is handled in combinatoial mathematics Tables ae aailable giing all the designs inoling at most 0 eplication and thei method of constuction Theoem: ( I) b ( II) λ( ) ( ) ( III) b

3 Poof: (I) Let N ( n ): b incidence matix i Obseing that the quantities E bne and E N E b ae the scalas and the tanspose of each othe, we find thei alues Conside E NE b Similaly, n n n n n n (,,,) n n n b b b n n (,,,) nb (,,,) b b n n n n n n E N Eb (,,) n n n b b b ni i (,,,) (,,,) n i i But E NE Thus b E N E b b as both ae scalas 3

4 Poof: (II) Conside n n nb n n n n n nb n n n N N n n n n n n b b b b ni nn i i nn i i i i i nn i i ni nn i i i i i nn i i nn i i ni i i i λ λ λ λ () λ λ Since so i n o 0 as n o 0, i n i Numbe of times τ i occus in the design fo all,,, of times occus in the design and nn Numbe of blocs in which τ and τ occus togethe N NE i i i λ fo all λ λ λ λ λ λ + λ( ) + λ( ) [ + λ( )] E () + λ( ) 4

5 Also N NE n n n n n n N n n n b b b n n N nb n n nb n n n b n n n i b b ni i n i i n i i E (3) ( ) ( ) Fom and 3 [ + λ( )] E E o + λ( ) o λ( ) ( ) Poof: (III) Fom (I), the deteminant of N N is det N N [ + λ( )]( λ) [ ( ) ]( λ ) + ( λ) 0 because since if λ fom (II) that This contadicts the incompleteness of the design 5

6 Thus N N is a nonsingula matix Thus an( N N) We now fom matix theoy esult an( N) an( N N) so an( N) But an( N) Thus b b, thee being b ows in N Intepetation of conditions of BIBD Intepetation of (I) b This condition is elated to the total numbe of plots in an expeiment In ou settings, thee ae plots is each bloc and thee ae b blocs So total numbe of plots ae b Futhe, thee ae teatments and each teatment is eplicated times such that each teatment occus atmost in one bloc So total numbe of plots containing all the teatments is Since both the statements counts the total numbe of plots, hence b Intepetation of (II) ( ) Each bloc has plots Thus the total pais of plots in a bloc Thee ae b blocs Thus the total pais of plots such that each pai consists of plots within a bloc b ( ) Thee ae teatments, thus the total numbe of pais of teatment ( ) Each pai of teatment is eplicated λ times, ie, each pai of teatment occus in λ blocs Thus the total numbe of pais of plots within blocs must be ( ) ( ) Hence b λ Using b in this elation, we get ( ) λ ( ) ( ) λ Poof of (III) was gien by Fishe but quite long, so not needed hee 6

7 Balancing in designs: Thee ae two type of balancing Vaiance balanced and efficiency balanced We discuss the aiance balancing now and the efficiency balancing late Balanced Design (Vaiance Balanced): A connected design is said to be balanced (aiance balanced) if all the elementay contasts of the teatment effects can be estimated with the same pecision This definition does not hold fo the disconnected design, as all the elementay contasts ae not estimable in this design Pope Design: An incomplete bloc design with b is called a pope design Symmetic BIBD: A BIBD is called symmetical if numbe of blocs numbe of teatments, ie, b Since b, so fom b Thus the numbe of pais of teatments common between any two blocs λ The deteminant of N N is N N [ + λ( )]( λ) ( ) ( λ ) + ( λ) When BIBD is symmetic, b and then using b, we hae Thus N N N ( λ), so N± ( λ) Since N is an intege, hence when is an een numbe, ( λ) must be a pefect squae So 7

8 N N ( λ) I + λe E, ( N N) N N λ I E E λ λ λ N I E E Post-multiplying both sides by, N, we get NN ( λ) I + λe E N N Hence in the case of a symmetic BIBD, any two blocs hae λ teatment in common Since BIBD is an incomplete bloc design So eey pai of teatment can occu at most once is a bloc, we must hae If, then it means that each teatment occus once in eey bloc which occus in case of RBD So in BIBD, always assume > Similaly λ < [If λ then λ( ) ( ) which means that the design is RBD] Resolable design: A bloc design of - b blocs in which - each of teatments is eplicated times is said to be esolable if b blocs can be diided into sets of b/ blocs each, such that eey teatment appeas in each set pecisely once Obiously, in a esolable design, b is a multiple of Theoem: If in a BIBD Dbλ (,,,, ), b is diisible by, then b + Poof: Let b n (whee n > is a positie intege) 8

9 Fo a BIBD, λ( ) ( ) o because b λ( ) o n ( ) o n λ( n ) ( ) n λ + λn Since n> and >, so λn> is an intege Since has to be an intege ( n ) λ is also a po sitie intege Now, if possible, let b< + n < + o n ( ) < ( ) ( ) o n ( ) < (because ) λ λ λ( n ) < which is a contadiction as intege can not be less than one b< + is impossible Thus the opposite is tue b + holds coect Intabloc analysis of BIBD: Conside the model y µ + β + τ + ε ; i,,, b;,,,, i i i whee µ β τ ε i i is the geneal mean effect; is the fixed additie i is the fixed additie th th bloc effect; teatment effect and is the iid andom eo with ε ~ N(0, σ ) i We don t need to deelop the analysis of BIBD fom stating Since BIBD is also an incomplete bloc design and the analysis of incomplete bloc design has aleady been pesented in the ealie module, so we implement those deied expessions diectly unde the set up and conditions of BIBD Using the same notations, we epesent the blocs totals by B i y, teatment totals by i V b y, i i adusted teatment totals by Q and gand total by b G y The nomal equations ae obtained i i by diffeentiating the eo sum of squaes Then the bloc effects ae eliminated fom the nomal 9

10 equations and the nomal equations ae soled fo the teatment effects The esulting intabloc equations of teatment effects in matix notations ae expessible as Q Cτˆ Now we obtain the foms of by b ni i c (,,, ν ) The off-diagonal elements of C ae gien by b c nn ( ;,,,, ν ) i i i λ The adusted teatment totals ae obtained as whee T Q i( ) b Q V nb (,,, ν ) i i i V i( ) B i C and Q in the case of BIBD The diagonal elements of C ae gien denotes the sum oe those blocs containing th teatment Denote i( ) B, then T V i The C matix is simplified as follows: N N C I I ( ) I E E λ + λ λ + I E E λ λ + I E E λv E E I ( i ) ( i ) 0

11 Since C is not as full an matix, so its unique inese does not exist The genealized inese of C is denoted as C which is obtained as Since E E C C+ λ E E C I C E E o I, λ the genealized inese of C λ is Thus E E C C+, λ λ C I E E E E I + I Thus an estimate of τ is obtained fom Q ˆ τ CQ λ Q Cτ as The null hypothesis of ou inteest is H0 : τ τ τ against the altenatie hypothesis H : at least one pai of τ s is diffeent Now we obtain the aious sum of squaes inoled in the deelopment of analysis of aiance as follows The adusted teatment sum of squaes is Teat( ad) ˆ τ Q QQ λν Q, λν The unadusted bloc sum of squaes is

12 Bloc ( unad) b Bi G b i The total sum of squaes is Total b G yi b i The esidual sum of squaes is obtained by Eo () t Total Bloc ( unad) Teat( ad) A test fo H0 : τ τ τ is then based on the statistic F T Teat( ad) /( ) /( b b + ) Eo () t b b + λ Q Eo () t If FT > F α,, b b + ; then H0( t) is eected This completes the analysis of aiance test and is temed as intabloc analysis of aiance This analysis can be compiled into the intabloc analysis of aiance table fo testing the significance of teatment effect gien as follows Intabloc analysis of aiance table of BIBD fo H0 : τ τ τ Souce Sum of squaes Degees of feedom Between (adusted) teatment Teat( ad) Mean squaes MS teat Teat( ad) F MS MS Teat E Between blocs (unadusted) Bloc ( unad) b - Intabloc eo Eo () t (by substaction) b b + MS E Eo () t b b + Total i Total y i G b b

13 In case, the null hypethesis is eected, then we go fo paiwise compaison of the teatments Fo that, we need an expession fo the aiance of diffeence of two teatment effects The aiance of an elementay contast ( τ, ) unde the intabloc analysis is τ V Va( ˆ τ ˆ τ ) Va ( Q Q ) λ [ Va( Q ) ( ) ( + Va Q Co Q Q )] λν ( c + c c ) σ λν λ σ λν + σ λ This expession depends on σ which is unnown So it is unfit fo use in the eal data applications One solution is to estimate σ fom the gien data and use it is place of σ An unbiased estimato of σ is () ˆ Eo t σ b b ν + Thus an unbiased estimato of V can be obtained by substituting ˆ σ in it as ˆ () Eo t V λ b b ν + If H 0 is eected, then we mae paiwise compaison and use the multiple compaison test In ode to test H0 : τ τ ( ), a suitable statistic is ( b b + ) Q Q t λ Eo () t which follows a t-distibution with ( b b + ) degees of feedom unde H 0 A question aises that how a BIBD compaes to an RBD Note that BIBD is an incomplete bloc design wheeas RBD is a complete bloc design This point should be ept is mind while maing such estictie compaison 3

14 We now compae the efficiency of BIBD with a andomized bloc (complete) design with eplicates The aiance of an elementay contast unde a andomized bloc design (RBD) is whee V σ σ unde RBD ˆ ˆ R Va( τ τ ) RBD Va( yi ) Thus the elatie efficiency of BIBD elatie to RBD is σ Va( ˆ τ ˆ τ ) RBD Va( ˆ τ ˆ τ ) BIBD σ λ λ σ σ The facto λ E (say) is temed as the efficiency facto of BIBD and λ E < (since > ) The actual efficiency of BIBD oe RBD not only depends on efficiency facto but also on the atio of aiances σ / σ So BIBD can be moe efficient than RBD as σ can be moe than σ because < Efficiency balanced design: A bloc design is said to be efficiency balanced if eey contast of the teatment effects is estimated though the design with the same efficiency facto If a bloc design satisfies any two of the following popeties: (i) efficiency balanced, (ii) aiance balanced and (iii) equal numbe of eplications, then the thid popety also holds tue 4

15 Missing obseations in BIBD: The intabloc estimate of missing (i, ) th obseation y is i y i ( ) B ( ) Q ( ) Q ( )( b b + ) i Q : sum of Q alue fo all othe teatment (but not the th one) which ae pesent in the i th bloc All othe pocedues emain the same Intebloc analysis and ecoey of intebloc infomation in BIBD In the intabloc analysis of aiance of an incomplete bloc design o BIBD, the teatment effects wee estimated afte eliminating the bloc effects fom the nomal equations In a way, the bloc effects wee assumed to be not maed enough and so they wee eliminated It is possible in many situations that the bloc effects ae influential and maed In such situations, the bloc totals may cay infomation about the teatment combinations also This infomation can be used in estimating the teatment effects which may poide moe efficient esults This is accomplished by an intebloc analysis of BIBD and used futhe though ecoey of intebloc infomation So we fist conduct the intebloc analysis of BIBD We do not deie the expessions a fesh but we use the assumptions and esults fom the intebloc analysis of an incomplete bloc design We additionally assume that the bloc effects ae andom with aiance σ β Afte estimating the teatment effects unde intebloc analysis, we use the esults fo the pooled estimation and ecoey of intebloc infomation in a BIBD In case of BIBD, 5

16 ni nn i i nn i i i i i nn i i ni nn i i N N i i i nn i i nn i i ni i i i λ λ λ λ λ λ ( λ) I + λe E ( N N) λe E λ I The intebloc estimate of τ can be obtained by substituting the expession on ( N N) in the ealie obtained intebloc estimate GE τ N N N B b ( ) Ou next obectie is to use the intabloc and intebloc estimates of teatment effects togethe to find an impoed estimates of teatment effects In ode to use the intebloc and intabloc estimates of conside the intebloc and intabloc estimates of the teatment contast τ togethe though pooled estimate, we The intabloc estimate of teatment contast l τ is l ˆ τ lcq lq λ lq λ l ˆ τ, say The intebloc estimate of teatment contast l τ is 6

17 l N B l τ (since le 0) λ b l nb i i λ i λ l τ lt The aiance of l ˆ τ is obtained as Va( l ˆ τ ) Va l Q λ l Va( Q ) + ll Co( Q, Q ) λ ( ) Since Va( Q ) σ, λ Co Q Q σ (, ), ( ), so λ Va( l ˆ τ) σ l l l σ λ ( ) λ l + l σ (since 0 being contast) λ Similaly, the aiance of [ λ λ + λ] l λ σ l λ ( ) (using ( ) ( )) τˆ is obtained as Va( l τ ) l Va( T) + ll Co( T, T ) λ ( ) σ fl + λσ f l l λ σ f l λ 7

18 The infomation on τˆ and τˆ can be used togethe to obtain a moe efficient estimato of τ by consideing the weighted aithmetic mean of τˆ and τ This will be the minimum aiance unbiased and estimato of τ when the weights of the coesponding estimates ae chosen such that they ae inesely popotional to the espectie aiances of the estimatos Thus the weights to be assigned to intabloc and intebloc estimates ae ecipocal to thei aiances as ( )/, λ σ f espectiely Then the pooled mean of these two estimatos is λ λ λ λ l ˆ τ + l ˆ τ ˆ lτ + l τ σ σ f σ σ f L λ λ λ λ + + σ σ f σ σ f λω l ˆ τ + ( λω ) l τ λ ω + ( λω ) λω lˆ τ + ( λω ) l τ λω + ( λω ) λωτˆ + ( λ) ωτ l λω + ( λω ) l τ λ/( σ ) and whee λνωτˆ ˆ + ( λ) ωτ τ λνω + ( λ) ω,, ω ω σ σ f Now we simplify the expession of τ so that it becomes a moe compatible in futhe analysis Since ˆ τ ( / λν ) Q and τ T / ( λ), so the numeato of τ can be expessed as ω λντˆ + ω ( λ) τ ω Q + ω T Similaly, the denominato of τ can be expessed as ωλ + ω ( λ) ( ) ( ) ω + ω (using ( ) ( )) λ [ ω ( ) + ω( ) ] Let W ( V ) ( ) T + ( ) G whee W 0 Using these esults we hae 8

19 whee ( ) ω Q + ωt τ ω( ) + ω ( ) ( ) ( V T ) T T ω + ω (using ) Q V [ ω ( ) + ω ( )] ω ( ) V + ( ω ω)( ) T [ ω ( ) + ω ( )] ω ( ) V + ( ωω) W ( V ) ( ) G ω ( ) + ω ( ) [ ] [ ] [ ω ( ) + ω ( ) ] ω ( ) ( ωω)( ) V+ ( ωω) W ( ) G ω ω + V + ξ { W ( ) G} { ( ) } V W G ω ( ) + ω ( ) ω ω,, ξ ω ω ω ( ) + ω ( ) σ σ f Thus the pooled estimate of the contast l τ is l τ lτ l V + W l ( ξ ) (since 0 being contast) The aiance of l τ is Va( l τ) l λω + ( λω ) ( ) l (using λ( ) ( ) [ ] ( ω + ( ) ω l σ E whee ( ) σ E ( ) ω + ( ) ω is called as the effectie aiance 9

20 Note that the aiance of any elementay contast based on the pooled estimates of the teatment effects is Va ( τi τ ) σe The effectie aiance can be appoximately estimated by [ ω ] ˆ + ( ) σe MSE whee MSE is the mean squae due to eo obtained fom the intabloc analysis as and Eo () t MSE b b + ω ω ω ( ) ω + ( ) ω The quantity ω depends upon the unnown σ and obtain the unbiased estimates of σ and ω To do this, we poceed as follows σ To obtain an estimate of ω, we can β σ β and then substitute them bac is place of σ and σ β in An estimate of ω can be obtained by estimating σ fom the intabloc analysis of aiance as ˆ ω [ ] σ MSE ˆ The estimate of ω depends on ˆ σ and ˆ σ To obtain an unbiased estimato of β σ β, conside fo which + Bloc ( ad) Teat( ad) Bloc ( unad) Teat( unad) E b b ( Bloc ( ad) ) ( ) σβ + ( ) σ Thus an unbiased estimato of σ β is ˆ σ ˆ β Bloc ( ad) ( b ) σ b Bloc ( ad) ( b ) MSE b b MSBloc ( ad) MSE b b MSBloc ( ad) MSE ( ) 0

21 whee Thus MS ˆ ω Bloc ( ad) Bloc ( ad) ˆ ˆ σ + σβ b ( ) ( b ) ( ) Bloc ( ad) Eo () t Recall that ou main obectie is to deelop a test of hypothesis fo H0 : τ τ τ and we now want to deelop it using the infomation based on both intebloc and intabloc analysis To test the hypothesis elated to teatment effects based on the pooled estimate, we poceed as follows Conside the adusted teatment totals based on the intabloc and the intebloc estimates as T T + ω W ;,,, and use it as usual teatment total as in ealie cases The sum of squaes due to T is S T T T Note that in the usual analysis of aiance technique, the test statistic fo such hull hypothesis is deeloped by taing the atio of the sum of squaes due to teatment diided by its degees of feedom and the sum of squaes due to eo diided by its degees of feedom Following the same idea, we define the statistics F S / [( ) MSE[ + ( ) ]ˆ ω] T whee ˆ ω is an estimato of ω It may be noted that F depends on ˆ ω The alue of ˆ ω itself depends on the estimated aiances ˆ σ and ˆ σ So it cannot be ascetained that the statistic F f necessay follows the F distibution Since the constuction of F is based on the ealie appoaches whee the statistic was found to follow the exact F -distibution, so based on this idea, the distibution of F can be consideed to be appoximately F distibuted Thus the appoximate distibution of

22 F is consideed as F distibution with ( ) and ( b b + ) degees of feedom Also, ˆ ω is an estimato of ω which is obtained by substituting the unbiased estimatos of ω and ω An appoximate best pooled estimato of V ˆ + ξw l l τ is and its aiance is appoximately estimated by λω ˆ + ( λ) ˆ ω l In case of the esolable BIBD, σ ˆβ can be obtained by using the adusted bloc with eplications sum of squaes fom the intabloc analysis of aiance If sum of squaes due to such bloc total is Bloc and coesponding mean squae is then MS b Bloc Bloc ( )( ) E( MS ) σ + σ b ( ) σ + σβ Bloc β and b ( ) ( ) fo a esolable design Thus and hence ( )( + ) E MSBloc MSE σ σ β [ MSE] MSbloc MSE ˆ ω, ˆ ω The analysis of aiance table fo ecoey of intebloc infomation in BIBD is descibed in the following table:

23 Souce Sum of squaes Degees of Between teatment (unadusted) S T feedom - Mean squae F F MS Blocs ( ad ) MSE Between blocs (adusted) Bloc ( ad) Teat( ad) + Bloc ( unad) b - MS Blocs( ad) b Bloc ( ad) Intabloc eo Teat( unad) Eo () t (by substaction) b b + Eo () t MSE b b + Total b - Total The incease in the pecision using intebloc analysis as compaed to intabloc analysis is Va( ˆ τ ) Va( τ) λω + ω ( λ) λω ω( λ) λω Such an incease may be estimated by ˆ ω( λ) λω ˆ Although ω > ω but this may not hold tue fo ˆ ω and ˆ ω The estimates ˆ ω ˆ and ω may be negatie also and in that case we tae ˆ ω ˆ ω 3