Open Journal of Statistics, 06, 6, 37-48 Published Online February 06 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/0.436/ojs.06.6006 Some Construction Meods of Optimum Chemical Balance Weighing Designs III Rashmi Awad *, Shati Banerjee School of Statistics, Devi Ahilya University, Indore, India Received 4 November 05; accepted 5 February 06; published 8 February 06 Copyright 06 by auors and Scientific Research Publishing Inc. This wor is licensed under e Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Meods of constructing e optimum chemical balance weighing designs from symmetric balanced incomplete bloc designs are proposed wi illustration. As a by-product pairwise efficiency and variance balanced designs are also obtained. Keywords Balanced Incomplete Bloc Design, Symmetric Balanced Incomplete Bloc Design, Ternary Balanced Bloc Design, Variance Balanced Design, Efficiency Balanced Design, Nested Balanced Incomplete Bloc Design, Weighing Design, Chemical Balance Weighing Design, Optimum Chemical Balance Weighing Design. Introduction Originally Yates (see []) gave e concept of weighing design. Afterward his wor was formulated by Hotelling (see []) and he gave e condition of attaining e lower bound by each of e variance of e estimated weights. Many statisticians did prominent wor in obtaining optimum weighing designs (see [3]-[7]). In recent years, different meods of constructing e optimum chemical balance weighing designs; using e incidence matrices of nown balanced incomplete bloc designs, balanced bipartite bloc designs, ternary balanced bloc designs and group divisible designs have been given in e literature (see [8]-[]). Construction meods of obtaining optimum chemical balance weighing designs using e incidence matrices of symmetric balanced incomplete bloc designs have been given by Awad et al. []-[4]; some pairwise balanced designs are also been obtained which are efficiency as well as variance balanced. In is paper; some oer new construction meods of obtaining optimum chemical balance weighing designs using e incidence matrices of nown symmetric balanced incomplete bloc designs are propose. Some more pairwise efficiency as * Corresponding auor. How to cite is paper: Awad, R. and Banerjee, S. (06) Some Construction Meods of Optimum Chemical Balance Weighing Designs III. Open Journal of Statistics, 6, 37-48. http://dx.doi.org/0.436/ojs.06.6006
well as variance balanced designs are also been proposed. Let us consider a bloc design in which treatments arranged in b blocs and elements of e incidence matrix N are denoted by n ij, for all i =,,, ; j =,,, b, such at e j bloc contains j ex perimental units and e i treatment appears r i times in e entire design. A balanced bloc design is said to be (a) binary when n ij = 0 or, i =,,, ; j =,,, b. Oerwise, it is said to be nonbinary (see [7]); (b) ternary if n ij = 0, or, i =,,, ; j =,,, b, and it has parameters, b, ρ, ρ, r,, Λ ; where ρ, ρ are e number of times, occurs in e incidence matrix, respectively (see [5]); (c) generalized binary if n ij = 0 or x, i =,,, ; j =,,, b and some positive integer x ( > ), and it has parameters, b, r,, Λ (see [6]). A balanced incomplete bloc design is an arrangement of symbols (treatments) into b sets (blocs) such at () each bloc contains ( < ) distinct treatments; () each treatment appears in r ( > λ ) different blocs and; (3) every pair of distinct treatments appears togeer in exactly λ blocs. Here, e parameters of balanced incomplete bloc design (, b, r,, λ ) are related by e following relations ( ) ( ) ( ) r = b, r = λ and b Fisher s Inequality A balanced incomplete bloc design is said to be symmetric if b = ( consequently, r = ). In is case, incidence matrix N is a square matrix i.e. N = N. In case of symmetric balanced incomplete bloc design any two sets have λ symbols in common. Balancing of design in various senses has been given in e literature (see [7] [8]). In is paper, we consider e balanced design of e following types: ) variance balanced bloc designs; ) efficiency balanced bloc designs and; 3) pairwise balanced bloc designs. ) A bloc design is said to be variance balanced if and only if its C-matrix, R NK N, satisfies R NK N = ψ I ( ), for some constant ψ (see [9]-[]); where ψ is e unique nonzero eigen value of e matrix C wi e multiplicity ( ), I is e identity matrix. ) A bloc design is said to be efficiency balanced if R NK N = µ I + ( µ ) n r, for some constant µ (see [0]-[]); where µ is e unique non zero eigen value wi multiplicity ( ). For e EB bloc C= µ R n rr ; see [3] design N, e information matrix is given as ( )( ( ) ) b 3) A bloc design is said to be pairwise balanced if nn j= ij i j = Λ (a constant) for all i, i ; where i i and ii,,,,. A pairwise balanced bloc design is said to be binary if n ij = 0 or only, for all i, j and it has parameters, b, r,, Λ (= λ, say) (in is case, when r = r and = b, it is a BIB design wi parameters, b, r,, λ ). A design is said to form a nested structure, when ere are two sources of variability and one source is nested wiin oer. Preece (see [4]) introduced a class of nested BIB designs wi treatments, each replicated r times, wi two systems of blocs, such at (a) e second system nested wiin e first, wi each bloc from e first system (called super blocs) containing exactly m blocs from e second system (called sub-blocs). (b) Ignoring e sub-blocs, leaves a BIB design wi parameters, b, r,, λ. (c) Ignoring e superblocs, leaves a BIB design wi parameters, b, r,, λ. The, b, b, r,,, λ, λ and m are called e parameters of a nested BIB design. The parameters satisfy e following conditions: so at r = b = bm = b ( ) = ( ) r λ ( ) = λ ( ) r ( )( λ λ ) = ( ) m m r The following additional notations are used = [ b ] is e column vector of bloc sizes, r = [ rr r ] is e column vector of treatment replication, K = bb diag[ b], Rv v diag[ rr r ] b ri = j = n is e total number of experimental units, wi is N b = r and N i= j= =, =, Where a is e a vector of ones. Furermore µ represents e loss of information, i.e., µ represents an effi- 38
ciency of e design, ( ψ ) represents e variance of any normalized contrast in e intra bloc analysis (see [0] [3] [5] [6]). For given p objects to be weighted in groups in n weightings, a weighing designs consists of n groupings of e p objects and e least square estimates of e weight of e objects can be obtained by e usual meods when n p. Using matrix notations e general linear model can be written as: Y = Xw + e () where Y is n column vector of e weights of e objects, w is e p column vector of e unnown weights of objects and e is n vector of error components in e different observations such at E( e ) = 0 n and E( ee ) = σ I n. Also X = ( x ij ), ( i =,,, n; j =,,, p) is e n p matrix of nown quantities called design matrix whose entries are +, or 0. Let matrix X taes e values as: + if e j object is in e left pan in e i weighing xij = if e j object is in e right pan in e i weighing 0 if e j object is not in e i weighing The normal equations for estimating w are as follows: X Xwˆ = X Y () where ŵ is e vector of e weights estimated by e least squares meod. Singularity or non-singularity of a weighing design depends on wheer e matrix XX is singular or nonsingular, respectively. When X is of full ran, en it is obvious at e matrix XX is non-singular. Then in is case e least squares estimate of w is given by and e variance-covariance matrix of ŵ is ( ) ŵ= X X XY (3) Var w ( ˆ ) σ ( X X ) = (4) A weighing design is said to be e chemical balance weighing design if e objects are placed on two pans in a chemical balance. In a chemical balance weighing design, e elements of design matrix X = ( x ij ) taes e values as + if e j object is placed in e left pan in e i weighing, if e j object is placed in e right pan in e i weighing and 0 if e j object is not weighted in e i weighing. Hotelling (see []) has shown at e precision of e estimates of e weight of e object increases furer by placing in e oer pan of e scale ose objects not included in e weighing and us using two pan chemical balance. He proved at if n weighing operations have been done to determine e weight of p = n objects, e minimum attainable variance for each of e estimated weights in is case is σ n and he also shown at each of e variance of e estimated weights attains e minimum if and only if ( X X ) = ni p. A design satisfying is condition is called an optimum chemical balance weighing design.. Variance Limit of Estimated Weights Cerana et al. (see [8]) studied e problem of estimating individual weights of objects, using a chemical balance weighing design under e restriction on e number of times in which each object is weighed. A lower bound for e variance of each of e estimated weights from is chemical balance weighing design is obtained and a necessary and sufficient condition for is lower bound to be attained was given. Then Cerana et al. (see [8]) proved e following eorem: Theorem.. For any n p matrix X, of a nonsingular chemical balance weighing design, in which maximum number of elements equal to and in columns is equal to m, where m= max m, m,, m p (where m j be e number of elements equal to and in j column of matrix X). Then each of e variances of e estimated weights attains e minimum if and only if X X = mi (5) ( ) p Also a nonsingular chemical balance weighing design is said to be optimal for e estimating individual 39
weights of objects; if e variances of eir estimators attain e lower bound given by, ( ) σ Var wˆ = m; j =,,, p (6) Preece [4] introduced a class of nested BIB designs. Using is concept Banerjee; (see [7]-[9]) constructed e nested EB as well as VB designs, where e sub-blocs form ternary design D wi parameters (, b, ρ i, ρ i, r,, Λ ) while e super-blocs form generalized binary design D wi parameters (, b, r,, Λ ); where i =,. Proposition.. Existence of BIB design D wi parameters (, br,,, λ ) implies e existence of a nested EB as well as VB design. The sub-blocs form an EBT as well as VBT design wi parameters =, b = λ, ( )( ) ρ = r, ρ = r( ), r r ( ) Λ = λ ( + )( ), µ = ( + ) ( ) ( ) ( ) =, =,, ψ = Λ while e super-blocs form a generalized binary EB as well as VB design wi parameters =, b = λ =,, r ( ) r =, ( ) λ Λ =, ( ) ( ), ψ = [ Λ ] µ = Proposition.3. Existence of BIB design D wi parameters (, br,,, ) λ implies e existence of a nested EB as well as VB design. The sub-blocs form an EBT as well as VBT design wi parameters =, b = 3λ, ρ = r( )( 3 4), ( ) ρ = r, r r ( ) ( 3 Λ = λ 3 4), µ = 4 + 3 ( ) 3( ) ( ) = 3, =,, ψ = Λ while e super-blocs form a generalized binary EB as well as VB design wi parameters b =, = λ, r r ( ) = 3, 3 =, Λ = λ( ) ( 3 ) ( ), ψ = [ Λ ] µ = 3. Construction of Design Matrix: Meod I 9, Consider a SBIB design D wi e parameters (,, λ ); each pair of treatments occurs togeer in λ blocs. Tae any pair of treatments, say, (θ, φ ) from is design. Then mae two blocs from e design which contains e pair (θ, φ ).. Give e negative sign to e treatment θ and eliminate e treatment φ while e oer ( ) remaining treatments of e same bloc remain as it is.. Give e negative sign to e treatment φ and eliminate e treatment θ while e oer ( ) remaining treatments of e same bloc remain as it is. The matrix N of design D is obtained. Now doing e same procedure for all possible pairs of treatments, we obtain matrix N i ; i =,,,. Then e incidence matrix N of e new design D so formed is e matrix having e elements, and 0; given as follows by juxtaposition 40
= (7) N N N N Then combining e incidence matrix N of SBIB design repeated s-times wi N we get e matrix X of a chemical balance weighing design as s-times X = N N N (8) Under e present construction scheme, we have n = λ + sb = λ ( ) + sb and p =. Thus e each column of X will contain δ = r ( )( ) + sr elements equal to, δ = r( ) elements equal to and r( )( + ) elements equal to zero. Clearly such a design implies at each object is weighted ( ) m = δ+ δ = r + sr times in n = λ + sb weighing operations. Lemma 3.4. A design given by X of e form (8) is non singular if and only if λ ( 3) ( 6λ 7) + + s( r λ) 0. Proof. For e design matrix X given by (8), we have and { ( )( 3) } { ( )( 5) } ( )( 5) XX = r + sr λ + sλ I + λ + sλj { ( 3) ( 6 7) } ( ) {( )( 5) } = λ λ+ + s r λi + λ + s J 3 {( 7 3 7) } λ { ( 3) ( 6λ 7) } ( λ) X X = r + + s + + s r e determinant (0) is equal to zero if and only if { + + } but 3 ( 7 3 7 ) r s ( ) ( ) s( r ) ( )( 3) λ( )( 5) r + sr = + sλ ( ) ( ) s( r ) λ 3 6λ+ 7 + λ = 0 or 3 ( ) r 7 + 3 7 + s = 0 is positive and en det ( X X ) = 0 if and only if λ 3 6λ+ 7 + λ = 0. So e lemma is proved. Theorem 3.5. The non-singular chemical balance weighing design wi matrix X given by (8) is optimal if and only if ( )( ) (9) (0) λ 5 + s = 0. () Proof. From e conditions (5) and (9) it follows at a chemical balance weighing design is optimal if and only if e condition () holds. Hence e eorem. If e chemical balance weighing design given by matrix X of e form (8) is optimal en σ Var ( wˆ j ) = ; j =,,, p r ( ) + s Example 3.6. Consider a SBIB design wi parameters = b = 4, r = = 3, λ = ; whose blocs are given by (,,3), (,,4), (,3,4), (,3,4). 4
Theorem 3.5 yields a design matrix X of optimum chemical balance weighing design as 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
Clearly such a design implies at each object is weighted m = 8 times in n = 3 weighing operations and Var ( wˆ ) = σ 8 for each j =,, 3, 4. Corollary 3.7. If e SBIB design exists wi parameters (,, λ ); en e design matrix X so formed in (8) is optimum chemical balance weighing design iff < < 5 and s. Remar: In SBIB design wi parameters (,, λ ); if = 0 en e design matrix given in (8) is perfectly optimum and s = 0 in is case. Corollary 3.8. If in e design D ; is replaced by zero en e new design D so formed is a BIB design wi parameters V =, B =, R= r( )( ), K =, Λ= λ. Then e struc- ture form a pairwise VB and EB design = V, b = B + sb, r = R + sr, sλ = ( ) + and sλ µ = ( λ ) r +. ψ λ D s-times N N N N = () wi parameters = λ, =, λ 4. Construction of Design Matrix: Meod II sλ =Λ+, Consider a SBIB design D wi e parameters (,, λ ); each pair of treatments occurs togeer in λ blocs. Tae any pair of treatments, say, (θ, φ ) from is design. Then mae ree blocs from e design which contains e pair (θ, φ ).. Give e negative sign to e treatment θ and eliminate e treatment φ while e oer ( ) remaining treatments of e same bloc remain as it is.. Give e negative sign to e treatment φ and eliminate e treatment θ while e oer ( ) remaining treatments of e same bloc remain as it is. 3. Give e negative sign to bo e treatments θ and φ while e oer remaining treatments of e same bloc remain as it is. The matrix N of design D is obtained. Now doing e same procedure for all possible pairs of treatments, we obtain matrix N i ; i =,,,. Then e incidence matrix N of e new design D so formed is e matrix having e elements, and 0; given as follows juxtaposition: = (3) N N N N Then combining e incidence matrix N of SBIB design repeated s-times wi N we get e matrix X of a chemical balance weighing design as: Under e present construction scheme, we have n = 3λ + sb s-times X = N N N (4) and p =. Thus e each column of X 43
will contain { 3 ( )( ) } δ = r + sr δ = r elements equal to and n δ δ elements equal to zero. Clearly such a design implies at each object is weighted r m = δ+ δ = ( 3 ) + sr times in n = 3λ + sb weighing operations. Lemma 4.9. A design given by X of e form (4) is non singular if and only if r λ ( 3 0) + sr ( 3 3 + 36) + sλ. Proof. For e design matrix X given by (4), we have and ( )( 3 0) elements equal to, ( ) r λ λ XX = + sr ( 3 3+ 36) + sλ I + ( 3 3+ 36) + sλ J r λ λ = ( 3 0) + sr ( 3 3+ 36) + sλ I + ( 3 3+ 36) + sλ J (5) r λ X X = ( 3 0) + sr + ( ) ( 3 3 + 36) + sλ r λ ( 3 0) + sr ( 3 3 + 36) + sλ (6) e determinant (6) is equal to zero if and only if r λ + sr = + + sλ ( 3 0) ( 3 3 36) r λ + sr = + + sλ or ( 3 0) ( ) ( 3 3 36) r λ + sr + + + sλ r λ ( 3 0) + sr = ( 3 3 + 36) + sλ. So e lemma is proved. Theorem 4.0. The non-singular chemical balance weighing design wi matrix X given by (4) is optimal if and only if but ( 3 0) ( ) ( 3 3 36) is positive and en det ( X X ) = 0 if and only if λ ( 3 3 + 36) + sλ = 0. (7) Proof. From e conditions (5) and (5) it follows at a chemical balance weighing design is optimal if and only if e condition (7) holds. Hence e eorem. If e chemical balance weighing design given by matrix X of e form (4) is optimal en σ Var ( wˆ j ) = ; j =,,, p r ( 3 ) + sr Example 4.. Consider a SBIB design wi parameters = b = 4, r = = 3, λ = ; whose blocs are given by (,,3), (,,4), (,3,4), (,3,4). 44
Theorem 4.0 yields a design matrix X of optimum chemical balance weighing design as X = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45
Clearly such a design implies at each object is weighted m = 30 times in n = 48 weighing operations and Var ( wˆ ) = σ 30 for each j =,, 3, 4. Corollary 4.. If e SBIB design exists wi parameters (,, λ ); en e design matrix X of e form (4) is optimum chemical balance weighing design iff 5 and s 4. Corollary 4.3. If in e design D ; is replaced by zero en e new design D so formed is a BIB design wi parameters V =, B = 3, R= 3r( )( ), K =, Λ= 3λ. Then e structure form a pairwise VB and EB design =, b = B + sb, r = R + sr, ( ) 3 3 s ψ = λ + µ ( ) and λ 3 3 s = +. r 5. Result and Discussion D s-times N N N N = (8) wi parameters =, =, λ =Λ+ sλ, The following Table and Table provide e list of pairwise variance and efficiency balanced bloc designs for Meods I and II respectively, which can be obtained by using certain nown SBIB designs. Table. For Meod I. S. No. b r λ ψ µ Reference No. ** 5 70 3 4.5 0.968 R (4) 7 98 3 4 8 0.3437 R () 3 3 8 3 4 4 9.5 0.3906 R (37), MH (3) Table. For Meod II. S. No. b r λ ψ µ Reference No. ** 5 0 3 4.5 0.968 R (4) 7 84 3 4.667 0.6 R (0), MH () 3 7 54 3 4 8 0.3437 R () 4 3 86 3 4 4 9.5 0.3906 R (37), MH (3) ** The symbols R(α) and MH(α) denote e reference number α in Raghavrao [30] and Marshal Halls [3] list. 6. Conclusion In is research, we have significantly shown at e obtained designs are pairwise balanced as well as efficiency balanced. The only limitation of is research is at e obtained pairwise balanced designs are all have large number of replications. Acnowledgements We are grateful to e anonymous referees for eir constructive comments and valuable suggestions. 46
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