E-OPTIMAL SEMI-REGULAR GRAPH DESIGNS AND PARTIALLY EFFICIENCY BALANCED DESIGNS

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1 ounal of Statistical Theoy and Applications ISS Volume, ume, 22, pp.7-6 E-OPTIMAL SEMI-REGULAR GRAPH DESIGS AD PARTIALLY EFFICIECY BALACED DESIGS D.K. Ghosh Depatment of Statistics, Sauashta Uniesity, Rajkot, India oseph O.C. and Alex Thannippaa Depatment of Statistics, St.Thomas College, Pala, Aunapuam, P.O , Keala, India S.C. Bagui Depatment of Mathematics and Statistics, Uniesity of West Floida, Pensacola, FL25, USA Astact acoux (985) extended the definition of Regula gaph designs of Mitchell and ohn (976) to Semi-egula gaph (SRG) designs, and studied the type optimality of lock designs. Hee, the constuction and optimality of some moe SRG designs ae discussed. Moeoe, in this inestigation, it is estalished that a class of two associate patially efficiency alanced design is, in fact SRG designs. Key wods: Semi-egula gaph (SRG) design, Symmetical alanced incomplete lock (SBIB) design, Patially efficiency alanced (PEB) design with two associate classes, E optimality.

2 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 8. ITRODUCTIO Block designs ae extensiely used in many fields of eseach actiities. A wide ange of alanced incomplete lock (BIB) designs and patially alanced incomplete lock (PBIB) designs ae aailale in the liteatue. Howee, these designs ae esticted to equi-eplicate and equi-lock sizes. Anothe new class of incomplete lock designs, temed as patially efficiency alanced (PEB) designs was deeloped y Pui and igam (977), which can e made aailale in aying eplications and unequal lock sizes. Let d e a lock design haing teatments aanged in locks of size k(> k) whose incidence matix is d whee enties n dij gie the nume of times the i th teatment occus in the j th lock. When n dij = o fo all i, j, the design is said to e inay. The i th ow of d is denoted y di and epesents the nume of times teatment i is eplicated in the design. The matix d d whee d is the tanspose of d is efeed as the concuence matix of d, and its enties ae denoted y λ dij. The mathematical model which is usually used to analyze the data otained fom d is the two-way additie model. This model specifies that all oseations y ij (the oseation otained afte applying the i th teatment to a unit occuing in the j th lock) ae uncoelated, hae constant aiance, and hae expectation α i + β j, whee α i and β j ae unknown paametes epesenting the effects of the i th teatment and j th lock, espectiely. Let T d and B d denote ectos of teatment and lock totals espectiely, then the educed nomal equation fo estimating the teatment effects in d can e witten in matix fom as C d α = T d d K - B d whee C d = diag( d,..., d ) d diag(/k, /k 2,..., /k ) d, α = (α,..., α ) and diag( d,..., d ) denotes a diagonal matix. The matix C d is called the infomation matix o C-matix of d, and is positie semi-definite with zeo ow sums and zeo column sum. In the susequent sections, we need to find the M o -matix, whee M o = M ( d,..., d ) /n and M = diag (/,.., / ) ( d diag(/k, /k 2,..., /k ) d ). Unde the two-way additie model gien aoe fo d, it is well known that a necessay condition fo a linea comination

3 E-Optimal Semi-Regula Gaph Designs 9 Ci i i C i i of the teatment effects to e estimale is that =. Such a linea comination of the teatment effects is called a teatment contast. A contast of the fom α i α j is called a teatment diffeence. A design is said to e connected poided all possile teatment diffeences ae estimale. Altenatiely, a design d is connected if and only if its C-matix has ank. Since connectedness is a desiale popety fo most lock designs to hae, only such designs ae consideed in this inestigation. Let D(,, k) denotes the class of all connected lock designs haing teatments aanged in locks of size k. In this pape, two diffeent methods fo the constuction of semi-egula gaph designs ae caied out in section. In the fist method, the SRG design is constucted y augmenting a new teatment in each of the locks of the BIB design, and then augmenting one moe lock which contains all the ( + ) teatments. In the second method, the SRG design is constucted y augmenting a new teatment in each of the locks of the BIB design, and then augmenting ( λ + ) locks which contain all the teatments of the BIB design only. Futhe, it is eified that these semi-egula gaph (SRG) designs elong to a paticula class of patially efficiency alanced (PEB) designs with two efficiency classes. This shows that a paticula class of PEB designs with two efficiency classes is also semi-egula gaph designs. Definition.. If in a design all the diagonal elements and off-diagonal elements of the concuence matix ae diffeing y at most one, then it is called a semi-egula gaph design. This definition is due to acoux (985). Definition.2. A design d(,, k, ) is said to e a PEB design with m-efficiency classes if (i) thee exists a set of ( ) linealy independent contasts s i, i =, 2,..., m, such that ρ i of them satisfy the equation so that the efficiency facto associated with eey contast of the i th class is ( µ i ) whee µ i (i =, 2,..., m) ae eigen alues of M o with multiplicities ρ i (Σρ i = ), and (ii) thee exists mutually othogonal idempotent matices L i (i =, 2,..., m) of anks ρ i such that, whee, M o = R - P / n, P = K -, and is the incidence matix, is the ecto of teatment eplications, k is the ecto of lock sizes, R and K denote the diagonal matices with diagonal elements as and k, and R - and

4 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 5 K - ae thei ineses, and n denotes the total nume of units. The paametes of PEB design with m-efficiency classes may now e witten as,,, k, µ i, ρ i, L i (i =, 2,..., m). This definition was gien y Pui and igam (977). 2. PRELIMIARY RESULTS Fo estalishing the E-optimality of designs in sections. and.2, the following esults ae needed. Theoem 2.. Let d є D (,..., ; ; k) hae C-matix C d and let m equal to the smallest offdiagonal enty occuing in d d = ((λ dij )). Then m / k Z d p (k ) / {( )k} with stict inequality on the ight-hand side whenee λ dpq p (k )/( ) fo some q p, whee Z d is assumed as the smallest nonzeo eigenalue of C d matix. Theoem 2.2. Let d є D (,..., ; ; k) hae infomation matix C d. If the enties of d d = ((λ dij )) satisfy the condition that λ dij p (k )/( ) fo all i j then Z d = p (k )/{( )k} and d is E-optimal in D(,..., ; ; k). Theoems 2. and 2.2 ae due to acoux (98).. METHODS OF COSTRUCTIO OF SEMI-REGULAR GRAPH DESIGS In this section, two methods of constuction of semi-egula gaph designs, which ae otained using BIB designs, ae discussed. These SRG designs ae also happened to e PEB designs with two efficiency classes... Constuction using BIB designs with = + Theoem... Let d(=, =k, λ) e a symmetical BIB design and let its paametes and satisfy the elation = +, whee e the nume of locks and the nume of eplications of the teatments and λ = -. Let d e the incidence matix of the BIB design d, and d* e the incidence matix of the augmented design d *. The incidence matix d* defined as:

5 E-Optimal Semi-Regula Gaph Designs 5 d* d gies a semi-egula gaph (SRG) design with paametes * = +, * = +, * = ( +,, +, +), k * = (k+,, k+, +), λ * = λ+, λ * 2 = +. Poof. Let us conside a symmetical BIBD with paametes =, = k, and λ poided =+. Using these paametes we otain the concuence matix of the augmented design d + as d* d* ( ) I ( ) ( ) ( ) ( ) Fo this concuence matix we oseed the following: (i) The diffeences of the diagonal elements ae: (+)-(+) = - = (since = +) (ii) The diffeences of the off-diagonal elements ae: (+)- (λ+) = - λ = (since fo a symmetical BIBD with paametes, = + and k= and then using the elation λ( ) = ( k ), we get λ = -). That is, the diagonal elements and the off-diagonal elements of the concuence matix ae diffeing y at most one. Hence the design d + is a semi-egula gaph (SRG) design. This completes the poof. Remak: The BIBD, discussed in section., can easily e otained fom a educed BIBD with paametes = =, = k = and λ = 2. Coollay... The class of semi-egula gaph (SRG) designs constucted using Theoem.. is in fact PEB designs with two associate classes. Poof. This Coollay is poed using the M o matix of the design d +. The M o matix of the design d + is otained fom the expession: ' M o =diag(/, / 2,..,/ + ) ( d* diag (/k, /k 2,,/k +) ) d* ) (, 2,..., + )'/n (.) whee is a column ecto of one s of ode + and n is the total nume of units. Using the paametes of symmetical BIBD discussed in Theoem.., we otain the M matix of design d * as M =

6 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 52 m k m k m k m k I k ) )( ( whee m = ( + ) + ( + ). We found that the eigen alues of M o ae ) )(k ( with multiplicity (-), and o 2 ) ( (M ) tace with multiplicity one espectiely. So the design d * is patially efficiency alanced (PEB) with efficiency facto -μ with multiplicity (-), and (- μ 2 ) with multiplicity one. This completes the poof. Theoem..2. Let d(=, k=, λ) e a symmetical BIB design and its paametes satisfy the elation = +. Let d e the incidence matix of the BIB design d, and d* e the incidence matix of the augmented design d *. The incidence matix d* defined as: ) ( ) ( * d O gies a semi-egula gaph (SRG) with paametes * = +, * = +(-λ+), * = (2-λ +,, 2-λ +, ), k * = (k+,, k+, ), λ * = +, λ * 2 =. Poof. Let us conside a symmetical BIB design with paametes =, k=, λ and we also oseed that fo this seies of BIB design = 2 λ. The concuence matix of the augmented design d * e I d d ) ( ) ( * * Fo this concuence matix, we oseed the following: (i) The diffeences of the diagonal elements of the concuence matix d* d* ae (-λ)+(+)- = 2-λ+- = (since = 2 λ) (ii) The diffeences of the off-diagonal elements ae (+)- =.

7 E-Optimal Semi-Regula Gaph Designs 5 That is, the diagonal elements and the off-diagonal elements of the concuence matix d* d* ae diffeing y at most one. Hence the design d * is a semi-egula gaph (SRG) design. This completes the poof. Coollay..2. The class of SRG designs constucted using Theoem..2 is in fact patially efficiency alanced (PEB) designs with two efficiency classes. Poof. This Coollay is poed using the M o -matix of the design d +. The M o -matix of the design d + is gien y the following expession: M o =diag (/, / 2,..,/ + ) ( d* diag (/k, /k 2,,/k +(-λ+) ) d* ' ) (, 2,., + )'/n whee is a column ecto of one s of ode + and n is the total nume of units. Futhe M is simplified as M o = m I (k )m m k m m (k ) m k m m k m whee m = 2-λ+. Hee eigenalues of M o ae with multiplicity (-), (k )(2 ) and 2 tace(m o) ( ) with multiplicity one. So the design d * is patially efficiency alanced (PEB) with efficiency facto (-μ ) with multiplicity (-), and (- μ 2 ) with multiplicity one. This completes the poof..2. umeical Examples As an application of the aoe two theoems, two numeical examples ae gien in this section. Example.2.. Conside the symmetical BIB design d(,,,, 2) such that locks (as column) of this design ae shown elow: = +, and d = 2 2 2

8 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 5 Using Theoem.., the augmented design d + with paametes * = 5, * = 5, * = (,...,, 5), k * = (,...,, 5), λ * = and λ * 2= is otained, whose incidence matix d* and the concuence matix d* d* of the design d * is gien y d* = d* d* 5 This class of designs is Semi egula gaph designs, ecause one can find that all the diagonal and the off-diagonal elements of the concuence matix d* d* ae diffeing y at most one. The M o -matix of the design d * is otained as follows:.7.55 Mo The eigen alues of M o ae μ =.625 with multiplicity (= -) and μ 2 =.25 with multiplicity one. So the design d * is patially efficiency alanced (PEB) with efficiency factos: (i) - μ =.975 with multiplicity (-) = and (ii) - μ 2 =.9975 with multiplicity. The following Tale.2. gies the examples of unequi-eplicated SRG designs with unequal lock sizes constucted using BIB designs.

9 E-Optimal Semi-Regula Gaph Designs 55 Tale.2. showing the paametes of BIB designs and SRG Designs Paametes of BIB design Paametes of esulting Semi egula Gaph designs k λ * * * k * λ * λ * ,,, 5,,, ,,5, 6 5,,5, ,,6, 7 6,,6, ,,7, 8 7,,7, ,,8, 9 8,,8, ,,9, 9,,9, ,,,,,, ,,, 2,,, 2 Futhe, it is found that all the aoe SRG designs elong to a paticula class of PEB designs with two efficiency classes. Example Conside the symmetical BIB design d(,,,, 2) such that = +. Blocks of this design ae shown elow: 2 d 2 2 Using Theoem..2, the augmented design d+ with paametes * = 5, * = 6, * = (5,..., 5, ), k * =, λ * = and λ * 2 = is otained, and its incidence matix d* is expessed as: d* Moeoe, the concuence matix of the design d * is gien y the expession:

10 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 56 d* d* This class of designs is Semi egula gaph designs, ecause one can find that all the diagonal and the off-diagonal elements of the concuence matix d* d* ae diffeing y at most one. The Mo -matix of the design d * is otained as follows: M o The eigen alues of Mo ae µ =.5 with multiplicity (= ) and µ2 =. with multiplicity one. So, the design d * is patially efficiency alanced (PEB) with efficiency factos: (i) - µ =.95 with multiplicity ( - ) =, and (ii) - µ2 =.9 with multiplicity. The following Tale.2.2 gies the examples of unequal-eplicated and pope SRG designs constucted using BIB designs. Tale.2.2 showing the paametes of BIB designs and SRG Designs Paametes of BIB design Paametes of esulting Semi egula Gaph designs k λ * * * k * λ * λ * ,,5, ,,6, ,,7, ,,8, ,,9, ,,, ,,, ,,2,

11 E-Optimal Semi-Regula Gaph Designs 57 Futhe, it is found that all the aoe SRG designs elong to a paticula class of PEB designs with two efficiency classes.. E-OPTIMALITY OF SRG DESIGS A design d * is E-optimal if and only if the maximal aiance among all est linea uniased estimatos of nomalized linea contasts is minimal unde d *. That is, a design d * Ω,,k is called E-optimal if Z d* Z d fo all designs d Ω,,k, whee Z d is the least nonzeo eigen alue of the infomation matix C d. acoux (98) otained chaacteization of E-optimal lock designs in the suclasses of pope designs, with unequal nume of eplications of the teatments. Using the esults of acoux (98), we otained the E-optimality of semi egula gaph design with unequal-eplicated and unequal lock sizes. In this section, the E-optimality of unequal-eplicated designs with unequal lock sizes and the E-optimality of unequal-eplicated pope designs ae consideed... E-optimality of unequi-eplicated designs with unequal lock sizes ow, the esults of acoux (98) is extended, and the E-optimality of unequi-eplicated designs with unequal lock sizes is consideed. Let d D(p, ; ks, ; n) with incidence matix d and infomation matix Cd ; whee p = min(, 2,..., ) and ks = min(k, k2,..., k ) and C d = Rd d K - d d ' = R d P d and let P d = (δii' ). ow, conside the matix T xd = ks C d x{( ) - I ( ) - } (.) whee x is any eal nume, I is the identity matix and is the matix of ones. The eigenalues of T xd ae < ks Z d x/( )... k s Z d- x/( ). ow, conside the following theoems: Theoem... Fo any design, d D(p, ; ks, ; n) and m = min ' i i (ks δii ' ) then m/ks Zd p ()(ks )/{( )ks } Futhe, if m = p (ks )/( ), then Zd = p ()(ks )/{( )ks } and then d is E- optimal in D(p, ; ks, ; n).

12 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 58 Poof. Consideing x = p (ks ), the poof follows on the lines of the poof of Theoem. of acoux (98) y taking Txd as defined in (.). Theoem..2. The class of SRG designs constucted using Theoem.., with paametes * = +, * = +, * = ( +,..., +, + ), k * = (k +,.., k +, + ), λ * = λ + and λ * 2 = + haing the incidence matix d* is in fact an E-optimal in D(p*, * ; ks*, * ; n * ). Poof. Let, Pd* = d K d - d ' Afte simplification, we otain P d* as P d* I k k k Hee the off-diagonal elements ae and k Also, min ii ( ii* ) = (Since, λ = ). k k k espectiely. k Since k <, and k * = (k + ) o * = +, the minimum lock size ks* = (k + ); and, min ii ( ks * ii* ) = ( k ), since λ = and k <. k Also, p* = + = ( since = + ). So, p* (ks* )/( * ) = k/ =. That is, min ii( ks* ii* ) = p* (ks* )/(* ). Theefoe, using the peliminay esults of Theoems 2. and 2.2, we oseed that Z d = p* (*)(ks* )/{(* )ks*} and hence the aoe class of designs is E-optimal in D(p*, * ; ks*, * ; n * ). Example... Conside the SRG design d * with paametes * = 5, * = 5, * = (,...,, 5), k* = (,...,, 5), λ* = and λ* 2 = gien in Example... ow, Pd* = d K d - d ' which afte soling, we hae

13 E-Optimal Semi-Regula Gaph Designs 59 P d* Fo this design the minimum lock size, ks* =, p* = and y using Theoems.. and..2, we get m = min ii( ks* ii* ) = 2.8 (m an intege). Also, p* (ks* )/(* ) = ( - )/(5 - ) =. i.e., m = p* (ks* )/( * ), then the minimum eigenalue of the infomation matix Cd*, is gien y Z d = p* (* )(ks* )/{( * )ks* }= (5)( )/{(5 )} =.75. Hence d * is E-optimal in D(, ;, ; 2)..2. E-optimality of unequal-eplicated pope designs ow, the esults of acoux (98) is extended, and the E-optimality of unequal-eplicated pope designs is consideed. Let d є D(, 2,..., ; ; k) hae incidence matix d and infomation matix Cd and p = min(, 2,..., ). ow, conside the matix Txd = kc d x{( ) I ( ) } (.2) whee x is any eal nume, I is the identity matix and is the matix of ones. The eigenalues of Txd ae < kz z x/( )... kz d- x/( ) ow, we conside the following theoems: Theoem.2.. Let d є D(, 2,..., ; ; k) hae C matix C d and let m equal to the smallest off-diagonal enty occuing in d 'd = ((λdij )). Then m/k Z d p ()(k )/{( )k} Futhe, if λdij p (k )/( ) fo all i j then Zd = p ()(k )/{( )k} and then d is E- optimal in d є D(, 2,..., ; ; k). Poof. Conside x = p (k ). ext we can easily poe this theoem on simila lines of the poof of Theoem. of acoux (98 y taking Txd as defined in (.2). Theoem.2.2. The class of SRG designs constucted using Theoem.2., haing incidence

14 D.K. Ghosh, O.C. oseph, A. Thannippaa and S.C. Bagui 6 matix d+ and paametes * = +, * = + ( λ + ), * = (2 λ +,..., 2 λ +, ), k * = (k +, k +,..., k +,,..., ), λ * = +, and λ * 2 = is E-optimal in D( *, 2*,..., * ; * ; k * ). Poof. Hee we otain the concuence matix of the augmented design d * as d* d* ( ) I ( ) Hee, the off-diagonal elements of the concuence matix ae ( + ) and, That is, λd*ij = +, fo all i j=,2,, and λd*ij =, fo all I =,2,., and j = +. Also, p+ (k+ )/(+ ) = k/ = k =, (since the design is a symmetical BIB design). So, λd*ij p* (k* )/( * ) fo all i j. Theefoe, using the peliminay esults of Theoems 2. and 2.2, we oseed that Z d* = p * ( * )(k * )/{( * )k * } and hence the class of SRG design is E-optimal in D( *, 2*,..., * ; * ; k * ). Example.2.. Conside the SRG design d * with paametes * = 5, * = 6, * =(5,, 5, ), k * =, λ * = and λ * 2 = gien in Example..2. The concuence matix of the design d * of example.2. is gien y the expession d* d* Hee, the off-diagonal elements of the concuence matix ae and, that is, λd*ij = fo all i j=,2,, and λd*ij = fo all i =,2,, and j = 5. Also, p* (k * )/( * ) = ()/ =. Theefoe, λd*ij p* (k* )/(* ) fo all i j. Hence, using the Theoems 2. and 2.2, we hae Z d* = p* (* )(k* )/{(* )k* }= ()(5)/() =.75. Hence we say that the design d * is E-optimal in D(5,...,5, ; 6; ).

15 E-Optimal Semi-Regula Gaph Designs 6 REFERECES [] Gupta, V.K. and Singh, R. (989). On E-optimal lock designs. Biometika, 76(), [2] acoux, M. (98). On the detemination and constuction of E-optimal lock designs with unequal numes of eplicates. Biometika, 67(), [] acoux, M. (985). Some sufficient conditions fo the type optimality of lock designs. ounal of Statistical Planning and Infeence,, [] Mitchell, T.. and ohn,.a. (976). Optimal Incomplete lock designs. ORL/CSD-8. Aailale fom the ational Technical Infomation Seice, U.S. Dept. of Commece, 5285 Pot Royal Road, Spingfield, VA.225. [5] Mukejee, R. and Kageyama, S. (99). Roustness of goup diisile designs. Communications in Statistics-Theoy and Method A, 9(9), [6] Pui, P.D. and igam, A.K. (977). Patially efficiency alanced designs. Communications in Statistics-Theoy and Method A, 6(8),