Stat'8tic~ - krishi.icar.gov.in · 1 Introduction . Designed experiments (tre generally conducted for making all possible parred com \ parisons among . the treatments. However, there

$Page 1: Stat'8tic~ - krishi.icar.gov.in · 1 Introduction . Designed experiments (tre generally conducted for making all possible parred com \ parisons among . the treatments. However, there$
Stat8tic~ and Applications Volume 32001 pp 133-146

Block designs for comparing test treatments with control treatments- an overview

VK Gupta and Rajender Parsad Indian Agricultural Statistics Research Institute New Delhi - 110 012 India

Abstract The problem of finding efficient block designs for comparing several test treatshy

ments vith one or more standard (control) treatments has received considerable attention in the recent past This paper presents a review of certain recent results in this area

Key w01ds TIeinforced block designs supplemented balanced desigm augmented clefigns BBrBUB designs of Type Gj GDBPUB designs of Type G A-uptimality

1 Introduction

Designed experiments (tre generally conducted for making all possible parred comshyparisons among the treatments However there are situations when the interest uf the experimenter is only in a subset of these paired comparisuns For example in genetic resources environment an essential activi1 y is to test or evaluate the new germ plasm I provenance I superior selections (test treatments) etc with the exshyistiJJg provenance or released varieties (control treatments) Similar situations may also occur in other disciplines of agricultural sciences industry etc The problem here is to design an experiment for the estimation of the test treatments versus conshytrol treatments contrast~ and the comparisons among the test treatments or among the control treatments are of lesser cOUBequence

It is welJ known that for the egttimation of elementary contrasts among test and control treatments conventional designs like balanced incomplete block (BIB ) designs are not the besl (~P( CDX 1958 Fpderer 1956 Sinha ] 980 1982) For these experi mental sit uaLions thereforeurogt tIlt problem of choosing an efficient design is important and needs attention For an earljer review on the designing problem for this experimental seLting see Hedayat Jacroux and Majumdar (1988) and Mashyjumdar (1996) This paper also ltiddresses the same problem Vve begill with the maLhematical formulatiotl of the problem

2 Problenl formulation

Vc bctyp W +11 = v treatments divided into two disjoint sets II and G of respective cardinality wand u The w test treatments in H are denoted by 12 11) and tIlt u control tn~atments iii C are denoted by w --r 1 W +2 v The problem here is to

i

134 VK GUPTA All) RAJENDER PARSAD [Vo13 [0$1amp2

design an experiment for estimating W1J test treatments verSUlti control treatmpllts contrasts with as high a precision as possible SUPPoltie that n experimental units avaiiable for experimentation can be arranged ill b blocks of sizes kl kl kb respectively kl + k2 + L kb = n Let Ytjl denote the observation of test or control treatment t(t = j v) on experimental unit 1(1 = 1 ntj) of block j (j = 1 b) Assume the Isual fixed effects additive linear model

Ytjl -= fJ + Tt + f3j + etj l (21)

where jt is Lhe overall mean Tt is the effect of test or ontrol treatment t OJ IS the effect of block j and etjl are the random errors normally distributed WIth zero mean and variance CT

2 kj Note that homoscedastid ly is not assumed Here Q ~ 0 IS a scalar constant generally unknown and Iltj is the number of replications of test or control treatlmmt t in block J The v x b incidence matrix N has elements ntJ For 0 = 0 we get the wmal homoscetlastic model

The contrasts of interest are Tg - TI 9 E G h E H Comparisons of treatments within G and with in H are of secondary importance The contrasts of major interest may be written in matrix notation as PT where the wu x v matrix P may be

(xpreslied ltUl Pc [1 It) [ - I (9 1u Wlth pl1v = O Here It is a i-component vector with all elements one II is an identity matrix of order l denoLPs the Kronecker product of maLrices and T is a v-component vector of tellt or control trea tment effects

Let C = (eft) = R- Nf-l N be the usual v xv C-matrix of a block design with v treatments Here R - diag(rl T2 Tv) denotes a diagonal matrix of replication numbers of treatmeuts and J( = diag (k l k2 kb) denotes a diagonal matrix of

block sizes Partition N as N = [NN~J where N J = ((nlj)) is a w x b incidence matrix of Lest trecltlnents and N2 = (nl1) is a 1L x b inciuelltP matrix of control treatshyments Similarly RI = diag(rlT2 T) and R2 = diag(r w+lrw+2) denote respectively the diagonal matriceH of replications of the test treatments and the control treatmellts The information matrix C can then be partitIOned a~

(22)c-(i ~] where

I b b

A - L)lRl i - k -1NljNLJ B - - 2kj I N1j ~) and D = 2klt [R2j

J

j =1 j=1 )=1

k j N 2j ~j Abo Hlj = diag(nij 1111) rilL) ) n~j = diag(n(W-lt- l j

IIg) Hvj) NI = gtV11 middot middotmiddot N1j NIb] N2 = [N21 middotmiddotmiddot N2j N2b N I ) = (nlj n2j middot n uj) and N 2j = (n( w+l)jn(w~2)j nlJ) Fora = 0 11 = R I shy

NJ 1(-1 N B - - Nlj(-I N2 and D =- R2 - N2J(-1 N~

LLt T - h deuote the best lillear unbiased estimator (BLUE) of T - Th E

Gh E If Th( DLlE llf contrais of interest PT is Pr- with Jispersion matrix Cuv(Plf) = cr 2 p( - P Here C- is a generalized inverse of C ie CC C = C

IL is assumed that the deign is connected and Rauk (C) = l - l It might tlso be desiraLlt that the comparisons of interest are estimated t hrough the design Ib

2001] DESIG

the same varia

tegtt treatment call such desi (BBPtB desig

Definjtion 2 i said to be a

b

(i) 2)-1 J=I

II

(ii) 2)jO-1 j=J

b

(Iii) 2)1 j= I

The C rna

c= [ A geneTalized in

c- == [ wI

or 1 8G design mated with sam~

Var(fg - Til)

BG designs h ror 11 == 1 BG ( fJf proper settlll Block Sizes (BT of type G for Q

Bipartite Block Designs with Gr n - (J

indeed there llf Test t reatltlell1 Divisible Bipart or type G henet i- ulfined ~elow

[YoJ3 Nos1amp2

llS control treatments n experimental units )f sizes kJ k2 middot kb )bservation of test or = 1 ntj) of block model

(21)

treatment t (3j is the bated with zero mean ned Here Q 2 0 is a replications of test or has elements ntj For

)arisons of treatments rasts of major interest ( v matrix P may be

It is at-component rder t0 denotes the tor of test or control

of a block design with 11 matrix of replication s a diagonal matrix of

)) is a w x b incidence batrix of control treatshy

~g (rU+IrtL+2 Tv) 3e test treatments ant be partitioned as

(22)

b

Id D = 2)) [R~J shyi=1

= diag(n(wll) bull middot

f bull N2b j N 1j = (nl) i cr = 0 A = R shy

~LUE) of T9 - Til g E

with dispersion roaLnx 7 ie CC C - C - v - 1 It might also lIough the design with

2001) DESIGNS FOR TEST TREATMENTS - CONTROL COMPARIS01S 135

the same variance A design is said to be variance balanced for the estimation of test Lreatmeut versus control treatment contrasts if it permits the estimation of these ontrasts with the same variance and the covariance between any two estimated test treatment versus conLrol treatment contrasts is also same In general we shall call such designs as Balanced Bipartite Block Designs with Gnequa1 Block Sizes (BBPlB desi~s) of type G and henceforth denote these as BG deSigns

Definition 21 An arrangement 0 v treatments in b blocks of $izes kl k2 bull kb is said to be a BG design if

b

i Lkj- Inhjnhj = L~ a constant Vhf- hi = 1 W

)=1

ii Lkj-lngjn9 j =- Lao a constant Vg t g = w + 1 v and J= 1

b

iii Lkj-O-Ingjnhj = La a constant IlL - 1 wg = w + 1 v j=l

The C matrix of a BG design is

c _ [ (wL + uLo)Jw - Lll~ -Lolwl~ ] (23) J - -Lolu 1~ (I1LOOL wLu)Iu - Loolul~

A generaliled inverse of C is

c- shy-

[ WL~Lo (Iw 7 (LluLo)l)~ )0 o

J [ lmiddotu 1 IlJ[oo+w[o ( - ( ) u ) ] (24)

tor 1 BG design t he wu test treatment versus control treatment contrasts are estishymated with same variance given by

A ~ 1_ [(V -l)Lo -L + (v -1)Lo + Lou] (72 1 9 E Gh E H Var(Tq - Th) -

__

vLo w L L uLu uLoa + w Lo

BG d(~signs have been studied exensively in the literature under rliffprent names For u = 1 BG designs are BalcUlced Treatment Incomplete Block (BTIB) designs for proper setting and Balanced Treatmpnt Incomplete Block Designs viUI Lnequal Block Sizes (BTICB designs) for nOll proper seiling for a - a and BTIlR df~Bign of type G for 0 f- U For U gt 1 the BG designs have been terffitmiddotd oS Balanced Bipartite Block (BBPB) designs for proper settings and Balanced Bipartite Block Designs with Cnequal Block Sibes (BBPlB designs) for non-proper settings with 0 = O

lndeed l here may exist designs other than BG designs thac permi t til estiIlation of test treatment versus conlrol treatment contrasts with the smne vananee (~rolp Diisiblp Bipartite Block Dpsiglls with l~neQual Block Sizes (GD13PBlB deSigns) of ry pe ( lWlIceforth denoted as GBG designs anc s11(h dpsigns The G DC desi~11 i-i cltgtTIlled llPlLl

((k +a

For 11

matrix

uf eartlill

Here lilldl a lt1(

treatm(ll

136 VK GUPTA AND RAJENDER PARSAD [Vol3 t-os1amp2 200l] D

Definition 22 An arrangement of v = u + w w = mn treatments in b blocks of 3 M sizes kJ k2 middot kb with parameters u W m n b kJ k2 kb Lo L L2 Loo i iaid to be a GBG design tf 1 2 W W + 1 w +u treatments can be partitioned into WfI10W

Tn + 1 disjoint groups 11 12 1m Vo of respective cardmalit~es VI V2 tmiddot m U 51ructioll such that Method

de~igJl wi(i) 0 = w + 1 W + 2 W + it

12 +(ii) VI = V2 = = vrn = n -lshy

correspOll J81 =rb _ Ll h yen hi E Vq q = 1 Tn

(iii) 2)CI lnh]llhj= L2ilh l hEV h EF q-j q=l Tn q q

j=1

b

(iv) Lkj CI-lngjngj =- Loo 9 -j 9 E Vo is a I3G ( j=1

b

(v) Lkj n-l TLgjHhJ = Lo9 E Fa h = 1 w j =

Here pound1 L2 Lo() Lo art some constants The C matrix of a GBG design is

_ [ (A - 8) 0 1m + B 01ml~ D]C (25)- D E

Here A - [nLJ -- n(m - I)L2 + llLojI - LJlnlE = (lULu + uLoo)lu - Loolu1 B = - L2Inl~ and D = -Lo1u1~

A generalized inverse of C is

C = [ X 0

J ___ _ oJ 1 I ] (26) tulo+uloc [ u - u lu1 ti]

wilh X - (4 - B) (si 1m -I- B (gtlt) 1m 1~1l X - I = (P - Q) 0 1 -+- Q euroI IT l~ Qshy-B(04- B)-J [A+ (m -] )B] 1 and P ~ [A+ (m - 2)B1(A - B)-1 [A+ (111- l)B]-I

For a GBG design the wu tesl treatment versus control treatmpnt comrasLs afl

stimateu with same variance given by

1 pound1 - )Var (79 - Tit) --------- j - ------- shy[7lLl + wL~ _ nLl + uLo (nLl - UiL~ - nL2 ~ uLo)(vLo)

L2 11 - 1 ] uLO(W[ 2 + uLo) ~ u(lJLoo wLo) (T

For LJ = pound these designs am same as BG desigru Somp interesting special cases of GBG designs have been studied in the li terature for u = 1 GBG designs are termed as Group Divisible Treatment Designs (GDTD) for proper setting and Group Disibk Treaimtnf Designs with Cnequal Block Si7es (GDTtB clesipns) for non proper ~etting fur n - () alld GDTlB designs of lype G for 0 f O

3 Noslamp2

b blocks of Loo is said ttoned into VtnmiddotU

m

lesigu is

(25)

~ - Loolu1~

(26)

lI1l1~ Q == m-l)Bt1

outrasts are

uLo)(uLo)

pecial cases lEO designs setting and design~) for

2001J DESIG]S FOR TEST TREATMENTS - CONTROL COMPARISONS 137

3 Method of construction of BG designs

We now give a general method of con~truction BG designs The methods of conshystruction hitherto known in the literature fall out as special cases of this method

Method 31 Suppose that there exists a pairwisE balanced binary block (PBBB) design with parameters w bj b2 bp k = (kll~ k21~2 kpli) gt b = bi +

b2 + + bp and incidence matrix N = [NI N2 Np] Ns heing the w x b~ matrix corresponding Lo the incidenc( of treatment~ in the blocks of size ks satisfying

11= 1 bull Then the design with incidence matrix

Nl Qlt) 10 N20102 N p 01e p ]

N= [ a)lu1b191 a21ul~282 ap 1u1~~8

is a BG desigll with parmneters w u b - bIO I + b202 + -r upBp k ((k l + al u) Ibj 8 (kl + a2u)lb2B (kp apu) l~pop) whee aI ah ap are some

p

non-Iwgativc integers such that La- = aI abeing a scalar and Ot O2 bull Op are 8=1

LakEl1 ill tht ratio

)+1 (k )u-t I (k )u+l)((k I+alu ~+a2u p-rapu

For 11 =- l an efficient block design for making test treatment versus con~rol treatshyment comparisons can be obtained by Laking as inL(k2) V s = 1 p For a review of the methods of constructiOll of PBBB designs see Parsad Gupta and Khallduri (200U)

Remark 31 If _V is the indJence matrix of a (1-1 IL2 bull 1-~ jLp) - resolvable PBBB desigll ie NlLmiddot = 1-81 S = 1 p Then the design with incidence matrix

N0 _ [Nl 2) l N~ 6-~ 1~2 Np ~~ l~p ]0

L+ as1~8

is a BG design with pamm~ters w U ulaquo = bl f)l + b2f)2 L bull bull + bpOp bull kmiddot = ((k] + Ul )1~01 (k2+a2 )1~202 ( kp+ap)1~8) where all (12middot ap are some nonnegative integers such that alt d = a S = 1 p a being a scalar and OJ B~ Op are take1 in the ratio

((k l (It rH 1 (k 2 + a)~ l (kp -r CLp)U 1)

Remark 32 SUPPOSL there exists a binary block design with lfIcidltlCe matrix

J - Nt 12 middot middot middot Np ] and parameters vb1b2 upk = (kll~ kpl~ )gtIA~ anu in which tl1f w test treatmems can be divided into m disjoint Ciets 11 12 bullbullbull It

of larclinali ~ n each such that

P x 1 bull h j hi E Vq q == 1 m2~gt~Ih = A~ h j h hE Vq hi E 1qq 1= q = 1 TIl = 1

Ilpre Ah is the concurrence of treatments l and hi in N 8 bull One way of obtai ning such it deCiipll is Lo replace every treatment in eUl equirfplicate PRBB design in m 1reatme1l1~ with a group of 1 new treatrntllts

138 VK GUPTA AID RAJENDER PARSAD [Vol 3 081 amp2

Following the procedure piven in Method 31 [or BG designs on N gives a GBG design

4 Optimality results

Ve now present some results on optimal designs for the problem Tn the present contex t Lhe most appropriate optimality criteria are -4- and MF-optimalit niteria A design d belonging to a cert aIn ciass of competing designs D is said middot0 hI Ashy

optimal if it rninimiws L LlEGI~fl Var(fg - Th) over all design d E D Lp

m 1 s L s wu be the Lth diagonal element of PCd - P the dispersion maLrix of FT under the design d Then one has to find a design da E D that minimizes L~ I mJ wu over V From the arithmetic mean - harmonic mpan inequahty it ollows that

U middot U tLU

L m )wu ~ wu L lmL 1 = 1 =1

The eqa a l ity is attained hell m L - m for all L = 1 IV) and thiH holds for GBG m d BG designs These designs are therefore canthdat(~s for tIl most efficient desIgns for tct treatment versus contrul treatment compansons according Lo 1shyoptimality cntenon

A design dmiddot btlonging to a certain daFS of competing designs D is said Iu h i AI1shyroptimal if d has the least value o f the maxhnum variance of BLeE of elellwTltary

contrasts among test Lreatments and control treatments as compareci to allY other design d E V It may be mentioned here that for the present probllIn all the A-optimal designs are IV-optimal agt well

R esul t 1 (Majumdar 1 986) For Olle-way heterogeneity setting for compari n~ lJ)

t est t reattnfllts with 11 wntrol treatlllPnts llsing n --- bk experimental uni ts urrtlllged in b hlocks of sitp k each a design is A-optimal if k = () (JJlOclw + jlIli) 11 is a perfect square k = () (modu 1 ~) and

nhj = jgt +kJwu ngj TlhijWlJ V h -1 w9 - w -I bull v ) = L b

For non-orthogonal clesigllH thp follow ing optimality results are avai la b ll~ in t Ill

literature

Result 2 COllstantine (1983 ) showed that a remforCld BTB design obtal1ed b~

acidinf a control treatment oncE 111 every bloc llt of a BID design is A-optimal in I Ill restrict ed class of bloek deRigns Il1IvlIlf a ~ingle replication oj t he control treatmellt in each block

R esult 3 Jacroux (1984) showed that in the restricted class of block deSigns with a single replication of control treatment in each block a design obtained by adding control treatment once to each block of a most balanced group divisib le ciesign is l -optimal

Desigru in which I standard treatment is reinforced in each block of the dlsi ll were tfnned as St andard reinfor(eJ (SH-) designs

2001J DESIGK

Result 4 Majt treatmCIlLs und (1984) utilized tl classified the B replication of L

BTIB designs (1 one) Stufkell (1 vlajumdar Stufk BTIE designs an

(1987) abo studi methods of their

Hedayat and in the form of an 1

sufficient collditi single replication the sufficient COil

of the (olltro] t Tillt

Result 5 (Stufk~ i mes to each bloc

is A-optimal when Gupta (1989)

dcmiddotsign alllong the of information rna

Sinha (1992) I ing treatments ill g(l1eral methods ( it) usiug sulficie Jacroux (1987a) in pUler intensive su Jacroux and ~1aj (1988) Giovagnoli interesting resul ts comparin telit t relt the llumber of test problem when the

All these H1Udie mem lur more I

ied by llajnmdar (2000) aud Solorz to obI am A-opti for small block liz stantine (1983) to A-optimality of BB treatments appear (LOOO) ~avl meUlO ciIIL orthogonal

[Vo3 Koslamp2

gnll on N gives a GBC

roblem In the present IV-optimality criteria gus D is said tu be shyI de~igns d E D Let the dispersion matrix d E D that minimies onie mean inequality it

1 and t his holds for GBC es for the most efficif1l1 larisons according 10 4shy

signsD is said Lo be MYshy~ of BLtE of elementary

compared to any other present problem all the

Isetting for comparing JI

erimental units arn1llged (modw ~ JWu) WU is l

+pound 1 j = I b

ults are availablp in the

BIB design obtai 11(~d b) gtsign is A-upimal in tlw 01 the control treatment

ass of block design itl ign obtained by adding

group divisible design is

each block of the design

2001J DESIGNS FOR TEST TREATMENTS - COtTROL COMPARlSOKS 139

Result 4 Majumdar and Kotz (1983) showed that a BTlB design binary in test treatments and satisfying certain conditions is A-optimal Hedayat and Majumdar (1984) utilized these conditions to gi ve a nronger definition of BTTB designs They classified t he BTIB designs as (a) Rectangular or R- type BTIE designs (equal replication of the control treatment in all the blocks) and (b) Step or (S-type) BTIB designs (the replications of the control t reatment in the blocks differs by one) StutKen (1988) fave the bounds to the A-efficiency of BTIB designs Cheng Majumclar StutKen and Ture (1989) studied the 1- and MV-optimality of S-type BTIB designs and gae an algorithm to obtain these designs Das (1986) and Kisan (1987) also studied the optimality aspects of these designs and gave some general methods of their constrnctiolJ

Hedayat and Majumdar (1985) obtained a sufficient condition for A-optimality iu the form of an inequality involvillg nwnber of test treatments and block size This sufficient condition is helpful ill obtaining A-optimal R-type BTlBdesigns ~laying single replication of the control treatment in each block Stuiken (19b7) ell-tended t he suflkient condition to the case of R-type BTID designs having t 1 replications of the control treatmlmt in each block

Result 5 (Stufken 19R7) A BTIB design obtained by adding a control reatnwllt t times to each block of il BIB design with parameters w b r k-t gt in test trealOlPnts is A-optimal whenever (k - t - If + 1 wt2 (k - tf

Gupta (1989) obtained a simpler sufficient condition to search an A-optimal design among the class of all connected binary block designs 111 lcrms of elpments of information matrix

Sinha (1992) gave general methods of construction of BTTB designs by mfrgshying treatments in a group divisible design Parsad Gupta and Prasad (1995) gav general methods of construction of BTlB desipns and investigated their optimalshyit y using sufficient condition of Hedayat and Majumdar (1984) and Gupta (1989) J acroux (1 987a) illtroduced Group Divisible Treatment Designs (GDTD) A comshyputer intensive sufficient condi~ion for a GDTD to be A-optimal is given by Hedayat Jacrollx and Majumdar (1988) Jacroux (1987b 1987c 1988 19B)) rill illld KOL L

(1988) Giovagnoli and Wynn (1985) and Stuiken (1991) have also provided some interesting results Jacroux aud Majumdar (1989) gave optimal block designs for comparing test treatments with a comrol treatment when b lock size is greener thoU I

the lumber of test treatm -t1tS Bhaumik (1990) and Cutler (199a) have studied tIll problem when the errors are correlated

All llte~e studies are restricted to shuittions when t here is a singh control tremiddot ment For more than one control treatment optimality aspects huve been Iudshyied by ~1ajumdar (la8G) Jaggi (1992) Jaggi Gupta and Parsad (HlJG) Jacroux (2000) and SolorJlano and Spurrier (2001) r-Iajumdar (1986) gcwe all algorithm to obtain A-optimal BBPB deSigns and a catalogue of A-optimal BBPB deSIgns for small block sizes Jaggi Parsad and Gupta (1996) extended the reult of ConshystanLine (1983) to more than one control treatment situation and also studied the A-optimality of BBPB designs in the rebtricted class of designs in htdl all control treltments appear equally frequently ill a block or do not appear al alL Jaerollx (2000) gave methods for determining and constructing MV-uptimal anel highly effishyCilllt orthogonal and nearly orthogonal block designs for comparin~ t(st t reatlt1enL

140 VK G[PTA AND RAJE1DER PARSAD [Vo13 1051amp2

middotith several control treatments under the restriction that replication number of coutrol treatments is fixed Solorzano and Spurner (2001) obtained some results on construction and A-optimality of BBPB designs for small values of u amiw

The studies just descri bed relale to proper setting under fixed effects model In an incomplete block design t he block effects may be random Pandey (1993) and Gupta Pandey and Parsad (1998) have obtained sufficient conditions for generatshying A-opt imal incomplete block designs for making teHt treatments versus control tramp1tment comparison under a two-way classified additive linear mixed eff(ds model It has been shown empirically that an A-optimalefficient deSIgn under a fixed effects model remains A-optimal efficient under a mixed effects Illodel also Catalogues of A-efficient optimal designs have also been given

T he problem of characterization and construction of A- and Ml -optimal deshysigns for making t est treatment versus control t reatment comparisons was till llOW

restricted to proper setting However non-proper experimental settings do exshyist and it is required to generate efficient designs under these situations ampgt well P rasad (1989) investigated the optimality of designs with wlequal block sizes in a very restri cted class of designs when the control replications are taken ilS constant and intra-block variances are assumed to be constant

For compMing test treatments with a control treatlIlent in block designH with unequal blocks the concept of Balanced Treatment Incomplete Block Designs with unequal block sizes (BTI1B) was given by Angelis and Moyssiadis (lJ91) as a natushyral extension of BTlB designs T hey also gave a sufficient condition for etablishing the A-optimality of BTIUB designs Angelis and Moyssiadis (1991) Angeligt Moysshysiadis and Kageyama (1993) and Gupta and Kageyama (1993) gave some methods of constructing A-efficient BTnB designs Jacroux (1992) studied the A-and Myshy

optimality of block designs with two distinct block sires where block sizel may be greater than the number of test ireltltments for comparing several test treatments with a control treatment These studies were also carried out under thegt assumption that intra block variances are constant Parsad (1991) Parsad and Gupta (1994a) inLroduced BTltB dEsigns of Type G and obtained a sufficient condit ion for Ashyoptimality of non - proper incomplete block designs for comparing test treaiments with a control treatment assuming that intrablock variances are proportional to non - negative real power of block cizes Parsad and Gupta (1994b) introduced GDTtB designs of type G mId a sufficient condition for A-optimality of GDTCB designs of type G in the class of block designs that are hinary in test treatments anti in which the control treatmellt is added same number of timp8 to each block of same size A catalogue of A-optimal GDTUB desigll- of tvpe G has also been given Srivastava Gupta and Parsad (2000) have studied the A-optimality of nonshyproper block designs for comparing test treatments with a control treatment whell t he block sizes may be larger than the number of test treatments Jaggi (1996) and Jaggi and Gupta (1997a 1997b) have studied the A-optimality aHpects of the designs for comparing several test treatments with several control treatments under a non-proper block design settiug where intra block variances have been assumed to be canst ant The results are obtained in a restricted dass of designs in which all controls appear equally frequent ly in a block or do not appear at all and block sizes a re largE For small block sizes the condition of Majumdar (1986) has been obtained fur Hon-proper settings Jaggi Parad and Gupta (1999) gave methods of

2001J DESIC

construction i

suflkieut cOile sets of treallll Result 6 A 1 treatments ill

if LoL = 1 other designs out as a part experimental suggesl ed IIs in

The problE several COnLr

been seell in 1

Further all tb compansOllS from different Divisible Hip k=2

5 Weig verSlU

In certain pro interest with d the BLlE of

restil1e tne pr We CUll sid

the class of lI

Laa L Var gEG hf li

and so

Lily L aCG hOi

fiud a (h~i

II

[Vo131os1amp2

cation number of ed some results on ofU amI w

d effects model In Pandey (1993) and

tions for feneratshyents versus control ear mixed effects

ient design under a effects model also

d MV -optimal deshyDarisons was till now rntal settings do exshyIe situatlOns as well ~qual block sites in a u-e taken as constant

block designs with e Block Designs wi til adis (1991) as a natushyClition for establishing 11991) Angelb MO)8 shy

3) gave some methods udied the A-and MV shy~e block sizes may be everal test treatments Wlder the assumption

ad and Gupta (1994a) cient condition for Ashyparing test treatments es are proportional to Ita (1994b) introduced optimality of GDTUB lary in test treatments of times to each block f type G has also beell 1e A-optimality of nonshyoutrol treatmtnt when ~atments Jaggi (1996) otimality aspects of the ontrol treatments under Ices have been rusumed lass of designs in which appear at all and block jumdar (1986) has been (1999) gave methods of

2001J DESIGNS FOR TEST TREATMENTS - CONTROL COMPAIUSONS 111

construction of BBPLB designs Parsad Gupta and Singh (1996) have obtained a sufficient condition for a block design to be A-optimal for comparing two disjoin t sets of trtatments under the above heteroscedastic set up Result 6 A BG design is A-optimal for comparing w test treatments with u control treatments in the class of designs binary in test treatments and control treatmentl

if Lo l L = 1 + Jwtu This is a fairly gentral condition and the condi tions for all

other designs useful for test treatment versus control treatment comparisons fall out as a particular case of this A procedure to obtain an efficient design for the experimenLal situations for which this condition does not hold well hal also beeu suggestcd Ising the concep of lower bound to the average variance

The problem of obtaining efficient designs for comparing test treatments with several control treatments under an unrestricted class is still unsolved and as ha been seen in the discussion above only partial solution to the problem is available Further all the results are available for a class of deisgns in which the pairwise comparisons within a set are made with same variance and between treatments from different sets with same variance Kuriakose (1999) has introduced Group Divisible Bipartite Block Designs and studied the A-optimality of thes( designs for k ==- 2

5 Weighted A-optimal designs for test treatments versus control treatments comparisons

Tn certain problems it is netessary to generate designs that estimate contrasts of interest with differential precision and minimi7e the weighted sum of variances of the BLlE of the contrasts of interest It may indeed be possible to obtain exalt optimal designs for these experimental seHlJlgt In the moST general set lip we may restaLe the problem as follows

We consider again tllf (xperimental settinp described in Section 2 Let Tgt denot the claclt of competing designs Find a design d E D that minimi1e~

w-1 lL~ 1-1 1

L3yL Var(fug-frlh)-tpoundtL E Var(fdlt f dh ) + I L L Var (fd~ - ToI~ gE C fiH h lh=lltl g=u-r-l y=y I

(ii l)

Here 8182 (Ju a ~ 0 are scalar constants or the w(gtights attltlched wiLh the precision of various comparisons and satisfy 81 + fh --- + (Ju TO + ~r =- 1

The comparisons among control treatments are of no intere~t to the experimemer and so) = O The problem then reduces to findmg a design d E D that mjnimiZf-_~

rL-1 lL

2 Bq 2 Var (Ttl - Tdll)U 2 2 Var (TlI - Tdl) fJl + flr- -+- 11J +Q =oc i yEe hElf 11=111=1+1

(5 2) Wnen all the controls have the same weights of importance then thl prnbJern is tli

find a design d~ E V thut minimizes

u-l U

o2E Var(fuv - itll)--oL L Var(i1I1- TJ)3co 1 (5~) 11 =1 h = + I gEG Ii

142 VK GUPTA AND RAJENDER PARSAD [Vo11 Ios1amp2

A special case of the problem in (52) is when Q = O III other words the different controls are given different weights according to their importance but he comparshyisons among test treatments are not considered The problem ill (52) now reduces to find a design dmiddot E V that minimizes

LSg L Var (fdg - Tdh)31 + (32 + +Bu =1 (54) gEG hEH

For U = 1 the problem reduces to finding a design d E V that minimizes

w It 1 w

8 L Var(Td(w+l) - Tdh) -- Ct L L Var (Tdh - TlIl ) Q + (3 = 1 (55) h~l h=1 =+1

The problems in (53) and (55) are the weighted sum of be variames of the BLUE of test treatments versus control treatments contrasts and contrasts among test t reatments respectively with weights as 8 and Q Since more precision is required for the test treatments versus control treatments comparisons than the comparisons among test treatments we insist B gt Q For a = 0 (3 = 1 the experimental Slttillgs in (53) and (55) reduce to the usual setting of A-optimality of test treatment verSllS Wfltrol treatment comparisoru It may be seen that for B = f) these experimental stlttings reduce to the usual setting for A-optimality of designs when all the possible paired comparisons among the v treatments are of equal intprest However there may be situations when more precision is required for comparisons among test treatments than the test neatments versus control treatments comparisons For this setting (3 lt a

The problem posed in (55) has been solved under the block design set up by Gupta Rarnana Clnd Parsad (1999) A catalogue of A-efficientoptimal designs h~ also been presented

The problems of obtaining weighted A-efficient designs for many control treatshyments have beell handled in two phases In the first phase the problem of obtCllJling weighwd A efficient designs for several COlltrol treatmellt~ has been attempted by gidng unequal weights to various control treatments In the choice of an optimal design no consideration is given to the comparisons among Iest treatments This refers to the situation In (5-1) [see Gupta Ramana and Paflad (2001) Tn lbe secshyond pha~e not only we consider the problem of obtainin~ weighted A-efficient design by giving eqlal importawe to all the control treatments but also con~iderillg the estimat ioll of comparisons among lest treatments t hrough the same desin thuugh wi Lh a smaller precision than that of the 1e8t treatments ersus colltrol treatmenL~ comp(l ri()n~ This (orresponds to problem in (53) sel Ilalnana (UJ95) The most general problem in (5 I) and the problem in (52) is st~fminly a difficillt problem These rna) however need attemion so as to bt~ able 10 solve completely thE problem of test Lreatmellts versus control treatments comparisons

This artide prondes a limited revIew on the designng probiem for making test rNnrnfnts verSlS (ontrol treaLments comparisons There are many otll(~r stutlips availahhmiddotgtWI this prohlem under rUfferent expenmental set ups 4 compltl bibliogshyraph 01 rlti subject is ltIvail ble with the authors

2001J DESIG

Acknowledg marllls(fipt III sidemhly impl immensely ihr

Referenc Angelis L a

unequal Plann 1

Angelis 1 A-efficier

Bhaumik DJ Dents wi Math 3

Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2

iher words the different ftance but the comparshyrn in (5 2) now reduces

(54) BlL =1

that minimizes

)a+ 3=L (55)

of the variances of l he jSts and contrasts among Since more precision is ~ comparisons than the For a = 0 B = 1 the

ttl setting of A-optimality It may be seen th at for

rtting for A-optimality of ~ the v treatments are of bore precision is required reatments versus control

e block design seL lip by cient joptimal designs hflS

s for many comrol treatshy the problem of obtaining Its has been attem pted by

the choice of an optimal ng test treatments This arsad (2001)] In the secshy

eighted A-efficient desi~lls s but also cOllsidering the h the same design though veriUS control treatments amcUla (1995)) The most ingly a difficult problem

lye completely the problem

19 problem for makinf test ere are many other studies I shyet ups Aw mplcte bibliogshy

2001J DESTGKS FOR TEST TREATMEKTS - COKTROL COMPARISOlS 143

Acknowledgements The authors are grateful to Dr Aloke Dey for reading the manuscript meticulously cmd making some very useful suggestions I hat have 1011shy

siderab ly improved the presentation in the manuscript The author also benefited immensely through discussions with Dr MN Das

References

Angelis L and Moyssiadis C (1991) A-optimal inwmplete block designs with unequal block sizes for comparing test treatments with a control J Statist Plann In 28 353-368

Angeli L lloyssiadis C and Kageyama S (1993) -Iethods of constructillg A-efficient BTILB designs Utilitas Math 44 5-15

Bhaumik D K (1990) Optimal incomplete block designs for rompariug treatshyments with a control under t he nearest neighbour correlation model Utilita8 Math 38 15-25

heng C S Majumdar D Stllfken I and Ture TE (1989) OplirnaJ step type ueigns [or comparing treatments with a control J Amer Statist Assoc 83 177-482

Constantine G11 (1983) On the efficiency for control of reinforced BIB designs J flay Statist Soc B45 31-36

Cutler RD (1993) Efficient block deSIgns for companng test tn~atments to a control when tlw errors are correlated J Statist Plann In 36 107-125

Cux D U (1958) Planning of Experimen l s [ew York Wiley

Das A (198G) Incompiele block designs for comparing treatments wit Ii a control Cnpublished )1 )c Thesis 1 ARI ew Delhi

Federer vT (195(j) Augmented design IIawazian Planter RCC01d 55 191-208

GiovagnoJi A and W) 1II1 HP (1985) Schur optimal continuous block desipns for lrtatments with a contro Proc of Berkeley Conference in hOllour of Jerzy J eYITlltl1l and lack KIefer 2 (i5 Hi66

Cupta S ( L989) EffieienL desipns for comparing test treatmpIlt~ dlll a control Biomct7lkn 76 783-787

GupLa S and Kageyama S (1093) TyplS designs in unequal blocks J Combtn inoTTn System Sci 18 97-112

Gupta K Paneley A and Parsad U (1998) A-optimal block designs under a mixed rnoclltgtl for making est trecltrnellts-control comparisons 8unkhyii B60 IHJG-510

Gupu 1 RtlllCln(L D-V ilnd Pusad H (1999) Weightpd A-eflicielHY If block designs ~or making treatmlllt-Clllllro] idO 1n~almeill-treat 11Itnt (middotOlllpilr isolls I StutiM Plaln in 77 J01-320

Plr~ild H

14~ 2001j DESVK GUPTA AND RAJENDER PARSAD [Vol 3 to1amp2

Gupta K Ramana D V and Parsad R (2001 ) Weighted A-optimal block designs for comparing tet treatments with controls with unequal pre(i~ioll J Statist Plann In (Special Issue in memory of Profesor Yamamoto) To appear

Hedayat AS and Majumdar D (1984) A-optimal incomplete block desifIls for tfst treatment - control comparisons Technometrics 26 363-370

Hedayat AS and Majumdar D (1985) Families of r1-op1imal block designs fur comparing test treatments with a control AnnStat~sl 13 757-7(j7

Hedayat AS J acroux M and Majumdar D (1988) Optimal designs for comshyparing test treatmenls with a control Statist Be 3 363-370

Jacroux M (1984) On the optimality and usage of relllforced block designs for companng test treatments w-ith a standard treatment J Roy Stati~t Soc B46 316-322

Jacroux 11 (1987a) On the determination and construel i ufllV-Oplimal block desiglls for comparing block designs with a standard trlltlnent J StatiRt Plann In 15 205-225

Jacroux 11 (19S7b) Some yIv -()ptilnal block designs for comparjn~ test trta1shymellts wit1 a standard treatment Sankhya B49 2W-261

Jacroux M (1987c) On A-optimality of block designs for comparill) te~t treatshyments wiLh a sl andard Technical Report Deptt of Math Sciences Washshyingtoll )late Cnivershy

1anoux -1 (1988) Some further results on the M) middotoptimality of block rksi)lls for comparing test treatments to a standard lrtHtment 1 8tnt~t Plnnn In 20 201 - 214

1acroux 11 (1989) The A-Optimality of block desi)ns for comparing tlst tnatshylllents wlth a control J Amer Statuto Assoc 84 nO-317

Jacroux 1 (1992) Ou lLlmparillg test treatments with a control using bloc desigm hilillg lllle(pal siz( d blocks Sallkhya B54 32 1-~45

Lwrl)UK 11 (2000) SOlDl o(Jtimal orthogonal and warly llrlhogonai blolk desi1~

for comparillg ltI set of tfst treatments 10 ltI set of stal(larrl treltmP1I1 SnTlkhlli B62 27G -289

JaCWILX 1 and laJllmdar D (lJ8U) Optimal hlock dl~)igmi for comparing J S(r

of Lest trpalnHmts wiLl 1 ~t of controls when k gt v I Statist 11111111 Ill 23 381-l(]

Jag~i S (1)D2) Study Oll optinwlirv of OIH-Vay lLetlwprgtnlity de~igns for cumshyplril1[ tWI) disjllint stL~ of treatllilIts Clpublishld PhD tltp~i s I TU ~l

Dflbi

[Vol3 1os1amp2

hted A-optimal block ith unequal precision essor Yamamoto) To

plete block designs for 6363-370

illnal block designs [or 13 757-767

timal desillS for COnshy

63-370

orced block designs for 1 Roy Statist Soc

)ll of MV-Optimal block i treatment Statist

~r comparing test treltshy261

r comparing Lest t reatshylath Sciences Washshy

mality of block dsigns nt J Statist Plann

comparillg tlst treatshy0-317

I a (ontrol using block 1345

thogonal block rlrsil~lls reatmpllt~ Sankh1a

~1S for comparil1g sel II Statist Plann InJ

eity de~igns for ClHll shy

D thesis IARl I

2001) DESIGNS FOR TEST TREATMENTS - CONTROL COMPARISONS 145

Jafgi S (1996) A-efficient block designs with unequal block sizes for comparing two se1s of treatments J Indian Soc Agric Statist 48(2) 125-139

Jaggi S G upta VK alld Parsad R (1996) A-efficient block designs for comshyparing two disjoil1l sets of treatments Comm Statist - Theory and Methods 25 967-983

Jaggi S and Gupta VK (1997a) A-optimal block designs with unequal bluck sizes fur comparing two disjoint sets or treatments Sarrkhlli B59 164-180

laggi S and Gupta middot K (l997b) A-optimality of block designs for comparing two di ~joint sets of treatments Technical Report LASRI few Delhi

J aAAi S Parsad rt and Gupta V K (1999) Construction of non-proper balanced bipartite block designs Calcutta Statist A~soc Bull 49 55-u3

Ki~ltUl P (t987) Optimal designs for comparing treatments with a control Unshyublished ]1S( thesi lAIll Kew Delhi

K l1 riako~( S (1999) A study on baluncer and partially balanced incomplete block dp~igll rnpublished PhD thesis LAR few Delhi

--lajulTldar D (19HG) Oprimal designs [or comparisons between two disjoint sets of treatments I Statist Pann InI 14359-372

--l iljllluJar D (199G) Optimal and efficient treatment-control designs In Handshybook of StIListirs 13 (S(1108h and CRllao Edgt) Elsevier Scieml BV

IluJulTIltlar Dand ll)t~ 1 (1983) Optimal iucomplete block d(lsigns for mlDshy

paring trea tments with ) COllLrol J S tatist Plann Inf 4 9 387-40n

Pandey A (1993) Stud) of optimality of block designs undm a Hllxecl effects models Ll1plblishf~d PhD thesis IkRI Kew DelhL

Parsad n (1991) Studiplgt 011 optiwality of incomplete block designs with UIl shy

IJqual block sizes for making test treatments - control comparisons uncirr a IWLlroscedastic model lnpublished PhD Theslh lARI Kew Delhi

Parsad H and Gupta VK (1994lt1) Optimal block designs with 1lT1PCjual block sillgt for making test lreatment~ (ontrol comparisons under a helfOScedatic IlOtit Srmkhyrl B56 ltJ -19-4Gl

Parsad H mel Gupta 1 (] 99middot1b) A-optimality of group uivisible dtsigns with lllllqual block sies fur making test treatments - control comparisons IlIlUtr a hetfroscedastic model Inter 1 MlLth Statist S(middot 5-31

ParsatlH Gupta Vh and Khand uri OP (2000) Cataloguiu) 111lt1 cnflstrurt lOll

of variance iJa(U1(ed block dfsigns computer algorithms for (onstfucliO1 L liB l p1lblication

ParsHI fl Cupla K and Praatl S G (HH)5) On (~OnstrunlOn of A-efficilaquo111 ba iar lced tremlDeI1t illCOlnplee block desill1s UtiltLas Math 471-HIO

146 VK GtPTA AND RAJEfDER PARSAD To13 Jus 1amp2

Parsad R Gupta VK and Singh VPf (1996) Tract- optimal desi~lls [or comparing two disjoint sets of treatments Sankhya B58 414-42G

Prasad KSG (1989) Some investiga1 ions on general efficiency balanced designs lnpubHshecl PhD thesis LAR I few Delhi

Ramana DVV (1995) Optimlt11ity )spects of desiglls for making test lreatmPlltsshycontrol treatments comparisons CnpubHshed PhD thesis IAIU -tw Delhj

Sinha BK (1980) Optimal block designs Cnpublisherl Seminar lotts Iudian Statistical Institute Calcutta

Sinha B K (1982) On complete classes of experiments for certain inarialli prdbshylems of linear inference J Statist Plann Inf 7 Ill-ISO

Sinha K (1992) Construction of balanced treatment incomplete block desllns CommStatist - Theory and Methodil 211377 - 1382

Solonallo E and Spurrier JD (2001) Comparing more than one lIpatllHlIt

to more cJan one control in incomplete blockli J Statiilt Pnrm In 97 385-398

Srivastava R Gupta VK and Parsad R (2000) Studieli on optilJalit) of block designs for making test treatments-control comparisons Technical Report IASJU Kew Delhi

Stllfken T (1987) A-optimal block designs for comparing test treatments with Cl

control AnnStatist 15 1()2) - 1G3

StnfkplJ T (1988) 011 bounds for the (fficienc) of hlork u(-gtsigns fllr comparillf TP IreatnwnLS witll (l control J8tati Plann In 19 3G1 - 72

SLUfknl J (1091) On group clivbibh~ treaLlllllll deiglls for comparin test trcltlt shy

ments with a standard tfearment in blocks of S11( 3 J 8tatiigtt Plann 111 28205-211

Ting CP 1ot 1 (Hl88) A-optimal complete block designs [or treatment - ((Jnt wl comparIsons In Ortinwl De9ign and Analyss of Experimcnts (YD()(ge V FeU()fov illld H P )nll Eds) AmSlerdam fortlt Holland pp 29- 37

i

134 VK GUPTA All) RAJENDER PARSAD [Vo13 [0$1amp2

design an experiment for estimating W1J test treatments verSUlti control treatmpllts contrasts with as high a precision as possible SUPPoltie that n experimental units avaiiable for experimentation can be arranged ill b blocks of sizes kl kl kb respectively kl + k2 + L kb = n Let Ytjl denote the observation of test or control treatment t(t = j v) on experimental unit 1(1 = 1 ntj) of block j (j = 1 b) Assume the Isual fixed effects additive linear model

Ytjl -= fJ + Tt + f3j + etj l (21)

where jt is Lhe overall mean Tt is the effect of test or ontrol treatment t OJ IS the effect of block j and etjl are the random errors normally distributed WIth zero mean and variance CT

2 kj Note that homoscedastid ly is not assumed Here Q ~ 0 IS a scalar constant generally unknown and Iltj is the number of replications of test or control treatlmmt t in block J The v x b incidence matrix N has elements ntJ For 0 = 0 we get the wmal homoscetlastic model

The contrasts of interest are Tg - TI 9 E G h E H Comparisons of treatments within G and with in H are of secondary importance The contrasts of major interest may be written in matrix notation as PT where the wu x v matrix P may be

(xpreslied ltUl Pc [1 It) [ - I (9 1u Wlth pl1v = O Here It is a i-component vector with all elements one II is an identity matrix of order l denoLPs the Kronecker product of maLrices and T is a v-component vector of tellt or control trea tment effects

Let C = (eft) = R- Nf-l N be the usual v xv C-matrix of a block design with v treatments Here R - diag(rl T2 Tv) denotes a diagonal matrix of replication numbers of treatmeuts and J( = diag (k l k2 kb) denotes a diagonal matrix of

block sizes Partition N as N = [NN~J where N J = ((nlj)) is a w x b incidence matrix of Lest trecltlnents and N2 = (nl1) is a 1L x b inciuelltP matrix of control treatshyments Similarly RI = diag(rlT2 T) and R2 = diag(r w+lrw+2) denote respectively the diagonal matriceH of replications of the test treatments and the control treatmellts The information matrix C can then be partitIOned a~

(22)c-(i ~] where

I b b

A - L)lRl i - k -1NljNLJ B - - 2kj I N1j ~) and D = 2klt [R2j

J

j =1 j=1 )=1

k j N 2j ~j Abo Hlj = diag(nij 1111) rilL) ) n~j = diag(n(W-lt- l j

IIg) Hvj) NI = gtV11 middot middotmiddot N1j NIb] N2 = [N21 middotmiddotmiddot N2j N2b N I ) = (nlj n2j middot n uj) and N 2j = (n( w+l)jn(w~2)j nlJ) Fora = 0 11 = R I shy

NJ 1(-1 N B - - Nlj(-I N2 and D =- R2 - N2J(-1 N~

LLt T - h deuote the best lillear unbiased estimator (BLUE) of T - Th E

Gh E If Th( DLlE llf contrais of interest PT is Pr- with Jispersion matrix Cuv(Plf) = cr 2 p( - P Here C- is a generalized inverse of C ie CC C = C

IL is assumed that the deign is connected and Rauk (C) = l - l It might tlso be desiraLlt that the comparisons of interest are estimated t hrough the design Ib

2001] DESIG

the same varia

tegtt treatment call such desi (BBPtB desig

Definjtion 2 i said to be a

b

(i) 2)-1 J=I

II

(ii) 2)jO-1 j=J

b

(Iii) 2)1 j= I

The C rna

c= [ A geneTalized in

c- == [ wI

or 1 8G design mated with sam~

Var(fg - Til)

BG designs h ror 11 == 1 BG ( fJf proper settlll Block Sizes (BT of type G for Q

Bipartite Block Designs with Gr n - (J

indeed there llf Test t reatltlell1 Divisible Bipart or type G henet i- ulfined ~elow

[YoJ3 Nos1amp2


(21)







(22)

b









b


)=1


b





c- shy-





__




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For 11

matrix

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de~igJl wi(i) 0 = w + 1 W + 2 W + it

12 +(ii) VI = V2 = = vrn = n -lshy



j=1

b


b



_ [ (A - 8) 0 1m + B 01ml~ D]C (25)- D E



C = [ X 0








3 Noslamp2


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(25)

~ - Loolu1~

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lI1l1~ Q == m-l)Bt1

outrasts are

uLo)(uLo)








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N= [ a)lu1b191 a21ul~282 ap 1u1~~8


p






N0 _ [Nl 2) l N~ 6-~ 1~2 Np ~~ l~p ]0

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w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































3 Noslamp2


m

lesigu is

(25)

~ - Loolu1~

(26)

lI1l1~ Q == m-l)Bt1

outrasts are

uLo)(uLo)








Nl Qlt) 10 N20102 N p 01e p ]

N= [ a)lu1b191 a21ul~282 ap 1u1~~8


p






N0 _ [Nl 2) l N~ 6-~ 1~2 Np ~~ l~p ]0

L+ as1~8














U middot U tLU

L m )wu ~ wu L lmL 1 = 1 =1








literature





2001J DESIGK













[Vo3 Koslamp2








+pound 1 j = I b






















2001J DESIC

construction i




been seell in 1


5 Weig verSlU



the class of lI

Laa L Var gEG hf li

and so

Lily L aCG hOi

fiud a (h~i

II

[Vo131os1amp2

















w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I











































U middot U tLU

L m )wu ~ wu L lmL 1 = 1 =1








literature





2001J DESIGK













[Vo3 Koslamp2








+pound 1 j = I b






















2001J DESIC

construction i




been seell in 1


5 Weig verSlU



the class of lI

Laa L Var gEG hf li

and so

Lily L aCG hOi

fiud a (h~i

II

[Vo131os1amp2

















w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































[Vo3 Koslamp2








+pound 1 j = I b






















2001J DESIC

construction i




been seell in 1


5 Weig verSlU



the class of lI

Laa L Var gEG hf li

and so

Lily L aCG hOi

fiud a (h~i

II

[Vo131os1amp2

















w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I












































2001J DESIC

construction i




been seell in 1


5 Weig verSlU



the class of lI

Laa L Var gEG hf li

and so

Lily L aCG hOi

fiud a (h~i

II

[Vo131os1amp2

















w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































[Vo131os1amp2

















w-1 lL~ 1-1 1


(ii l)



rL-1 lL




u-l U






w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I









































w It 1 w






2001J DESIG



unequal Plann 1



Cunstantillt J Roy

Cutler RD (HILml w

Cox DB (l(

c IIptl -

BiflTlltt

GlIptl 1

GUpLIl VI

is(lIJ

[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































[Vol3 Nos1amp2


(54) BlL =1

that minimizes

)a+ 3=L (55)













References















Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































Plr~ild H


















Dflbi

[Vol3 1os1amp2





63-370











D thesis IARl I





































[Vol3 1os1amp2





63-370











D thesis IARl I





















































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Stat'8tic~ - krishi.icar.gov.in · 1 Introduction . Designed experiments (tre generally conducted for making all possible parred com \ parisons among . the treatments. However, there