Design Workshop Lecture Notes ISI, Kolkata, November 25-29, 2002, pp. 139-182
Designs for Making Test Treatments Versus Control Treatments Comparisons1
V.K. Gupta IASRI, Library Avenue, New Delhi – 110 012, India
1 Introduction
Designed experiments are generally conducted for making all the possible paired comparisons among the treatments. Comparing treatments with one or more controls is an integral part of many areas of scientific experimentation. In such situations the interest is only in a subset of all possible paired comparisons. In genetic resources environments, which is a field in the forefront of biological research, an essential activity is to test or evaluate new germplasm / provenance / superior selections (test treatments), etc. with the existing provenance or released varieties (control treatments). In pharmaceutical studies new drugs are the test treatments and a placebo and/or a standard treatment is (are) the control(s). These experimental situations may also occur in the fields of entomology, pathology, chemistry, physiology, agronomy and perhaps others for screening experiments on new material and preliminary testing of experiments on promising material. In some other cases (e.g. physics), a single observation on new material may be desirable because of relatively low variability in the experimental material.
For the experimental settings considered here, all the possible paired comparisons among treatments are not of equal interest. The comparisons of interest comprises of comparisons among test treatments and control treatments. The within group comparisons though necessary need not have the same precision levels as required for inter group comparisons. For this experimental setting the usual variance-balanced designs may not be useful. We illustrate this fact through an example.
Example 1.1: Consider an experiment to compare w = 6 test treatments with u = 3 control treatments. n = 72 experimental units available can be grouped into b = 12 blocks of size k = 6 each. Among the large number of treatment connected block designs for 9(= 6 + 3) treatments arranged in 12 blocks of size 6 each, two are the following:
D1: A BIB design with parameters v* = 9 (w+ u), b* = 12, r* = 8, k* = 6, *= 5. D2: A reinforced design obtained by adding the three control treatments in all the blocks of a BIB design w = 6, b* = 10, r* = 5, k* = 3, * = 2 and then adding two more blocks containing each of the 6 test treatments. The design so obtained has parameters v = 9, b = 12,
r (716 , 1013 ) ), k = 6, hh = 4, gh = 5, gg = 10. Here, hh( gg ) is the number
1 The talk is essentially based on the paper Gupta, V.K. and Parsad, R. (2001). Block designs for comparing test treatments with control treatments – an overview. Statistics and Applications, 3(1&2), 133- 146. 140 V.K. Gupta [November of blocks in which any two test treatments (control treatments) concur and gh is the number of blocks in which a test treatment and a control treatment concurs.
9 2 In D1 all the = 36 elementary treatment contrasts are estimated with variance (4/15) 2 6 0.2672. In D2 the = 15 elementary contrasts among test treatments are estimated with 2 3 variance (4/13)2, while the = 3 elementary contrasts among control treatments are 2 estimated with variance (1/5)2 = 0.202. Each of the 18 elementary contrasts of interest between test treatments vs control treatments is estimated with variance (17/65)2 0.2612.
The total variance of all the 36 estimated elementary contrasts for D1 is (624/65) 2, while for D2 it is (645/65) 2. It is known that D1 is the most efficient design for estimating all the paired differences among treatments according to a family of efficiency criteria. This is also evident from the total variance of the estimated elementary contrasts. On the other hand, the total variance of the 18 estimated elementary contrasts between test treatments and control treatments is (312/65)2 for D1 and (306/65)2 for D2. Hence we see that D2 is better than D1 for comparing test treatments with control treatments.
The above examples clearly emphasize that the traditional designs, like BIB designs, are not useful for the experimental settings under consideration. In fact, the experimenter may fail miserably by using these designs, as these are not appropriate for the present experimental setting in which the controls play a distinguished role. Indeed Cox (1958, p.238) noted that BIB designs are not appropriate because of the role played by a control treatment. Sinha (1980, 1982) demonstrated that in the class of equireplicated designs with w test treatments and a control treatment, b blocks and a common block size k, a BIB design whenever existent, is the most efficient design for making test treatments vs control treatment comparisons according to various efficiency criteria. But in the unrestricted class of all connected designs, the limiting efficiency of a BIB design with respect to most efficient design is only 50% for tests vs control comparisons. Cox (1958) suggested a design in which control appears an equal number of times in each block and the test treatments form a BIB design in remaining experimental units. Pesek (1974) provided the analytical details of the Cox's designs. It is easy to prove that reinforced design suggested by Cox is always efficient than a BIB design for test treatments vs control treatment comparisons. It may be important to mention here that for more than one control situation also it has been established empirically that this reinforcement provides an efficient design for w 200, u 10 and k 20. However, it may be possible that the result holds for other parametric combinations also.
From the above discussion, it is clear that for the experimental situations described here, the problem of choosing an appropriate design is important and needs attention. This talk addresses this problem of obtaining efficient designs for the above experimental situations. In the sequel we describe the mathematical formulation of the problem.
2 Problem formulation 2002] Test Treatments Vs Control Treatments Comparisons 141
We have v treatments divided into two disjoint sets, H and G of cardinality respectively w and u, u + w = v, G H . The set H has w test treatments labelled as 1,2,⋯,w while the set G has u control treatments labelled as w 1, w 2,⋯, v . The problem here is to design an experiment to compare test treatments belonging to the set H with control treatments belonging to the set G with as high a precision as possible. Suppose that n experimental units are available for experimentation and these n experimental units can be arranged in b blocks of sizes th k1 , k2 ,⋯, kb , respectively, k1 k2 ⋯ kb n . If the treatment t is assigned to the l th plot of the j block (1 t v; 1 l ntj ; 1 j b) , we shall have ytjl denote the corresponding random variable: we assume the usual fixed effects, additive, and heteroscedastic linear model
ytjl t j tjl (2.1) where is the overall mean, t is the effect of treatment t, j is the effect of block j of size
k j and tjl is the random error present in ytjl and is normally distributed with zero mean and 2 variance k j . Here 0 is a scalar constant, generally unknown. This model is in fact a generalization of Fairfield Smith’s Variance Law. The value of may be estimated from the uniformity trial data. Lee and Jacroux (1987), Gupta, Das and Dey (1991), Das, Gupta and Das (1992), Parsad and Gupta (1994a, 1994b), Gupta (1995), Parsad, Gupta and Singh (1996) and Srivastava, Gupta and Parsad (2000) have earlier studied this model. For 0 we get the usual homoscedastic model. is usually taken as zero.
As described above, the contrasts of interest in the experiment are g h , g G, h H . Comparisons of treatments within G and within H are of secondary importance, though in many practical situations, the comparisons among treatments within the groups are also important, though not as important as the between group comparisons. The contrasts of interest then are also g g , g gG and h h , h h H . We rewrite the estimable parametric contrasts of interest in matrix notation as P , where uw x v matrix P may be expressed as
P 1u I w ⋮ I u 1w with P1 0 . is a v-component vector of treatment effects, 1t is a t-component vector of ones, It is an identity matrix of order t, 0t is a t-component vector of zeros, and denotes the Kronecker product of matrices.
Suppose that for some design d, N d ndtj denotes v x b incidence matrix and 1 ' Cd Rd NdK Nd be its C-matrix, Cd ((cdtt )), t, t 1,⋯,v. Here the non-negative th th integer ndtj denotes the number of times the t treatment occurs in the j block of d, R =
diag(r1 , r2 ,⋯, rv ) denotes a diagonal matrix of replication numbers of treatments and K =
diag(k1 , k2 ,⋯, kb ) denotes a diagonal matrix of block sizes. We may partition Nd as N ((n )) N ((n )) N d Nd1 ⋮ Nd 2 , where d1 dhj is a w x b matrix and d2 dgj is a u x b 142 V.K. Gupta [November
th th matrix, ndhj and ndgj being the number of times h test treatment and g control treatment th appears in the j block, respectively. Similarly, R 1 diag(r1 , r2 ,⋯, rw ) and
R 2 diag(rw1 , rw2 ,⋯, rv ) , denotes respectively the diagonal matrices of replications of the tests and the controls. The information matrix Cd can thus be partitioned as
A d B d Cd (2.2) Bd Dd where
b b 1 1 A d k j R 1 j k j N d1 j Nd1 j , B d k j N d1 j Nd 2 j , and j1 j1 b 1 Dd k j R 2 j k j N d 2 j Nd 2 j . Also R 1j diag(nd1 j ,⋯, ndhj ,⋯, ndwj ) , j1
R 2j diag(nd w1 j ,⋯, ndgj ,⋯, ndvj ) , N d1 N d11 ⋮⋯⋮N d1 j ⋮⋯⋮N d1b , N N ⋮⋯⋮N ⋮⋯⋮N d 2 d 21 d 2 j d 2b , N d1j nd1 j , nd 2 j ⋯, ndwj and N d 2j nd (w1) j , nd (w2) j ,⋯ndvj . For homoscedastic set up, 0 , and then 1 1 1 A R 1 N d1K Nd1 , B N d1K Nd 2 and D R 2 N d 2 K Nd 2 .
Suppose that ˆg ˆh denotes the Best Linear Unbiased Estimator (BLUE) of g h , g G, h H. Similarly ˆg ˆg and ˆh ˆh are respectively the BLUE of g g ,
g gG and h h , h h H . The BLUE of parametric contrasts of interest P is Pˆ with dispersion matrix Cov(Pˆ) 2 PC P . Here C is a generalized inverse of C and satisfies CCC C .
A design that permits the estimation of all the v – 1 linearly independent treatment contrasts is called connected and for a connected design Rank (C) = v - 1. Further, it would also be desirable that the comparisons of interest are estimated through the design with the same variance. The precision of other comparisons is of no consequence to the experimenter although these are also estimable through a connected design. A design is said to be variance balanced for the estimation of test treatments vs control treatments comparisons if it permits the estimation of these comparisons with the same variance and the covariance between any two estimated test treatments vs control treatments comparisons is also the same. In general we shall call such designs as Balanced Bipartite Block Designs with Unequal Block Sizes (BBPUB designs) of type G and henceforth denote these as BG designs.
Definition 2.1. An arrangement of v treatments in b blocks of sizes k1 , k2 ,⋯, kb is said to be a BG design if 2002] Test Treatments Vs Control Treatments Comparisons 143
b 1 (i) k j nhj nhj L , a constant h h 1,⋯,w , j1 b 1 (ii) k j ngj ngj L00 ,a constant g g w 1,⋯, v , j1 b 1 (iii) k j ngj nhj L0 ,a constant h 1,⋯, w; g w 1,⋯,v . j1 The C matrix of a BG design is
wL uL0 I w L1w1w - L0 1w1u C . (2.3) L0 1u1w uL00 wL0 Iu L00 1u1u
A generalized inverse of C is
1 I (L / uL )1 1 0 wL uL w 0 w w C 0 1 (2.4) 0 I (1/ u)1 1 u u uL00 wL0
It is easily seen that for the BBPUB designs of type G 2 2 I. Var(ˆh ˆh ) , h,h H , wL uL0 2 2 II. Var(ˆg ˆg ) , g,gG , wL0 uL00
1 (v 1)L0 L (v 1)L0 L00 2 III. Var(ˆg ˆh ) ,g G, h H. vL0 wL uL0 uL00 wL0
The equivalent expressions for Var(ˆg ˆh ) available in literature are
wL0 L00 w 1 2 L uL0 u 1 2 and . wL0 uL00 wL0 wwL uL0 uL0 wL uL0 uwL0 uL00
For a BBPUB design of type G all the wu test treatments vs control treatments comparisons are w estimated with same variance. Further, all the possible paired comparisons among the test 2 u treatments and possible paired comparisons among the control treatments are also 2 estimated with same variance. BBPUB designs of type G have been studied extensively in the literature under different names. A brief account of these is given in Table 1. 144 V.K. Gupta [November
Indeed there may exist designs other than BG designs that permit the estimation of test treatments vs control treatments comparisons with the same variance. These designs are called Group Divisible Bipartite Block Designs with Unequal Block Sizes (GDBPBUB designs) of type G. Henceforth these designs will be denoted as GBG designs. The GBG design is defined as below:
Definition 2.2. An arrangement of v u w , w mn treatments in b blocks of sizes k1 , k2 ,⋯, kb with parameters u, w, m, n,b,k 1, k2 ,⋯, kb , L0 , L1 , L2 , L00 is said to be a GBG design if the v treatments can be partitioned into m + 1 disjoint groups V0 ,V1 ,⋯,Vm of respective cardinalities u, v1 , v2 ,⋯, vm such that (i) V0 w 1,w 2,⋯,w u, (ii) v1 v2 ⋯ vm n , b 1 (iii) k j ngj ngj L00 , a constant g gV0 , j1 b 1 (iv) k j ngj nhj L0 , a cons tan t g V0 , h 1,⋯, w, j1 b 1 (v) k j nhj nhj L1 , a cons tan t h hVq , q 1,⋯,m , j1 b 1 (vi) k j nhj nhj L2 , a cons tan t h h, hVq , hVq , q q 1,⋯,m . j1
Here L0 ,L1 ,L2 ,L00 are some constants. The C matrix of a GBG design is given as
A B I m B 1m1m D C . (2.5) D E
Here A nL1 n(m 1)L2 uL0 I n L11n1n , E (wL0 uL00 ) Iu L00 1u1u , B L2 1n1n , D L0 1w1u , X 1 0 A generalized inverse of C is C 1 1 , (2.6) 0 Iu 1u1u wL0 uL00 u 1 with X A B I m B 1m1m , X P Q I m Q 1m1m , Q BA B1 A m 1B1 and P A m 2BA B1 A m 1B1 .
For a GBG design the test treatments vs control treatments contrasts are estimated with the same variance given by 2002] Test Treatments Vs Control Treatments Comparisons 145
1 L1 L2 Var(ˆg ˆh ) nL1 wL2 nL2 uL0 nL1 wL2 nL2 uL0 uL0
L2 u 1 2 uL0 wL2 uL0 uuL00 wL0
For L1 = L2, these designs are same as BG designs. Some interesting special cases of GBG designs have been studied in the literature and are given in Table 1.
3 Method of construction of BBPUB designs of type G
The purpose of this Section is to give a general method of construction of BG and GBG designs. The methods of constructions, hitherto known in the literature, fall out as a special cases of this method.
Method 3.1. Suppose that there exists a equi replicated pairwise balanced binary block (PBBB) w,b , b ,⋯,b , k (k 1 , k 1 ,⋯, k 1 ) design with parameters 1 2 p 1 b1 2 b2 p bp , r r1w , ,
b b1 b2 ⋯ b p and incidence matrix N N1 ⋮N 2 ⋮⋯⋮N p , N s , s 1,⋯, p being th the w x bs matrix corresponding to the incidence of treatments in the s block having blocks of sizes k s satisfying N s 1w rs 1w and r r1 r2 rp . Then the design with incidence * matrix N is a BG design with parameters w, u, b* b1 1 b2 2 ⋯ b p p , p p r {( s rs )1w ,( asbs s )1u } , s1 s1 k* k a u 1 , k a u 1 ,⋯, k a u 1 1 1 b11 2 2 b22 p p bp p , where a1 , a2 ,⋯, a p are some non-negative integers and 1 , 2 ,⋯, p are taken in the ratio 1 1 1 k1 a1u : k2 a2u :⋯: k p a pu :
N1 1 N 2 1 ⋯ N p 1 N* 1 2 p a 1 1 a 1 1 ⋯ a 1 1 1 u b11 2 u b2θ2 p u bp p
For u = 1, an efficient block design for making test treatments-control treatment comparisons can be obtained by taking as int(ks / 2); s 1,..., p . For a review of the methods of construction of PBBB designs, see Parsad, Gupta and Khanduri (2000).
Remark 3.1. Let N is the incidence matrix of a μ1 ,μ2 ,⋯, μs ,⋯, μ p -resolvable PBBB design, i.e., N s 1w s 1w , s 1,2,⋯, p . Then the design with incidence matrix
N 1 N 1 ⋯ N 1 1 1 2 2 p p N** a 1 s bsθs 146 V.K. Gupta [November
is a BG design with parameters w, u, b* b1 1 b2 2 ⋯ b p p k* k a 1 , k a 1 ,⋯, k a 1 1 1 b11 2 2 b22 p p bp p , where a1 , a2 ,⋯, a p are some p , ,⋯, non-negative integers such that ass a, a being a scalar constant and 1 2 p are s1 taken in the ratio 1 1 1 k1 a1 : k2 a2 :⋯: k p a p .
Remark 3.2. Suppose that there exists a binary block design with parameters w,b , b ,⋯,b , k (k 1 , k 1 ,⋯, k 1 ) 1 2 p 1 b1 2 b2 p bp , λ1 ,λ2 , b b1 b2 ⋯ b p and incidence matrix N N1 ⋮N 2 ⋮⋯⋮N p , N s , s 1,⋯, p being the w x bs matrix
th corresponding to the incidence of treatments in the j block having blocks of sizes ks satisfying
N s 1w rs 1w . The w test treatments can be divided into m disjoint sets V1 ,V2 ,⋯,Vm of cardinality n each such that p 1 , h hVq , q 1,2,⋯, m shh , h h, hV , hV , q q 1,2,⋯, m. s1 2 q q
Here shh is the concurrence of treatments h and h in N s . One way of obtaining such a design is to replace every treatment in an equireplicate PBBB design in m treatments with a group of n new treatments. Following then the procedure given above in Method 3.1 for BG designs on N gives a GBG design.
The experimental situations just described can be classified into two broad categories viz. CATEGORY A and CATEGORY B experiments. In category A experiments the test treatments have only single replication in the design. However, in category B experiments the test treatments are replicated in the design.
The replication of controls, however, is always possible in both the situations. In Table 1 below we describe in brief the historical development of the designs in the two categories of experiments:
Table 1: Summary of block designs developed for the experimental settings of interest
Category A experiments Category B experiments SINGLE CONTROL Augmented design (Federer, 1956, 1961, Supplemented balanced or Type-S 1963; Federer and Raghavarao, 1975; design (Pearce, 1960) Federer, Nair and Raghavarao, 1975; BTIB design (Bechhofer and Singh and Dey,1979; Parsad and Gupta, Tamhane, 1981) 2000) R - type and S - type BTIB design Designs with minimum number of (Hedayat and Majumdar, 1984) observations or minimally connected GDTD design (Jacroux, 1987a) design (Dey, Shah and Das, 1995) BTB design (Jacroux and Majumdar, 2002] Test Treatments Vs Control Treatments Comparisons 147
1989) BTIUB design (Angelis and Moyssiadis, 1991) BTUB design (Jacroux, 1992) BTIUB design of type G (Parsad and Gupta, 1994a) GDTUB design of type G (Parsad and Gupta, 1994b) BTUB design of type G (Srivastava, Gupta and Parsad, 2000
MANY CONTROLS Augmented design Inter and Intra group balanced design (Federer, 1956, 1961, 1963; Federer and (Nair and Rao, 1942) Raghavarao, 1975; Federer, Nair and Reinforced design (Das, 1954, 1958; Raghavarao, 1975; Parsad and Gupta, 2000b) Giri, 1957, Kulkarni, 1960; Pearce, An augmented design is any standard design in 1994) controls or standards to which additional new Balanced block designs with two treatments (tests) have been added. different number of replicates (Corsten, 1962) Balanced block designs with variable replications (Adhikary, 1965) BBPB design (Kageyama and Sinha, 1988; Sinha and Kageyama, 1990) GDBPB design (Kuriakose, 1999) BBPUB design (Parsad, Gupta and Singh, 1996) BBPUB design of type G (Parsad, Gupta and Singh, 1996) Note: For notations, abbreviations and details of the various designs described in the table please refer to the papers mentioned against the designs.
4 Category A experiments with single replication of tests
In many agricultural situations experiments are often conducted where the existing practices called control treatments or check varieties are to be compared with new varieties or germplasms collected through exotic or domestic collections, called test treatments. The experimental material for test treatments is scarce and it is not possible to replicate them in the design. However, enough material is available for replicating control treatments in the design. These kind of experimental situations came to be known to Federer (1955) in screening new strains of sugarcane and soil fumigants used in pineapples. Augmented (Hoonuiaku) designs were introduced by Federer (1956) to fill a need arising in screening new strains of sugarcane at Experimental Station of Hawaiian Sugarcane Planters Association on the basis of agronomic characters other than yield. An augmented experimental design is any standard experimental design in standard (control) treatments to which additional (new or test) treatments have been added. The additional treatments require enlargement of either the complete block or the incomplete block in the block design setting and the complete or incomplete rows and (or) columns in the row-column design settings, etc. The groupings in an augmented design may be of unequal sizes. Federer (1956, 1961) gave the analysis, randomization procedure and construction of these designs by adding the new treatments to the blocks of randomised 148 V.K. Gupta [November complete block (RCB) design and balanced lattice designs. Federer (1963) gave procedures and designs useful for screening material inspection and allocation with a bibliography. Federer and Raghavarao (1975) who obtained augmented designs using RCB design and linked block designs for one-way heterogeneity setting gave a general theory of augmented designs. They also gave a method of construction of augmented row - column designs using a Youden Square design and also provided formulae for standard errors of estimable treatment contrasts. Federer, Nair and Raghavarao (1975) gave systematic methods of construction of augmented row column design and an analytic procedure for these designs. The estimable contrasts in augmented designs are (i) among test treatments, (ii) among control treatments, (iii) among all test treatments and control treatments simultaneously. They also provided formulae for standard errors of estimable treatment contrasts.
A user-friendly program AUGMENT1 developed at the Documentation Unit of National Bureau of Plant Genetic Resources, New Delhi by Agarwal and Sapra (1995) may be used to analyze the data from an Augmented RCB design. Work at IASRI, New Delhi is in progress to extend the scope of this software to a general situation to cover all the augmented designs generated by augmenting test treatments in the blocks of incomplete block designs, row-column designs, etc. in control treatments and also when data on some co-variates is available.
4.1 Some other considerations Augmented designs are very popular with the experimenters particularly in germplasm experiments. However, in these experiments a question that is frequently asked by the experimenters concerns the choice of the number of replications of the control treatments in each block of the augmented design so as to maximize the efficiency per observation for making test treatments vs control treatments comparisons? To make the idea clearer, consider the following example:
Example 4.1: 36 test treatments are to be compared with 2control treatments. The experiment can be laid out in a block design with three blocks. The blocks may have unequal number of experimental units. There is enough material on control treatments to replicate them but the test treatments can be replicated only once. Consider three possible designs, D1, D2 and D3, in which the replication of the control treatment in each block is one, two and three, respectively. Thus in the three designs each of the control treatments is replicated three, six and nine times. The average variance of the BLUE of the tests vs control contrasts and efficiency per observation for D1, D2 and D3 is given in the following table.
Design Variance Efficiency per observation 2 -2 D1: w 36,u 2,b 3,r 1,r0 3,n 42 (15/9) 0.0143 2 -2 D2: w 36,u 2,b 3,r 2,r0 6,n 48 (12/9) 0.0156 2 -2 D3: w 36,u 2,b 3,r 3,r0 9,n 54 (11/9) 0.0152
This example clearly shows that D2 is better than D1 and D3 according to efficiency per observation criterion. It is also observed that efficiency per observation doesn't increase with the increase in the number of replications of the control in each block.
In the sequel, we work out the optimum number of replications of control treatments in each block so as to maximize the efficiency per observation. 2002] Test Treatments Vs Control Treatments Comparisons 149
4.2 Designs with Maximum Efficiency Per Observation b Suppose that w k j test treatments are to be compared with u control treatments. Each of j1 the u control treatments is replicated r times in each of the b blocks. The block sizes are k j ur , j=1, …, b. The total number of experimental units is w ubr and the total number of treatments is w + u. If we denote by 1,⋯,w the test treatments and by w 1,⋯,w u the control treatments, then the coefficient matrix of reduced normal equations for estimating treatment effects is given by:
b A B 2 1 C , where A diag ( A1,⋯,A j ,⋯,Ab ) , D brI r 1 1 , B D u u u j1 k j ur
r r r B 1 1 , 1 1 , ⋯, 1 1 u k1 u k2 u kb , and k1 ur k2 ur kb ur
1 A I 1 1 j k j k1 k j ; j=1, …, b. A generalized inverse of C is given by k j ur
1 I 1 1 0 . . . 0 0 k1 ur k1 k1 1 0 I k 1k 1k . . . 0 0 2 ur 2 2 ...... ...... C ...... 1 0 0 . . . I 1 1 0 kb kb kb ur 1 1 0 0 . . . 0 Iu 1u1u br u It is easy to see that the design is variance balanced with respect to the contrasts of interest and the variance of BLUE of these contrasts is given by
ubr b u 1 2 Var (ˆg ˆh )= ; h 1,..., w; g w 1,..., w u (4.1) ubr
2 2 ubr b u 1 ubr Therefore, average variance, V and Efficiency = . ubr ubr b u 1 150 V.K. Gupta [November
2ubr Efficiency/Observation = = 2 b, w, u, r (4.2) ubr b u 1w ubr
For fixed b, u and w, behave as a function of r and is independent of k1 ,...., kb separately. It b depends on the k j ' s only through w k j . j1 b u 1 w b, w,u, r w b u 1ubr b u 1 w b u 1w ubr
1 b u 1 w w b u 1 ubr b u 1 w ubr '(r) 0 w u b 1 0 (w ubr)2 (ubr b u 1)2
1 u b 1 w u b 1 w r u b b (4.3) ub
''(r) | 1 ub1 w 0 r u b b requires that b u 1 w . Therefore, the efficiency per observation is maximized when r is as given in (4.3) provided b u 1 w. For example, consider the problem of obtaining the optimum number of replications of the control treatments in an experiment with w = 24, u = 3, b = 4. We have 1 6 24 r 1. 3 4 4 Similarly, for w=98, u=2, b=7, we have 1 8 98 r 2 . 2 7 7 w Remark 4.1: For a single control situation, i.e. u = 1, the expression (4.3) reduces to r b and it can easily be seen that for u = 1, b w, which is always true. 2002] Test Treatments Vs Control Treatments Comparisons 151
In practice, however, the solution of the expression (4.3) may not always yield an integer value of r. A natural question then is as to what integer value of r should be taken? To answer this question, efficiency per observation was calculated for w 100, b 25 and u 10 such that b + u –1 w and r was taken as r* = int (r) and int (r) + 1 in the expression (4.3) besides taking r = 1. A close scrutiny reveals that if value of r > #.42 then take r* = int (r) + 1 and for values of r * smaller than or equal to #.42 take r = int (r) for u 2. For u = 1, the same rule applies but the value of r is taken as #.45 instead of #.42.
Augmented designs with u = 1 and r = 1 are the minimally connected designs (i.e., designs with minimum number of observations, number of observations (n) = number of treatments (w+1) + number of blocks (b) – 1). Allocating the control treatment once to each of the b blocks and allocating the w test treatments to the remaining n – b units of the b blocks such that a test treatment appears in only one block gives such augmented designs. For example, a design with w+1=9, b=4, k=3 is
9 9 9 9
9 1 3 5 7 2 4 6 8
These designs have been shown to be A- and MV- optimal for making test treatments-control treatment comparisons in a proper block design set up by Dey, Shah and Das (1995). Through simple algebra on the lines of Lemma 4.1 and Theorem 5.2 of Dey, Shah and Das (1995), it can be shown that the above procedure gives A- and MV-optimal designs for making test treatments control comparisons under a non-proper block design set up as well. One may be interested to know whether such A- and MV-optimal augmented designs for making test treatments-control comparisons also maximize the efficiency per observation or not. To answer this question, the combinations of w and b which maximizes the efficiency per observation have been obtained through empirical investigations as described earlier for obtaining the value of r, and are given in the following table.
b w b W b w 2 3 to 4 10 11 to 20 18 19 to 36 3 4 to 6 11 12 to 22 19 20 to 38 4 5 to 8 12 13 to 24 20 21 to 40 5 6 to 10 13 14 to 26 21 22 to 42 6 7 to 12 14 15 to 28 22 23 to 44 7 8 to 14 15 16 to 30 23 24 to 46 8 9 to 16 16 17 to 32 24 25 to 48 9 10 to 18 17 18 to 34 25 26 to 50
5 Category B Experiments With Many Replications Of The Tests Category B experiments are in fact a follow up of the Category A experiments. Several authors have investigated the optimality and construction aspects of designs for these experiments useful for making test treatments vs control treatments comparisons. For a critical and excellent review on the subject, reference may be made to Hedayat, Jacroux and Majumdar (1988), Majumdar (1996) and Gupta and Parsad (2001). Methods of construction and the catalogues of 152 V.K. Gupta [November these designs along with their efficiencies can be used by experimenters for choosing an experimental design.
This category of experiments may be with a single control treatment or more than one control treatment. We describe both types of experiments in the sequel.
5.1 Single Control Treatment 5.1.1 Multiple Comparison Procedure
Multiple comparison procedures involve optimum allocation of experimental units to control treatments and test treatments to maximize the probability connected with the joint confidence statement concerning many to one comparison between the means of control treatments and means of the test treatments {see e.g., Roessler (1946); Paulson (1952); Dunnett (1955, 1964); Robson (1961); Bechhofer and his co-workers (1969, 1970, 1971); Hochberg and Tamhane (1987); Bechhofer and Dunnett (1988); Hoover (1991), Dunnett, Horn and Vollandt (2001).
Bechhofer and Tamhane (1981, 1983a, 1983b, 1985), Notz and Tamhane (1983) introduced and studied BTIB designs that are optimal for simultaneous confidence intervals for test treatment vs control treatment contrasts and gave some tables of optimal allocation of observations for comparing test treatments with a control
The analysis of multiple comparison procedure of Dunnett can be performed by using options available under the MEANS statement in PROC ANOVA and PROC GLM of SAS system for linear models. These are DUNNETT, DUNNETTL and DUNNETTU, respectively, a two- tailed, a left tailed and a right tailed test to test if the effect of any test treatment is significantly different from that of control.
5.1.2 Supplemented Balance
The problem of comparing several tests with the control has also been studied using the concept of supplemented balance (type S) designs. Hoblyn, Pearce and Freeman (1954) suggested this device in connection with the changing of treatments in successive experiments in cases where it can be assumed that the new treatments will not interact with residual effects from those previously applied. According to them a design is supplemented balance if all treatments are replicated same number of times and concurrence of each of the test treatments with that of control treatment is same. Pearce (1960) provided real life examples for these supplemented balanced designs that were conducted at East Malling Research Station in 1953. The experimental situation 2 described in Section 1 in fact refers to this type of designs. The basic approach in these designs is to supplement any standard design in test treatments with replications of control treatment. Nigam, Gupta and Narain (1979) have written a monograph “Supplemented Balanced Block Designs”.
It can easily be seen that type S designs and BTIB (BTB) designs are similar and estimate all the elementary contrasts between test treatments and control treatment with same variance and covariance and also variances of estimates of elementary contrasts between test treatments are same.
Both the above approaches concern mainly with comparing several test treatments with a single control treatment. It is possible, though, to extend these approaches to comparing a set of test treatments with a set of control treatments. 2002] Test Treatments Vs Control Treatments Comparisons 153
5.2 More than one control treatment
The problem of comparing two disjoint sets of treatments for more than one replication of tests has been studied with different names in the literature. Nair and Rao (1942) termed the existence of such designs as inter and intra group balanced designs. Rao (1947) gave the analysis of such designs. Corsten (1962) investigated combinatorial aspects of these designs and termed them as balanced block designs with two different numbers of replicates. Adhikary (1965) also studied these designs with the name of balanced block designs with variable replications. Kageyama and Sinha (1988) and Sinha and Kageyama (1990) gave systematic methods of construction of these designs along with their catalogues. They called such designs as balanced bipartite block (BBPB) designs. The tests for simultaneous comparisons of multiple comparisons with more than one control have been developed by Hoover (1991); Kwong (2001) and Solorzano and Spurrier (2001).
5.2.1 Reinforced Designs The technique of reinforcement was initiated to control large intra block variances but if looked from another angle, this may also be used to compare a set of test treatments with a set of control treatments. Reinforced designs were developed to meet the need of experimental situations where it may not be possible to use a BIB design, a Lattice design or a partially balanced incomplete block (PBIB) design because of non-availability of large number of replications required in these designs. Das (1954) suggested a design that may be obtained by adding some extra treatments to each of the blocks of a BIB design. Some extra blocks are also added containing all the treatments, the treatments in the BIB design as well as the treatments added to each block of the BIB design. With a given number of treatments v, the u extra treatments can always be so adjusted that w (=v - u) treatments form a BIB design. Due to limitation on w, it may not always be possible to keep in such designs the total number of plots at any desired level having at the same time smaller blocks. This problem to some extent was solved by Giri (1957), who used the base designs in w treatments as a PBIB design and termed the resulting design as reinforced partially balanced incomplete block design.
A design obtained by reinforcing (i) each block of an incomplete block design in w treatments, b blocks and block size k with single replication of each of the u (>0) extra treatments, (ii) t (0) new blocks containing each of the u + w treatments are added to the b blocks of the incomplete block design, has been termed as a reinforced incomplete block design by Das (1958). The parameters of the reinforced design are v* v u w, b* b t, * * * * * k1 k2 ⋯kb k u, kb1 ⋯ kbt v . It is obvious that no general criterion can be set for choice of u. It is also clear that the reinforced incomplete block designs can be used for making comparisons between a set of test treatments and a set of control treatments.
Kulkarni (1960) has shown that in a reinforced incomplete block design, the mean variance of the differences between all pairs of treatments is a decreasing function of block size k. In other words, by adding a few extra treatments to an incomplete block design, the mean variance of all the pairs of treatments belonging to the original design is reduced. Pearce (1994) studied Reinforced lattice designs for both block and row-column designs set up.
It can be seen easily that information matrices of reinforced balanced incomplete block design, inter and intra group balanced block design, balanced block design with two different replications and balanced bipartite block design are similar to that of a BBPUB design of type 154 V.K. Gupta [November
G. Pearce (1983, pp. 128-129) through a discussion has shown that reinforced and supplementation techniques are related. It can easily be seen that with t = 0 and u = 1, reinforced BIB designs of Das (1954), supplemented balanced designs of Pearce (1960) and BTIB designs of Bechhofer and Tamhane (1981) are same. The reinforced group divisible design of Giri (1957) is the same as GDBPUB design of type G.
The analysis of the data generated from the above design is similar to that given in Section 4. Generalized inverses of C-matrix in (2.4) and (2.6) may be useful in this regard.
6. Optimality Results The experimental situations addressed here demand that all test treatments are tested against every control treatment. It, therefore, seems logical that in the choice of an appropriate design we put more emphasis on the allocation of control treatment(s) to the plots within a block. Intuitively it appears desirable to have more replications of the control treatments as compared to the test treatments. The statistical problem, therefore, is to obtain a suitable arrangement of treatments in the plots within blocks such that test treatments vs control treatments comparisons are estimated with as high a precision as possible. The most appropriate criteria for searching most efficient (optimal) design for making test treatments vs control treatment(s) comparisons, P , are A- and MV-optimality criteria. A design d* belonging to a certain class of competing Var ˆ ˆ designs D is said to be A-optimal if d* minimizes g h over all designs d D gGhH . In other words, a design d* is A-optimal over D if it has the minimum average variance of th Pˆ as compared to any other design d D . Let mL , L 1,⋯,uw be the L diagonal element of PCd P , the dispersion matrix of Pˆ . Then one has to find a design d*D that uw mL minimizes L1 over all D. From the arithmetic mean – harmonic mean inequality it follows uw that uw mL uw L1 . uw uw 1 L1 mL
The equality is attained when mL m L 1,⋯,uw and this holds for BG and GBG designs. These designs are, therefore, the candidate for the most efficient designs for test treatments vs control treatments comparisons according to A-optimality criterion.
A design d* belonging to a certain class of competing designs D is said to be an MV-optimal design if d* has the least value of the maximum variance of BLUE of elementary contrasts between test treatments and control treatments as compared to any other design d D. It may be mentioned here that for the present problem all the A-optimal designs are MV-optimal as well.
In the sequel we give some optimality results for orthogonal designs. We shall generally give A- optimality results. 2002] Test Treatments Vs Control Treatments Comparisons 155
Result 1: In a 0-way heterogeneity setting, for comparing w test treatments with u controls through n experimental units a design is A-optimal if n = 0 mod (w + uw ) and uw is a perfect w square and n 0 mod(u uw) when each control is replicated times the replication of u n the test treatments. For an A-optimal design r1 ... rw and w uw w n r ... r th w1 v , where rt is the replication number of the t treatment. u w uw
Thus, if w = 8 test treatments are to be compared with u = 2 control treatments using 36 experimental units then the A-optimal design assigns 3 experimental units to each of the test treatments and 6 to each of the control treatments. For u = 1, this result was first noticed by Fieler (1947) and possibly even earlier (see also Finney, 1952). For u = 1, the above result demands that for an A-optimal design the ratio of the replication of the control treatment to that of the test treatments should be equal to the square root of the number of test treatments.
Result 2: {Majumdar (1986)}. For one-way heterogeneity setting, for comparing w test treatments with u control treatments using n = bk experimental units arranged in b blocks of size k each, a design is A-optimal if k 0 mod(w uw) and uw is a perfect square and k 0 mod(u uw) k w and nhj , n n h 1,⋯, w; g w 1,⋯, v; j 1,⋯,b . w uw gj u hj
For example, if the experimenter is interested in comparing 4 test treatments with one control treatment using a complete block design in 4 blocks, then the most efficient orthogonal block design is
Block -1 : (0,0,1,2,3,4); Block -2 : (0,0,1,2,3,4); Block -3 : (0,0,1,2,3,4); Block -4 : (0,0,1,2,3,4).
Result 3: For two-way elimination of heterogeneity setting, for comparing w test treatments with u control treatments via n = bk experimental units arranged in b rows and k columns, a design is A-optimal if k 0 mod(w uw) ; k 0 mod(u uw) and b 0 mod(w uw) and b 0 mod(u uw) and uw is a perfect square when each control w treatment is replicated times the replication of the test treatments in each of the b rows u and also in each of the k columns.
For example, if the experimenter is interested in comparing w = 4 test treatments with u = 2 control treatments using a row-column design then the most efficient design is obtained by writing a Latin square of order v v 4 2 6 and merging treatments 5 and 6 to form the control, say 0. The A-optimal orthogonal row-column design is 156 V.K. Gupta [November
1 2 3 4 0 0 2 3 4 0 0 1 3 4 0 0 1 2 4 0 0 1 2 3 0 0 1 2 3 4 0 1 2 3 4 0
For non-orthogonal designs the following optimality results are available in the literature.
Result 4: Constantine (1983) showed that a reinforced BIB design obtained by adding a control treatment once in every block of a BIB design is A-optimal in the restricted class of block designs having a single replication of the control treatment in each block.
Result 5: 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 of block of a most balanced group divisible design is A-optimal.
All such designs in which a standard treatment is reinforced in each block of the design were termed as Standard reinforced (SR-) designs.
Result 6: Majumdar and Notz (1983) initiated a rigorous treatment to the optimality aspects of designs for test treatment vs control treatment comparisons in a general class of connected designs. They showed that a BTIB design, binary in test treatments, and replications of control b b th in the j block as n0 j int(n0 j /b) or int( n0 j / b) 1 with 1 no j int (k/2), j = j1 j1 1, …, b, block size as k and with least value of the trace of the variance-covariance matrix of BLUE of test treatment vs control treatment comparisons in the class of competing designs is A- optimal. Hedayat and Majumdar (1984) utilized these conditions to give a stronger definition of BTIB designs. They classified the BTIB designs as (a) Rectangular or R- type BTIB designs (equal replication of the control treatment in all the blocks) and (b) Step or (S-type) BTIB designs (the replications of the control treatment in the blocks differs by one). Stufken (1988) gave the bounds to the A-efficiency of BTIB designs. Cheng, Majumdar, Stufken and Ture (1989) studied the A- and MV-optimality of S-type BTIB designs and gave 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 construction.
Hedayat and Majumdar (1985) obtained a sufficient condition for A-optimality in form of an inequality involving number of test treatments and block size. This sufficient condition is helpful in obtaining A-optimal R-type BTIB designs having single replication of the control treatment in each block. Stufken (1987) extended the sufficient condition to the case of R-type BTIB designs having t 1 replications of the control treatment in each block.
Result 7: (Stufken, 1987). A BTIB design obtained by adding a control treatment t times to each of the blocks of a BIB design with parameters w, b, r, k-t, in test treatments is A-optimal whenever k t 12 1 wt 2 k t2 .
For example, the following design is most efficient (A-optimal) for comparing 7 tests (1,2,...,7) with one control (0), when 28 experimental units are arranged in 7 blocks of size 4 each. 2002] Test Treatments Vs Control Treatments Comparisons 157
Block -1 : (0, 1, 2, 4); Block -2 : (0, 2, 3, 5); Block -3 : (0, 3, 4, 6) Block -4 : (0, 4, 5, 7); Block -5 : (0, 5, 6, 1); Block -6 : (0, 6, 7, 2) Block -7 : (0, 7, 1, 3)
Gupta (1989) obtained a simpler sufficient condition to search an A-optimal design among the class of all connected binary block designs in terms of elements of information matrix.
Result 8: A BTIB design is said to be A-optimal in the class of designs that is binary in test treatments as well as control treatment for comparing tests with one control if 1/2 0 / = 1+ (w + 1) , where 0 is the number of blocks in which a control treatment and test treatments concur and is the number of blocks in which any two test treatments concur. Parsad, Gupta and Singh (1996) have shown that this condition is never satisfied by an R-type BTIB design and hence, is useful only to check whether a given S-type BTIB design is A-optimal or not.
Sinha (1992) gave general methods of construction of BTIB designs by merging treatments in a group divisible design. Parsad, Gupta and Prasad (1995) gave general methods of construction of BTIB designs and investigated their optimality using sufficient condition of Hedayat and Majumdar (1984) and Gupta (1989). Jacroux (1987a) introduced Group Divisible Treatment Designs (GDTD). A computer intensive sufficient condition for a GDTD to be A-optimal is given by Hedayat, Jacroux and Majumdar (1988). Jacroux (1987b, 1987c, 1988, 1989), Ting and Notz (1988), Giovagnoli and Wynn (1985) and Stufken (1991) have also provided some interesting results. Jacroux and Majumdar (1989) gave optimal block designs for comparing test treatments with a control treatment when block size is greater than the number of test treatments. Bhaumik (1990) and Cutler (1993) have studied the problem when the errors are correlated.
All these studies are restricted to situations when there is a single control treatment. For more than one control treatment optimality aspects have been studied by Majumdar (1986), Jacroux (1990); Jaggi (1992), Jaggi, Gupta and Parsad (1996), Jacroux (2000) and Solorzano and Spurrier (2001). Majumdar (1986) gave an algorithm to obtain A-optimal BBPB designs and a catalogue of A-optimal BBPB designs for small block sizes. Jacroux (1990) studied the optimality aspects in 0-way elimination of heterogeneity settings. Jaggi, Gupta and Parsad (1996) extended the result of Constantine (1983) to more than one control treatment situation and also studied the A-optimality of BBPB designs in the restricted class of designs in which all control treatments appear equally frequently in a block or do not appear at all. Jacroux (2000) gave methods for determining and constructing MV-optimal and highly efficient orthogonal and nearly orthogonal block designs for comparing test treatments with several control treatments under the restriction that replication number of control treatments is fixed. Solorzano and Spurrier (2001) obtained some results on construction and A-optimality of BBPB designs for small values of u and w.
The studies just described relate to proper setting under fixed effects model. In an incomplete block design, the block effects may be random. Pandey (1993) and Gupta, Pandey and Parsad (1998) have obtained sufficient conditions for generating A-optimal incomplete block designs for making test treatments vs control treatment comparisons under a two-way classified, additive, linear, mixed effects model. It has been shown empirically that an A-optimal/efficient design under a fixed effects model remains A-optimal/efficient under a mixed effects model also. Catalogues of A-efficient/optimal designs have also been given. 158 V.K. Gupta [November
The problem of characterization and construction of A- and MV -optimal designs for making test treatment vs control treatment comparisons was till now restricted to proper setting. However, non-proper experimental settings do exist and it is required to generate efficient designs under these situations as well. Prasad (1989) investigated the optimality of designs with unequal block sizes in a very restricted class of designs when the control replications are taken as constant and intra-block variances are assumed to be constant.
For comparing test treatments with a control treatment in block designs with unequal blocks, the concept of Balanced Treatment Incomplete Block Designs with unequal block sizes (BTIUB) was given by Angelis and Moyssiadis (1991) as a natural extension of BTIB designs. They also gave a sufficient condition for establishing the A-optimality of BTIUB designs. Angelis and Moyssiadis (1991), Angelis, Moyssiadis and Kageyama (1993) and Gupta and Kageyama (1993) gave some methods of constructing A-efficient BTIUB designs. Jacroux (1992) studied the A-and MV-optimality of block designs with two distinct block sizes when block sizes may be greater than the number of test treatments for comparing several test treatments with a control treatment. These studies were also carried out under the assumption that intra block variances are constant. Parsad (1991), Parsad and Gupta (1994a) introduced BTIUB designs of Type G and obtained a sufficient condition for A-optimality of non - proper incomplete block designs for comparing test treatments with a control treatment assuming that intra block variances are proportional to non - negative real power of block sizes. Parsad and Gupta (1994b) introduced GDTUB designs of type G and a sufficient condition for A-optimality of GDTUB designs of type G in the class of block designs that are binary in test treatments and in which the control treatment is added same number of times to each block of same size. A catalogue of A-optimal GDTUB designs of type G has also been given. Srivastava, Gupta and Parsad (2000) have studied the A-optimality of non-proper block designs for comparing test treatments with a control treatment when the block sizes may be larger than the number of test treatments. Jaggi (1996) and Jaggi and Gupta (1997a, 1997b) have studied the A-optimality aspects of the designs for comparing several test treatments with several control treatments under a non-proper block design setting where intra block variances have been assumed to be constant. The results are obtained in a restricted class of designs in which all controls appear equally frequently in a block or do not appear at all and block sizes are large. For small block sizes, the condition of Majumdar (1986) has been obtained for non-proper settings. Jaggi, Parsad and Gupta (1999) gave methods of construction of BBPUB designs. Gupta, Parsad and Singh (1996) have obtained a sufficient condition for a block design to be A-optimal for comparing two disjoint sets of treatments under the above heteroscedastic set up.
Result 9: A BBPUB design of type G is A-optimal for comparing w test treatments with each of the u control treatments in the class of designs binary in test treatments and control treatments if 1/ 2 L w u 0 1 L u This is a fairly general condition and the condition for all other designs useful for tests vs controls comparisons fall out as a particular case of this. A procedure to obtain an efficient design for the experimental situations for which this condition does not hold good has also been suggested using the concept of lower bound to the average variance.
The problem of obtaining efficient designs for comparing several tests with several controls under an unrestricted class is still unsolved. Only partial solution to the problem is available. The problem in fact has been solved only for orthogonal block design settings and for smaller 2002] Test Treatments Vs Control Treatments Comparisons 159 block sizes, i.e. when k min (u,w), under non-orthogonal block design settings. The results of Parsad, Gupta and Singh (1996) are applicable only to class of binary block designs. Jaggi, Gupta and Parsad (1996) obtained efficient designs for comparing two disjoint sets of treatments in the class of connected block designs under the restriction that all controls appear equally frequently in a given block. Some results have also been obtained under the restriction that total number of replications of controls is fixed. Moreover, all these results are available 2 when the pairwise comparisons within a set are made with same variance, say Vi . i 1(2) and between treatments from different sets with variance V3. However, such a design may not exist for all parametric combinations or even if it exists may require large number of experimental units. Therefore, an extension of Group Divisible Treatment Designs is needed. Kuriakose (1999) has made a beginning in this direction for block designs with k = 2.
In on farm experiments generally the farmers’ practice is to be compared with that of treatments identified from research stations. What treatment is to be taken as farmers’ practice is a problem as this varies from farmer to farmer. The one possible solution to this problem is that we take as many controls as there are farmers in the trial and add the farmers’ practice to each of the blocks of one replication of resolvable incomplete block design. The problem still needs attention.
8. Weighted A-optimal designs for tests vs controls comparisons
Jones and Eccleston (1980) described an interesting problem of obtaining weighted A-optimal designs by assigning different importance to various comparisons of interest. Mathematical formulation of the problem is as follows: Choose an optimal block design from D that minimizes the weighted sum of variances of a set of estimated treatment contrasts of interest. In other words, from a class of competing designs, choose a design that minimizes tracePC Pw , where PC P 2 is the dispersion matrix of the BLUE of the contrasts of interest t t , t t 1,⋯, v and w is a diagonal matrix of weights attached with the variances of the estimated treatments comparisons. These authors gave computer algorithms to search for optimal designs. Computer algorithms, however, may not always lead to an optimal design and sometimes one may end up with a nearly optimal design. A question that arises now is the following: Are there any actual experimental situations where such a type of problem exists and is it possible to generate exact designs rather than working on computer algorithms to generate designs?
Ture (1994) described the following experimental situation: "A certain type of synthetic fibre is used in the production of various consumer goods. The research team of the manufacturer of this fibre has developed three new types of synthetic fibres that can be used for the same purpose. Each of these alternative fibres is more cost-efficient than the present one and can replace it if any one of them is proved to be stronger. An experiment will be conducted to compare the breaking strengths of all these synthetic fibres. Although it is also desired to compare the three alternative fibres among themselves, more precision will be required for pairwise comparisons between the currently used fibre (the control treatment) and each of the new fibres (the test treatments). How should we design an experiment to give us high precision for treatment-control comparisons as well as good precision for treatment-treatment comparisons"? 160 V.K. Gupta [November
Bechhofer and Tamhane made the following comments on Hedayat, Jacroux and Majumdar (1988): "In future work it would be desirable to keep some important practical features of the problem in mind. One such feature is that the comparisons with different controls may not be required to be of equal precision. For example, in a clinical trial for a new drug it is not uncommon to include two controls, a placebo and an existing active drug. For regulatory purposes, it often is necessary to demonstrate the magnitude of the activity of the new drug, and therefore the comparison with the placebo is more important. It is not always necessary to demonstrate to the regulatory agency that the new drug is more effective than the existing drug. But for the purposes of the pharmaceutical company's marketing efforts, in fact, the second comparison is likely to be the more important. This latter comparison would generally be two- sided".
The problems just outlined fit into the set up of Jones and Eccleston (1980) where it is desired to generate designs that estimate contrasts of interest with differential precision and the design generated minimizes the weighted sum of variances of the BLUE of the contrasts of interest. It may indeed be possible to obtain exact optimal designs for these experimental settings. In the most general set up we may restate the problem as follows:
Suppose D denotes some appropriate class of treatment connected competing designs. We first describe the problem in the most general form. Find a design d*D that minimizes w1 w v1 v g Varˆdg ˆdh Varˆdh ˆdh Varˆdg ˆdg (8.1) gG hH h1hh1 gw1 gg1 over all designs d D .
Here 1 2 ⋯ u 1 and 1 , 2 ,⋯, u ,, 0 are scalar constants or the weights attached with the precision of various comparisons.
The comparisons among control treatments are generally of no interest to the experimenter and so = 0. The problem then reduces to finding a design d*D that minimizes
w1 w g Varˆdg ˆdh Varˆdh ˆdh , 1 2 ⋯ u 1. (8.2) gG hH h1hh1 over d D . When all the controls have the same weights of importance, then the problem is to find a design d*D that minimizes
w1 w Varˆdg ˆdh Varˆdh ˆdh , 1 . (8.3) gGhH h1hh1 over d D .
A special case of the problem in (8.2) is when = 0. In other words the different controls are given different weights according to their importance but the comparisons among tests are not considered. The problem in (8.2) now reduces to find a design d*D that minimizes 2002] Test Treatments Vs Control Treatments Comparisons 161
g Varˆdg ˆdh , 1 2 ⋯ u 1 (8.4) gG hH over d D .
For a single control treatment, i.e. u = 1, the problem in (8.1) reduces to finding a design d*D that minimizes
w w1 w Var(ˆd (w1) ˆdh ) Var(ˆdh ˆdh) , 1 (8.5) h1 h1hh1 over d D .
The problems in (8.3) and (8.5) are the weighted sum of the variances of the BLUE of tests vs controls contrasts and contrasts among tests, respectively with weights as and . Since more precision is required for the tests vs controls comparisons than the comparisons among tests, we insist , and for this setting 1 . For = 0, = 1, the experimental settings in (8.3) and (8.5) reduce to the usual setting of A-optimality of tests vs controls comparisons. It may be seen that for these experimental settings reduce to the usual setting for A-optimality of designs when all the possible paired comparisons among the v treatments are of equal interest. For this setting, =1. However, there may be situations when more precision is required for comparisons among tests than the tests vs control comparisons. For this setting, < and > 1.
The problem posed in (8.5) has been solved under the block design as well as row-column design set up (see, Gupta, Ramana and Parsad, 1999; Gupta, Ramana and Agarwal, 1998). For block design set up the problem has been solved for both the cases when 1 and > 1, though with some restrictions on the values of . For the row-column design set up, however, the problem has been solved for all values of < 1. In the sequel we present some results for block design setting under a mixed model.
Consider an experimental setting where w test treatments are to be compared with a single control treatment using a block design with b blocks of size k each. In obtaining efficient designs for this problem, only comparisons between test treatments and the control are considered and the paired comparisons among the test treatments have no role to play in the choice of the efficient designs. Since the optimal design for test treatments - control treatment comparisons is connected, this design also permits the estimation of the test treatment vs test treatment contrasts, though these comparisons may be required with a lesser precision than the test treatments - control treatment comparisons. We, therefore, study the weighted A- optimality of block designs for comparing test treatments with a control treatment and test treatments among themselves. For the sake of notational convenience and algebraic manipulations we shall henceforth let v denote the number of test treatments in place of w and the treatments will be assumed to be arranged in the order that control treatment comes first and then the v test treatments. 162 V.K. Gupta [November
Let D = DB v,b,k denote the class of all connected block designs with v test treatments and a control treatment arranged in b blocks of size k each, the ( v 1) treatments being indexed as 0,1,…,v with 0 denoting the control treatment and 1,…,v the test treatments. We design the experiment in incomplete blocks and hence the assumption k v . If the t th treatment is th th assigned to the l plot of the j block ( 0 t v, 1 j b , 1 l k ), let ytjl denote the corresponding random variable. We assume the usual two-way classified, mixed effects, homoscedastic model,
ytjl t j etjl , (8.6) th where is the overall mean, t the effect of the t treatment, etjl the uncorrelated error 2 th variables with mean zero and common variance e , and j the j block effect, a random 2 Cov , 0, j j 1,...,b. variable with mean zero, common variance b and j j
Further, j ' s are uncorrelated with the error variables etjl ' s. For any design d D, let
N d ndtj be the incidence matrix, where non-negative integers ndtj denote the number of times the t th treatment occurs in the j th block and
b b v ndij rdi , for i 1,...,v, nd0j rd0 , ndtj k, j1 j1 t0
th where rdi is the replication of the i test-treatment and rdo is the replication of the control treatment in d. The C matrix is given by
1 1 1 C R - N N ( N N r r ), d d k d d k d d bk d d
2 2 2 where Rd diag (rd0 ,rd1,...,rdv ), rd (rd0 ,rd1,⋯,rdv ) and = e ( e k b ) . = 0, which corresponds to the fixed effects model, is also of interest.
((c )) Cd can also be written as Cd= dtt' , where
1 b r2 c r n2 dt , t t' dtt' dt dtj k j1 bk (8.7) b r r 1- dt dt' - ndtjndtj , t t' k j1 bk As per the purpose of the experiment, we formulate the problem mathematically in the tradition of Kiefer. The criterion is weighted sum of variances of the BLUE of test treatments - control treatment contrasts and test treatment – test treatment contrasts, similar to the one proposed by 2002] Test Treatments Vs Control Treatments Comparisons 163
Jones and Eccleston (1980). Two non-negative weights and ) are given to the variances of the estimates of test treatment – test treatment contrasts and test treatment - control treatment contrasts, respectively. The problem then is to choose an experimental design d D that minimizes v v v var(ˆd0 ˆdi ) Var(ˆdi ˆdi) , (8.8) i1 i1 i(i)1 as d varies over D. This criterion has been called the weighted A-optimality criterion. Here (ˆd0 ˆdi ){(ˆdi ˆdi )} denotes the BLUE of ( d0 di ){( di di )} under the strong assumption that is known. We shall, however, show that the weighted A-optimal or A- efficient designs are robust to the value of . The expression in (8.8) is the weighted sum of the variances of the estimates of test treatments – test treatment contrasts and test treatment - control treatment contrasts, respectively with weights as and . Since more precision is required for the test treatment - control treatment comparisons than the test treatments - test treatments comparisons, we insist , and for this setting 1 . Remark 8.1. For = 0, the experimental setting in (8.8) reduces to the usual setting of A- optimality of test treatment - control treatment comparisons. For the experimental setting reduces to the usual setting for A-optimality of designs when all the possible paired comparisons among the (v + 1) treatments are of equal interest. For this setting, =1. However, there may be situations when more precision is required for test-test comparisons than the test - control comparisons. For this setting, < and > 1. We address to these problems, though a complete solution of the problem in (8.8) for any arbitrary values of and is desirable.
Write Pc 1v ⋮ I v a v x (v+1) matrix in partitioned form and
PTZ 0Z ⋮0Zx(vZ 1) ⋮1Z ⋮ IZ , a Z x (v+1) matrix in partitioned form, Z = 1,2,…,v-1. If
PT PT 1⋮⋯⋮PT (v1) , then (8.8) can be written as a function of Cd in the form
(Cd ) c (Cd ) T (Cd ) , (8.9)
where c (Cd ) trace(PcCd Pc ) , T (Cd ) trace(PT Cd PT ) and Cd is any generalized inverse of Cd. Then following is a subclass of matrices:
C C : C is non negative definite,rank(C) v, C1v1 0, for some scalars p, q, r p q1' and s, C can be written as C= v . (8.10) q1v rIv sJvv
For each d D, we define the following: 164 V.K. Gupta [November
1 v 1 v v r s cdii , s cdii v i1 v(v 1) i1i(i)1 (8.11) 1 v b q cdoi , tt' ndtjndt' j for 0 t,t' v v(v 1) i1 j1
Bechhofer and Tamhane (1981) gave the following definition:
Definition 8.1 : A design d D is a BTIB design if
do1 ... dov 0 (say) , and
d12 d13 ... d (v1)v (say).
The following lemmas are useful for proving the main results. Since our main purpose is to minimize ( C d ), we shall first obtain a lower bound on it.
Lemma 8.1 : For a design d D and for fixed values of and (Cd ) f (r, q) , ( v)(v 1) where f (r, q) . r q Proof : Let denote a permutation of test treatments and be the set of all such permutations. Let Q( ) be the corresponding permutation matrix. Then it is easy to see that
(Cd ) =c {Q( )Cd Q( )} and T (Cd ) T {Q( )Cd Q( )}.
Moreover, c (Cd ) and T (Cd ) are convex in Cd. These two properties imply that * * c (Cd ) c (Cd ) and T (Cd ) T (Cd ) (8.12)
1 where C* Q( )C Q( ) with the summation over all in . d v! d * q q q Clearly, C d C with p = cd00, r r , s s, and , where r ,s and are given in
(8.11). By using the fact that C1v1 0 and thereby p vq 0 and r vs q 0, we can * * express Cd in its spectral form as Cd rE1 (v 1)qE 2 ,
2 ' 0 0' v v1v where E1 = 1 , v(v 1)E2 . 0 Iv v Jvv v1v Jvv * The Moore-Penrose inverse of Cd is * 1 1 Cd r E1 {( v 1)q } E 2 . (v 1) * * 1 Hence, (C ) = trace(PcCd Pc ) - , (8.13) c d r q 2002] Test Treatments Vs Control Treatments Comparisons 165
v1 * * ' * ' and T (Cd ) = trace(PT Cd PT ) trace(PTZ C d PTZ ) z1 v1 1 v(v 1) ' = 2Z {since PTZ E 2 PTZ 0 ZZ }. (8.14) r Z 1 r
Since (Cd ) = c (Cd ) T (Cd ) and both c (Cd ) and T (Cd ) are non-negative quantities, by using (8.12), (8.13), (8.14) we have * * * f (r, q) (Cd ) (Cd ) = c (Cd ) T (Cd ) = , which proves the lemma.
From Lemma 8.1, the problem of minimizing (Cd ) reduces to the minimization of f r,q . We now give some more lemmas that will culminate in a general theorem from which weighted A-optimal designs may be obtained.
Lemma 8.2: For any design d D there is a design d* D with f (rd* , qd* ) f (rd , qd ) and nd*0j int(k / 2); 1 j b.
Proof: The proof follows on the similar lines of (A.2) of Jacroux (1992) pp.342 with suitable modifications. A sketch of the proof is as follows:
Consider a design d D. If in d, nd0 j int(k/2) , we replace the design d by d* D, where d* satisfies nd*0 j nd0 j if nd0j int(k / 2) (8.15) k - nd0j if nd0j int(k / 2) and is obtained from d by replacing nd0j nd*0 j 2nd0 j k replications of the control by
different test treatments such that nd*ij ndij or ndij 1 in those blocks where
nd0 j int(k/2) and retaining those blocks in which nd0 j int(k/2) as such.
With d* as defined above, we clearly have nd*0 j int( k / 2 ) and then rd*0 rd0 .
Now using (8.7), (8.8) and (8.15), for d* D, it can be shown that qd* qd .
Also using (8.7) and (8.8), it can be shown that for d D
v b 2 1 (1 ) 2 rd*i qd* rd* rd*i nd*ij (v 1) i1 k j1 bk (v 1) 166 V.K. Gupta [November
Therefore, to show that rd* rd it is enough to show that
b 2 b 2 1 2 rd*i 1 2 rdi (8.16) rd*i nd*ij rdi ndij k j1 k k j1 k Using the relations in (8.12), it may be easy to show that for those tests treatments for which
nd*ij ndij , equality is achieved in (8.16) while for those tests treatments for which
nd*ij ndij 1 , (8.16) simplifies to 2 b 1 b b = n n2 n d*ij d*ij d*ij j1 k j1 bk j1
2 b 1 b b = (n 1) (n 1) 2 (n 1) dij dij dij j1 k j1 bk j1 b b 1 2 2 1 2ndij = rdi ndij rdi (1 ) k j1 bk j1 k k 1 2n dij n int(k/2) Now, 1 > 0 (because dij ). k k
Hence, f ( rd* ,qd* ) f ( rd ,qd ).
Lemma 8.3: For fixed rd 0 and nd 0 j ,r is maximized when ndij 0 or 1 and (bk r ) r d0 r (say) for all 1 i v ; 1 j b, and an upper bound on r is given by di v d 2 2 r* vk(v 1)rd (1 )v rd (1 )(bk Q 2krd0 )/ kv(v 1) , where b 2 Q nd0 j . j1 Proof: Using (8.7) and (8.8) it can also be shown that
v b v 1 1 1 2 1 2 r rdi ndij (k nd0 j ) v i1 k j1(v 1) i1 v(v 1) (8.17) v 2 bk rd0 rdi bk(v 1) i1 v 2002] Test Treatments Vs Control Treatments Comparisons 167
b v b v 2 Now ndij ndij , with equality whenever ndij 0 or 1, for all iand j. Also j1i1 j1i1 v 2 bk rd0 rdi 0, (8.18) i1 v with equality whenever rdi rd , a fixed number, for all i = 1,…,v. These two relations, along with (8.17) and (8.18), yield the upper bound r * on r and hence the lemma is proved.
Lemma 8.4: For fixed rd0 and ndoj ,q reduces to the form
q b(1 )(krd0 Q) rd0 (bk rd0 )/ bkv
Proof: The proof easily follows from (8.7) and (8.8). q Lemma 8.5: For fixed rd 0 with rdi rd as given in Lemma 8.3, and as given in Lemma 8.4,
f ( r ,q ) is minimized when the nd0 j are either int(rd0 / b) or int(rd0 / b) 1 , for all values of satisfying
2 2 2 2 2 (2vk 2v k 1) (k 1) (v 1) (2v k 1) 2(2v k 1)(2vk 2v k 1) v[(k 1)(v 1)]2 when k is odd, and
(2vk 2v k)2 k 2 (v 1)2 2 (2v k) 2 2(2v k)(2vk 2v k) v[k(v 1)]2 when k is even.
Proof: For fixed rd0 and rd , as in Lemma 8.3, and q as given in Lemma 8.4, we have (v 1)( v) f (r, q) r q
Differentiating with respect to Q and putting the result over a common denominator yields a ratio whose denominator is positive and whose numerator is a function of Q , given by 168 V.K. Gupta [November
2 2 2 2 h(Q) vb k(1 )kv(v 1)rd (1 )v rd (1 )(bk Q 2krd0 ) 2 2 ( v)(v 1) k(1 )b(1 )(krd0 Q) rd0 (bk rd0 ) vb 2 k(1 )3 ( v)(v 1)2 Q 2 lower order terms in Q.
Clearly the coefficient of Q 2 is negative and hence h( Q ) is a concave quadratic in Q . From the definition of q in (8.11), it is a negative quantity and hence from Lemma 8.4 we have
b(1 )krd0 rd0 (bk rd0 ) Q Q (say), b(1 ) 2 2 b 1 b r2 Also, Q n2 n d0 Q (say), d0 j d0 j 1 j1 b j1 b 2 2 2 2 2 h(Q1 ) (1 )kv rd bk(v 1) (1 )(b rd )v - (1 v)(v 1) rd0 , . For h(Q1 )0, we should have 2 1/ 2 (8.19) bv(k 1) b(v k) rd0 rd0 rd0 (1v)(v 1) 0,
1/ 2 bv(k 1) b(v k) rd0 (1 ) (v 1)(1v) . (8.20)
b(k 1) Using Lemma 2 it can be seen that for odd values of k, r and for even values of d0 2 bk k, r . It is not possible to show that h(Q1 ) 0 for all values of , so we obtain an d0 2 upper bound on for which one can show that h(Q1 ) 0 . Hence from (8.20) one can see that
(i) for odd k
2 2 2 2 2 1 (2vk 2v k 1) (k 1) (v 1) (2v k 1) 2 v[(k 1)(v 1)] 2(2v k 1)(2vk 2v k 1) (8.21)
(ii) for even k
1 (2vk 2v k)2 k 2 (v 1)2 [ 2 (2v k)2 (8.22) 2 v[k(v 1)] 2(2v k)(2vk 2v k)
Now, h( Q2 ) = a positive constant a squared term, hence h( Q2 ) 0. 2002] Test Treatments Vs Control Treatments Comparisons 169
For Q1 Q Q2 , h(Q1 ) 0, under the conditions (3.21) and (3.22) and h( Q2 ) 0, it follows that h(Q) 0, for all Q satisfying Q1 Q Q2 . This implies that f (r, q) is an increasing function of Q and is minimized for fixed rd 0 when Q is as small as possible. For fixed rd 0 , Q is minimized when ndoj is equal to either int(rd0 / b) or int(rd0 / b) 1 .
Remark 8.2. The values of upper bounds on satisfying (8.20) have been computed using (8.21) and (8.14) for v 100 and k 11. The problem of weighted A-optimality of designs could be solved for these values of only. We have reported values of 0.2 only in Table 2. However, for any arbitrary value of , the problem of minimization of (Cd ) requires a solution.
Lemma 8.6: For d D and for fixed values of , and rdo
{ v}(v 1)2 k f (r, q) 2 2 vk(v 1)rd (1 )[v rd bk G(rd 0 ) 2krd0 ] bk b(1 - ){krd0 G(rd 0 )} rd0 (bk rd0 )
bk r where r d0 , as given in Lemma 3, and d v
2 G(rd0 ) rd0 (2rd0 b)int(rd0 / b) bint(rd0 / b)] .
Although we have defined a BTIB design in Definition 8.1, in view of Lemma 8.6, we give a subclass of BTIB designs which seems to be stronger definition than Definition 1 to search an A-optimal design as it incorporates the conditions obtained in the above Lemmas. This definition will be useful in the statement of Theorem 8.1.
Definition 8.2: For integers t {0,1,...,int(k/2)-1} and s {0,1,...,b}, with s > 0 whenever t=0, a design d D is a BTIB design with parameters v, b, k, t, s, if and only if it is a BTIB design with the additional property that
ndij {0,1}, 1 i v; 1 j b;
ndo1 = ... = ndos = t +1 ; ndo(s+1) ... ndob t.
Using Definition 8.2 of BTIB (v, b, k; t, s) design, we have rdo= bt + s and b 2 2 ndoj = bt + 2ts+ s. We now have the main theorem of this section which can be derived j1 along the lines of Hedayat and Majumdar (1984) by using Lemma 6 and Definition 2. 170 V.K. Gupta [November
Theorem 8.1: A BTIB (v, b, k; t, s) design with parameters v, b, k, t and s is A-optimal over D for fixed values of and , satisfying the conditions of Lemma 8.5, if
min g(x, z) g(t, s) = (x, z) where = {(x, z); x=0,1,…,int(k/2) - 1; z = 0,1,…,b with z >0 when x = 0},
( v)(v 1) 2 b g(x, z) , A(x, z) B(x, z)
A(x, z) k(v 1)[b(k x) z] (1 )[v{b(k x) z} bk 2 bx2 2xz z 2k(bx z)] B(x, z) b(1 )[k(bx z) (bx 2 2xz z)] (bx z)(bk bx z).
Theorem 8.1 is useful for obtaining optimal designs for treatment - control comparisons with high precision and treatment - treatment comparisons with good precision by appropriately choosing the value of (<) under a mixed effects model. It is also useful for obtaining weighted A-optimal designs for treatment - control comparisons and treatment - treatment comparisons by arbitrarily choosing the values of and under a mixed effects model for the range of values of satisfying the conditions of Lemma 8.5. One can question the applicability of this theorem under mixed effects model, as the value of is not known in advance. However, for = 0, i.e., under a fixed effects model, we can obtain optimal designs for different values of and then the efficiency of these designs can be inspected for varying values of . In other words, we can study the robustness of optimal BTIB designs under varying values of , for fixed value of .
Theorem 1 gives a condition for a BTIB(v, b, k; t, s) design to be weighted A-optimal in D for fixed values of and . It is difficult to give general methods of construction of weighted A- optimal designs satisfying the condition in Theorem 1. Therefore, we look for an indirect approach and settle for a design that, though possibly not weighted A-optimal, performs well under the weighted A-criterion. In Section 6 it was shown that a BG design is a strong candidate for A-optimality (or A-efficient). Method 3.1 describes a general procedure of obtaining BG designs. The weighted A-efficiencies of the BG designs satisfying the conditions of Lemma 5 are computed using the definition of A-efficiency given by Stufken (1988).
Definition 8.3: Stufken (1988): The efficiency E(d) of a design d D is defined as
- trace( PC d* P) E(d) (8.24) - trace( PC d P) where d* is a hypothetical A-optimal design in D for fixed values of and for which trace(PCd*P) is minimum and is obtained by minimizing g over in the condition of 2002] Test Treatments Vs Control Treatments Comparisons 171
Theorem 1. We attempt to find designs with efficiencies close to one. The designs with efficiency equal to one are weighted A-optimal.
The weighted A-efficiencies of the BG designs mentioned above have been computed for different values of (=0, 0.2, 0.5, 0.8) and (=0, 0.1, 0.2, 0.3, 0.4) satisfying the upper bound on (<1). For each of these designs, the coefficient of variation (CV) of the efficiencies over different values has been worked out. It is observed that some designs are optimal / highly efficient for all values of . In some cases designs first increase and then decrease in efficiency as increases. We also observe that the value of has very little influence on the efficiencies, as CVs are quite small. In other words, the designs that are optimal or highly efficient under fixed effects model remain optimal or highly efficient under a mixed effects model also.
The weighted A-efficiency of the BG designs have also been computed for different values of (=0, 0.2, 0.5, 0.8, 1.0) and (=0.5 and 0.6) satisfying the upper bound on ( 1). In this case the test vs test comparisons get more weight than the test vs control comparisons. It may be mentioned here that for = 0.5, all the possible paired comparisons are equally important and hence a balanced incomplete block design, if existent, will be A-optimal. For = 0.6, the efficiency increases for the increase in the value of , though the efficiency is smaller to the one for = 0.5.
So we can conclude that the BG designs do not always remain optimal for varying values of , where as these designs remain optimal or highly efficient for varying values of . In other words, for different precision that are assigned to treatment-control comparisons or treatment- treatment comparisons, the experimenter has to select designs depending upon the precision required for the two types of comparisons, whereas the same design performs equally well in both the fixed and mixed effects models.
The minimisation has been done using Theorem 8.1 and the method of construction of BG designs has been given as Method 3.1. Indeed it is possible to solve this minimization problem to obtain weighted A-optimal or A-efficient designs by using an intelligent algorithm. This approach has an advantage that the search may not be restricted to BG designs only and may work for any value of . In the sequel we describe an algorithm that enables to generate a large number of A-optimal/A-efficient designs that are not BG.
Algorithms in general or Working Rule of an Algorithm
The most common steps involved in any typical Optimisation Algorithm are the following:
1. Introduce a naïve design as a starting design and call it OLD. 2. Evaluate the criterion of optimisation on the OLD design. 3. Introduce a NEW design. 4. Evaluate the criterion of optimisation on the NEW design. 5. Compare the two evaluated values. 6. Accept the design that favours Optimality Criterion. 7. Call the design accepted at step 6 as OLD. 8. Go to step 3 and continue until no improvement is achieved on the criterion of evaluation or all possible configurations have been tested. 9. Accept the design that favours the optimality criterion the most. 172 V.K. Gupta [November
The Proposed Algorithm Step1. We introduce a naïve design as a starting design randomly. The starting design should be connected and binary. The starting design can be obtained by any of the following two approaches:
Full Randomisation: Allocate test treatments and control treatments randomly to blocks with randomly selected replication numbers of both the test treatments as well as the control treatments. Partial Randomisation: First allocate control treatments to each block once deliberately. Then allocate test treatments randomly in blocks with their replication numbers as close as possible.
Step2. Delete the weakest observation and replace it by the strongest observation. This step is called exchange procedure.
Step3. Continue with Step2 until no improvement is achieved.
Step4. When no further improvement is achieved then start a new procedure named as Interchange Procedure.
Step5. John and Eccleston (1980) gave an Interchange Procedure, which selects and interchanges a pair of treatments whose interchange yields an improvement with respect to optimality criterion, whereas we used Strongest Interchange for individual observation. In this algorithm we find the strongest interchange for each observation.
What are exchange and interchange steps? . Exchange Step: refers to exchanging treatment among the blocks. i.e. i. Increase nij by 1, ii. Decrease nij by 1 nij 1 0. Given, i i and j j , select i,i, j , randomly. This will change the replication vector but the vector of block sizes remain unchanged.
. Interchange Step: It is a combination of two exchange steps, such that it leaves replication vector unchanged, i.e.
i. Increase nij by 1.
ii. Decrease nij by 1.
iii. Decrease nij by 1.
iv. Increase nij by 1. Given i i and j j , select i,i, j, j randomly.
What are the weakest and strongest observations? Weakest Observation: This is an observation for which 2002] Test Treatments Vs Control Treatments Comparisons 173
Trace pwpCn1 - Trace pwpCn is minimum for all possible Cn1 .
Strongest Observation: This is an observation for which
Trace pwpCn1 - Trace pwpCn is maximum for all possible Cn1 .
Note: Trace (AB) = Trace (BA), so Trace pwpCn1 = Trace wpCn-1p.
Criterion to Minimize
In the above discussion the criterion for selecting the Weakest and the Strongest observation was Trace wpCn-1p, but for our problem of making test treatments vs control treatment comparisons the weights and are associated with contrasts. So as defined above the criterion to minimize now is
* - * - TraceP1C P1 TraceP2C P2 where C Matrix is same as defined for single control treatment situations under a two-way classified mixed effects model above and the coefficient matrices for the contrasts are P1 1w I w and 0 1 1 0 0 0 ⋯ 0 1 0 1 0 0 ⋯ 0 1 0 0 1 0 ⋯ P2 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ 0 0 1 1 0 0 0 0 1 0 1 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ Now consider that v = 4 and b=7 then
P1= 1.000000 -1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 -1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 -1.000000 0.000000 1.000000 0.000000 0.000000 0.000000 -1.000000
P2= 0.000000 1.000000 -1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 -1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 -1.000000 0.000000 0.000000 1.000000 -1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 -1.000000 174 V.K. Gupta [November
0.000000 0.000000 0.000000 1.000000 -1.000000
Exchange Procedure:
Starting Design BLOCK DIAGRAM BLOCK DIAGRAM BLOCK DIAGRAM Block 1 (1,2,5) Block 1 (1,2,5) Block 1 (1,2,5) Block 2 (1,2,3) Block 2 (1,2,3) Block 2 (1,2,3) Block 3 (1,2,3) Block 3 (1,2,3) Block 3 (1,2,3) Block 4 (1,3,5) Block 4 (1,3,5) Block 4 (1,3,5) Block 5 (1,2,5) Block 5 (1,2,5) Block 5 (1,2,5) Block 6 (1,2,3) Block 6 (1,2,3) Block 6 (1,2,3) Block 7 (1,4,5) Block 7 (1,4,*) Block 7 (1,4,2) Block 8 (1,4,5) Block 8 (1,4,5) Block 8 (1,4,5) Block 9 (1,2,5) Block 9 (1,2,5) Block 9 (1,2,5) Block 10 (1,3,4) Block 10 (1,3,4) Block 10 (1,3,4) Block 11 (1,3,4) Block 11 (1,3,4) Block 11 (1,3,4) Block 12 (1,4,5) Block 12 (1,4,5) Block 12 (1,4,5) Weakest Observation: Strongest Observation = Block 7, Treatment 5 Treatment 2
Interchange Procedure: Interchange Tried: (Block Final Design 1, Treatment 4) with Design Trace=1.1428571428571 (Block 2, Treatment 3) Minimization =1.1428571428571 Efficiency=1.0000 BLOCK DIAGRAM BLOCK DIAGRAM BLOCK DIAGRAM Block 1 ( 1 4 5 ) Block 1 ( 1 3 5 ) Block 1 ( 1 3 5 ) Block 2 ( 1 2 3 ) Block 2 ( 1 2 4 ) Block 2 ( 1 2 4 ) Block 3 ( 1 2 3 ) Block 3 ( 1 2 3 ) Block 3 ( 1 2 3 ) Block 4 ( 1 3 5 ) Block 4 ( 1 3 5 ) Block 4 ( 1 3 5 ) Block 5 ( 1 2 5 ) Block 5 ( 1 2 5 ) Block 5 ( 1 2 5 ) Block 6 ( 1 2 3 ) Block 6 ( 1 2 3 ) Block 6 ( 1 2 3 ) Block 7 ( 1 2 4 ) Block 7 ( 1 2 4 ) Block 7 ( 1 2 4 ) Block 8 ( 1 4 5 ) Block 8 ( 1 4 5 ) Block 8 ( 1 4 5 ) Block 9 ( 1 2 5 ) Block 9 ( 1 2 5 ) Block 9 ( 1 2 5 ) Block 10 (1 3 4 ) Block 10 ( 1 3 4 ) Block 10 (1 3 4 ) Block 11 (1 3 4 ) Block 11 ( 1 3 4 ) Block 11 (1 3 4 ) Block 12 (1 4 5 ) Block 12 ( 1 4 5 ) Block 12 (1 4 5 )
Using the algorithm many A-efficient designs have been generated. A few designs are given in Appendix.
The problems of obtaining weighted A-efficient designs for many control treatments have been handled in two phases. In the first phase, the problem of obtaining weighted A-efficient designs for several control treatments has been attempted by giving unequal weights to various control treatments. In the choice of an optimal design no consideration is given to the comparisons among test treatments. This refers to the situation in (8.4) [see Gupta, Ramana and Parsad (2002)]. In the second phase not only we consider the problem of obtaining weighted A- efficient designs by giving equal importance to all the control treatments but also considering 2002] Test Treatments Vs Control Treatments Comparisons 175 the estimation of comparisons among test treatments through the same design, though with a smaller precision than that of the test treatments versus control treatments comparisons. This corresponds to problem in (8.3) [see Ramana (1995)]. The most general problem in (8.1) and the problem in (8.2) is seemingly a difficult problem. These may, however, need attention so as to be able to solve completely the problem of test treatments versus control treatments comparisons.
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Appendix: Some A-efficient designs developed by using the algorithm v=5, b=4, r1=4, r2=3, r3=3, r4=3, r5=3, k=4, 01 =3, 1=2, ß=0.0, Eff=1.0000
( ß ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ( 0.4 0.5 0.9 )
(Ro=0.0) 1.00 1.00 1.00 1.00 1.00 0.98 0.96 . . . (C.V.= 0.00, 0.62, 1.35) (Ro=0.1) 1.00 1.00 1.00 1.00 1.00 0.98 0.96 . . . (C.V.= 0.00, 0.60, 1.31) (Ro=0.2) 0.99 1.00 1.00 1.00 1.00 0.98 0.96 . . . (C.V.= 0.06, 0.57, 1.26) (Ro=0.3) 0.99 1.00 1.00 1.00 1.00 0.98 0.96 . . . (C.V.= 0.23, 0.55, 1.20) (Ro=0.4) 0.99 1.00 1.00 1.00 1.00 0.98 0.97 . . . (C.V.= 0.39, 0.59, 1.16) (Ro=0.5) 0.99 1.00 1.00 1.00 1.00 0.99 0.97 0.95 . . (C.V.= 0.56, 0.66, 1.83) (Ro=0.6) 0.98 1.00 1.00 1.00 1.00 0.99 0.97 0.95 . . (C.V.= 0.72, 0.76, 1.78) (Ro=0.7) 0.98 0.99 1.00 1.00 1.00 0.99 0.97 0.95 . . (C.V.= 0.86, 0.84, 1.72) (Ro=0.8) 0.97 0.99 1.00 1.00 1.00 0.99 0.97 0.95 . . (C.V.= 1.01, 0.95, 1.67) (Ro=0.9) 0.97 0.99 1.00 1.00 1.00 0.99 0.97 0.95 . . (C.V.= 1.17, 1.08, 1.64)
(CV's=>) 1.06 0.34 0.00 0.00 0.00 0.13 0.29 0.23 N.A. N.A. (Mean= 0.50, 0.72, 1.49)
Block Diagram : (0, 1, 3, 4); (0, 1, 2, 4); (0, 2, 3, 4); (0, 1, 2, 3).
Two Distinct lamdas differing by one but not necessarily Group Divisible v=9, b=8, r1=8, r2=5, r3=5, r4=5, r5=5, r6=5, r7=5, r8=5, r9=5, k=6, 01=5, 1=3, 2=2, ß=0.2, Eff=0.9990
( ß ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ( 0.4 0.5 0.9 )
(Ro=0.0) 0.97 0.99 0.99 0.99 0.99 0.97 0.96 . . . (C.V.= 1.26, 1.29, 1.57) (Ro=0.1) 0.96 0.99 0.99 0.99 0.99 0.97 0.96 . . . (C.V.= 1.36, 1.35, 1.59) (Ro=0.2) 0.96 0.99 0.99 0.99 0.99 0.97 0.96 . . . (C.V.= 1.45, 1.42, 1.62) (Ro=0.3) 0.96 0.99 0.99 0.99 0.99 0.98 0.96 . . . (C.V.= 1.55, 1.49, 1.65) (Ro=0.4) 0.95 0.99 0.99 0.99 0.99 0.98 0.96 . . . (C.V.= 1.65, 1.57, 1.69) (Ro=0.5) 0.95 0.99 0.99 0.99 0.99 0.98 0.96 . . . (C.V.= 1.75, 1.64, 1.74) (Ro=0.6) 0.95 0.99 0.99 0.99 0.98 0.98 0.96 . . . (C.V.= 1.85, 1.72, 1.78) (Ro=0.7) 0.95 0.99 0.99 0.99 0.99 0.98 0.96 . . . (C.V.= 1.94, 1.80, 1.83) (Ro=0.8) 0.94 0.99 1.00 0.99 0.99 0.98 0.96 . . . (C.V.= 2.04, 1.88, 1.88) (Ro=0.9) 0.94 0.99 1.00 0.99 0.99 0.98 0.96 . . . (C.V.= 2.14, 1.97, 1.94) (CV's=>) 0.78 0.32 0.03 0.04 0.08 0.13 0.19 N.A. N.A. N.A. (Mean= 1.70, 1.61, 1.73)
Block Diagram : (0,1,3,4,6,8); (0,3,4,5,6,7); (0,1,2,3,7,8); (0,2,3,5,6,8); (0,1,5,6,7,8); (0,1,2,4,6,7); (0,2,4,5,7,8); (0,1,2,3,4,5). 2002] Test Treatments Vs Control Treatments Comparisons 181
v=9, b=6, r1=6, r2=3, r3=3, r4=3, r5=3, r6=3, r7=3, r8=3, r9=3, k=5, 01 =3, 1=1, 2=3, 3=2, ß=0.0, Eff=0.9916
( ß ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ( 0.4 0.5 0.9 )
(Ro=0.0) 0.99 0.99 0.99 0.97 0.95 0.93 0.91 . . . (C.V.= 1.45, 2.12, 2.88) (Ro=0.1) 0.99 0.99 0.99 0.98 0.96 0.94 0.92 . . . (C.V.= 1.37, 2.04, 2.79) (Ro=0.2) 0.99 0.99 0.99 0.98 0.96 0.94 0.92 . . . (C.V.= 1.30, 1.96, 2.71) (Ro=0.3) 0.99 0.99 0.99 0.98 0.96 0.94 0.92 . . . (C.V.= 1.24, 1.89, 2.63) (Ro=0.4) 0.99 0.99 0.99 0.98 0.96 0.95 0.93 . . . (C.V.= 1.20, 1.83, 2.56) (Ro=0.5) 0.99 0.99 0.99 0.98 0.96 0.95 0.93 . . . (C.V.= 1.16, 1.78, 2.49) (Ro=0.6) 0.99 0.99 0.99 0.98 0.97 0.95 0.93 . . . (C.V.= 1.13, 1.73, 2.43) (Ro=0.7) 0.98 0.99 0.99 0.98 0.97 0.95 0.93 . . . (C.V.= 1.12, 1.68, 2.36) (Ro=0.8) 0.98 0.99 0.99 0.99 0.97 0.95 0.93 0.91 . . (C.V.= 1.11, 1.65, 2.98) (Ro=0.9) 0.98 0.99 0.99 0.99 0.97 0.95 0.93 0.91 . . (C.V.= 1.13, 1.63, 2.92)
(CV's=>) 0.38 0.31 0.36 0.43 0.52 0.60 0.70 0.07 N.A. N.A. (Mean= 1.22, 1.83, 1.83)
BLOCK DIAGRAM
(0, 1, 2, 7, 8); (0, 1, 4, 5, 7); (0, 3, 4, 5, 8); (0, 2, 3, 5, 6); (0, 2, 4, 6, 8); (0, 1, 3, 6, 7);