NEAREST NEIGHBOURHOOD DESIGNS

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1 NEAREST NEIGHBOURHOOD DESIGNS Rajender Parsad and Cini Verghese I.A.S.R.I., Library Avenue, New Delhi Introduction This talk is mainly based on a review paper on "Nearest Neighbour Designs for Comparative Experiments: Optimality and Analysis" by G.K.Shukla and P.S.Gill, that appeared in the proceedings of the Symposium on Optimization, Designs of Experiments and Graph Theory (1986). It is well known that the theory of designs is based on three principles viz., replication, randomization and local control. Replication provides an estimate of the experimental error that acts as a basic unit of measurement for assessing the significance of observed treatment differences. Randomization is supposed to eliminate the bias by removing the differences in the characteristics exhibited by the experimental plots. Local control refers to the balancing, blocking and grouping of the experimental units so as to reduce the magnitude of the experimental error, thereby, making the experimental design efficient. The traditional ways of local control such as blocks, Latin and Lattice squares were developed in the third and fourth decades of this century. These methods have been a subject of mathematical study and are well understood by now. They serve the purpose well where fertility variations are smooth and well known so that the experimental area can be divided into homogeneous blocks or rows and columns. However, these methods are often applied uncritically in a cookbook approach, under the conditions where crop yield variations and fertility variations are complicated and often change from one season to another. In such cases it is likely that a bad choice of blocks may actually increase the error variance instead of decreasing it. Recently, there has been a considerable interest in the use of some alternative methods of local control called spatial or nearest neighbour (abbreviated as NN) methods. Surprisingly, these methods have evoked little interest along the Indian statisticians so far. This is because in agricultural field trial experiments, plots occurring close together within a field area are well known to be similar than plots occurring far away from each other. The article is divided into two sections. Section 1, introducing the currently available NN methods. Section 2 deals with the more sophisticated problem of optimality of experimental designs. 2. Nearest Neighbour Methodology 2.1 PAPADAKIS Covariance Method (The Genesis) It was first suggested by Papadakis (1937) and further discussed by Bartlett (1938) and Papadakis (1940), but thereafter it had been neglected until recent resurgence. The method relies on the assumption that the yield from a plot is closely related to the yields from its immediate neighbours due to inherent positive correlation between the fertility of

2 neighbouring plots. The basic method is as follows. Let y i denote the yield from i th plot and y i be the mean yield of all the plots that receive the treatment applied in the i th plot. 1. Let ei = yi yi be a residual from the i th plot. 2. Compute a covariate for each interior plot by taking the mean of the residuals of adjacent plots. For an edge plot, the covariate is the residual from the adjacent interior plot. 3. Analyze the data by using the method of analysis of covariance. Bartlett (1938) examined the Papadakis method and suggested the use of blocks among with NN adjustment. It was also seen that the expected contribution of the regression term to the sum of squares in the analysis of covariance is about two times the true error variance when the regression coefficient is nearly zero. Consequently, Bartlett suggested two degrees of freedom for the covariate instead of the usual one. Papadakis (1940) suggested the use of two regression coefficients, one for interior plots, and one for edge plots. Iterative method of calculating residuals, the use of second NN residuals as an additional covariate and an extension of the method for two-dimensional data were also given. Uniformity trial data were used to check the properties of the Papadakis method. Papadakis continued the development of his method and suggested many changes (Papadakis, 1954, 1970, 1984). Papadakis method can be applied to analyze any type of data from any design and assumes the least about the fertility pattern of the field. The method has always been open to the logical objection that covariates are calculated using treatment effects derived immediately from the data, yet it has been the feeling of theoreticians as well as practitioners that the method has potential for increasing the precision of comparative experiments. Pearce and Moore (1976), Pearce (1978, 1980) and Kempton and Howes (1981) reported substantial gains (as high as 50% in some cases) in the efficiency of treatment comparisons when Papadakis method was used. The method is suitable for variety trials where one-dimensional blocks are very long so those distant plots have little in common. 2.2 Correlation Methods (The reincarnation of Papadakis method) Another technique (closely related to Papadakis Method) of analyzing experimental data is based on the use of correlation among observations through generalized least squares (GLS) analysis. In situations, where the structure of correlation is known or can be postulated adequately, it may be advantageous to use this information at the stages of design and analysis of experiment. Williams (1952) considered the case when observations are assumed to follow the first order and second-order autoregressive (AR) processes and experimental plots are arranged in time or along a line. Atkinson (1969) investigated the connection of Papadakis procedure with the analysis based on first-order AR errors. He showed that the Papadakis estimator is the first approximation to the ML estimator. Variances of the two estimators were also shown to be nearly the same. Theoretical results were also 538

3 confirmed through simulation. Draper and Farraggi (1985) also established the relationship of the two estimators. In the light of recent developments spatial statistics (e.g., Whittle (1954), Besag (1974), Bartlett (1978)) re-examined Papadakis method theoretically for both one-dimensional and two-dimensional layouts making use of various forms on correlation models for observations. Iteration of Papadakis estimator, by using the residuals from one stage to obtain the estimator at the next stage, was also suggested. The adjusted error sum of squares was shown to be a conservative measure of the accuracy of treatment estimation. For large number of treatments, the additional advantage of using blocking along with NN adjustment was shown to be nonsignificant. Some numerical examples from field experiments and simulation experiments were given for illustrative purposes and an appreciable increase in efficiency to treatment estimation was reported. Bartlett's paper stimulated a lively and useful discussion of the issue as is evident from a series of papers that followed it. In the following sections we continue the discussion of further developments of the Papadakis-type methods. 2.3 WILKINSON's Methods Wilkinson, Eckert, Hancock and Mayo (1983) described the results of extensive Monte- Carlo randomization studies of the Papadakis method on uniformity trial data. The results showed that while a non-iterated Papadakis analysis is reasonably valid under randomization, iterated estimator is more efficient but leads to upward bias in treatment F-ratio. The most serious defect of method, however, was found to be its inherent efficiency when trend effects are appreciable. They proposed a smooth trend + independent error model. y i =τ i +ξ i +η i (2.3.1) where τ i represents the effects of treatment applied in the i th plot, ξ i is the smooth trend 2 component on plot i with var(ξ i ) = σξ, and ηi s are independent local errors with 2 var(η i )= σ η. The smoothness-of-trend assumption here is that the residual trend ' i components, after local linear detrending, ξ = ξ ξ, are small relative to the standard deviation of local errors. Here plots of plot i. ξni ' i Ni denotes average of trend components of neighboring Based upon this model, they suggested the analysis comprising two phases termed the intra - N and inter - N analysis analogues of intra - block and recovery of inter - block information analyses in the classical fixed block methodology. Validity of new method under randomization (for example, approximate unbiasedness of treatment F-ratio) was demonstrated empirically through Monte Carlo studies. The net efficiency of the method was shown to be comparable with the efficiency factors for lattice and lattice square designs with similar replications. A concept of partial nearest neighbour balance for large experiments was also formulated. Street and Street (1985) 539

4 considered the construction of nearest neighbour designs, which satisfy the balance conditions of Wilkinson et al (1983). The new methodology has been reported to be used in field trials in Australia, New Zealand and Canada (Wilkinson, (1984)). An important note made in the discussion of Wilkinson et al. (1983) by Patterson, Besag and others was that this method yields estimates very similar to those obtained by generalized least squares estimation assuming correlated observations. 2.4 Least Squares Smoothing Method Green, Jennison and Seheult (1983, 1985) proposed a purely data - analytic approach called least squares smoothing (LSS), in which an explicit use of smoothness assumption is made. The model is the same as proposed by Wilkinson et al.(1983), and can be written in the matrix notation as y = Xτ + ξ + η (2.4.1) Where y and τ are respectively the vectors of observations and treatment effects; X is the design matrix; components of ξ are spatially smooth trend effects; and η is the random error vector. The smoothness - of - trend in one dimension means that in some sense, the second differences ξ 1 ξi ξi 1 should be small. i A least squares approach leads to estimates of τ and ξ which minimize the penalty function φξ' ' ξ +(y - Xτ - ξ)' T(y - Xτ - ξ) (2.4.2) subject to a side condition 1' τ = 0 so that the overall mean is included in ξ. Here, ξ is the vector of second differences of ξ, φ is a tuning constant which can be varied to control the degree of smoothness in the estimate of ξ. It can be seen that τ is the same as that obtained by generalized least squares on y, assuming E ( y) = x τ and Var( y)=σ 2 (φ -1 + '). The choice of the tuning constant φ, estimation of the error variance and some generalizations of LSS were discussed by Green et al (1985). One advantage of the LSS method is that it allows an examination of the fitted ξ, and the residuals η, which may be helpful in detecting any outliers in the data and patterns in ξ. Green et al (1985) illustrated the LSS method using data from variety trials and experiment on mildew control. Green and et al.(1985) suggested the use of cross-validation both to choose the degree of smoothing required, and to select the appropriate smoother. There are some more NN methods available for example, Exponential variance, linear variance and errors in variable modules, the bifurcation of treatment etc. 2.5 Some Comments on NN Methods Recently there has been a rapid growth of NN methods, none of which has yet achieved an acceptance for routine use in field experimentation. Perhaps it is too early to expect such a radical change because the well established classical methods can not be discarded 540

5 unless some new method is clearly superior, on average, to the best of the classical methods. In 1984, the British region of Biometric Society conducted a one-day Workshop to compare the methods currently available, in the hope of making practical recommendations for the guidance of applied statisticians (Kempton, 1984). These methods are often recommended for increasing the efficiency of treatment estimation. But in the absence of proper randomization theory, assessment of the precision of estimates poses a formidable problem. The main advantage of a randomized design and analysis is the simplicity. The analysis can be justified by permutation tests thus making the inference model-free. 3. Experimental Designs For Correlation Models In section 2, we discussed the currently available nearest neighborhood analysis methods. While some of them can be viewed as alternatives to the traditional methods for local control, others can be used in conjunction with the traditional methods. It is also important to note that almost all the NN methods can be embraced by generalized least squares analysis assuming some correlation pattern for random errors. Atkinson (1969), Draper and Faraggi (1985), and Martin (1982) showed the equivalence of Papadakis and correlation methods. Patterson, Besag and others, while discussing Wilkinson et al (1983), pointed out that Wilkinson s procedure yields treatments very similar to those obtained through GLS analysis. Similarly least squares smoothing (though not the generalized version), first differences method of Besag and Kempton (1986) can be shown to be equivalent to GLS analysis. There exists a wide spectrum of models, which can be used to incorporate correlation among neighboring observations. Although the estimates of treatment contrasts are reasonably robust to the assumed correlation structure, yet the variance of an estimated treatment contrast may heavily depend on the assumed correlation structure. In the following we discuss some of the models which have been used in NN methodology. Autoregressive models: In one-dimensional case, let ε i denote the error component of the observation from the i th plot. The simplest model, frequently used in time series analysis, is the first-order autoregressive (AR(1)) or Markov model ε i = ρ ε i-1 +η i (3.1) where ρ <1 and η i are IIDN (0,σ 2 ). For AR(1) model, the correlation between ε i and i j ε j is ρ, and it dies exponentially. The model (3.1) can easily be generalised to higher orders. A two-dimensional extension of this model is doubly geometric model introduced by Quenoulie(1949). Denoting by ε ij the error component at the site (i,j) of rectangular lattice, we have ε ij = ρ1ξ i 1 j + ρ2ξi j 1 + ρ1 ρ2ξi 1 j 1 + ηij (3.2) where ρ 1 <1, ρ 2 <1 and η ij are IIDN(0,σ 2 ). Moving Average Models: Another simple model, also borrowed from time series analysis, i.e. the moving average model, ε i = ρ η i-1 +η (3.3) 541

6 where ρ <1 and η ij are IIDN(0,σ 2 ). The equation (3.3) is called first order moving 2 average (MA(1)), model for which the correlation between ε i and ε j is / ( 1 ρ ) i j = 1 but zero otherwise, i.e. only the nearest neighbour errors are correlated. ρ + if A two-dimensional extension of MA(1) model, called first-order spatial moving average (SMA(1) model by Wynn(1978) and nearest neighbour correlation model by Kiefer and Wynn(1981), is the one in which only the errors from nearest row and column neighbour plots are correlated. Conditional or Autonormal Model: Besag(1974) proposed an autonormal model where component has conditional variance as E[ε i all other values] = ρ ( ε + ε ) i 1 i+ 1 2 Var ([ε i all other values) = σ (3.4) The above prescription can be generalized to higher dimensions in an obvious way. Kempton and Howes (1981) used a variety of conditional models to compare the performance of Papadakis method with that of conventional blocking method by analyzing a series of variety trials and uniformity data. Most of the designs used for field experimentation are dictated by conditions other than the specific models for analysis. Theoretically, every design can be analyzed under any correlation model. If any of the NN analysis method is to be used, a thought must be given to the appropriate designs. With this view in mind, many workers have recently applied some sort of "optimality" criterion for the choice of experimental design. It has been observed that different designs turn out to be optimal under different situations. In the sequel we discuss the cases of block designs and two-dimensional designs. 3.1 One-Dimensional Designs Williams(1952) considered experimental designs when experimental units are arranged along a line and maximum likelihood (ML) estimation procedure is used assuming AR(1) and AR(2) models for observations. The design criterion was the simplicity of analysis. For AR(1) model, Williams suggested (see appendix) nearest neighbour balanced designs(nnbd) called type II designs in which every treatment occurs equally often next to every other. In a next II(a) design a treatment never occurs next to itself. Whilst in II(b) design every treatment is allowed to occur equally often next to itself. Appropriate designs for AR(2) model are type III designs which are type II(a) designs with an additional property that every treatment occurs equally often as second neighbour to every other treatment but not to itself. Williams calculated the relative efficiency of II(a) with respect to II(b) designs and noted that II(a) design is more efficient if ρ is positive, whilst II(b) design is preferable if ρ is negative. Kiefer(1960) proved that for AR(1) model with positive type II(a) designs are asymptotically (as the number of experimental units tends to infinity) optimal whereas type III designs are asymptotically optimal for AR(2) model. 542

7 For finite number of experimental units, Cox(1951) and Atkinson(1969) both conjectured A-optimality of II(a) designs for AR(1) model with positive ρ. Going through tedious and lengthy algebra, Kunert and Martin(1986a) proved the truth of this conjecture and also proved the D-optimality but found that II(a) designs are not E-optimal for all ρ ( 0 1, ). Kunert and Martin(1986b) investigated the optimality of type-ii designs under various forms of AR(1) model including circular structure and abnormal models of Besag(1978). It was found that optimality ordering of designs changes under different optimality criteria and correlation structures. It may be noted that Williams(1952) did not allow the block structure of plot. In many situations, blocks are needed for, say administrative convenience and may represent effects not necessarily related to local fertility variability, Kiefer and Wynn(1981) considered optimality of block designs under ordinary least squares (OLS) estimation and moving average (MA(1)) correlation model. Their optimal criterion was weak universal optimality which included A- and E-optimalities, but not D-optimality, as special cases. They showed that a design is optimal for MA(1) model if in all treatment pairs i and j, 1 i, j t, the quantities tx { q + { q i and j are nearest neighbour in block q} + { q i is applied to an edge plot of block q} j is applied to an edge plot of block q} are the same. Here denotes the cardinality of a set. Construction of designs satisfying this condition has been discussed by Kiefer and Wynn(1981) and Cheng(1983). The optimality of binary block designs under GLS estimation with AR(1) correlation model was discussed by Gill and Shukla(1985) and in more details by Gill(1986). Exact and approximate universally optimal designs were constructed (see appendix). The optimality conditions for complete block designs are that (i) (ii) (iii) every treatment be applied to the edge in the same number of blocks; every pair of distinct treatment be applied in adjacent blocks; and every pair of distinct treatment be applied in edge plots of the same number of blocks. Because of condition (iii) the optimal design needs an impractical number of replications, viz, a multiple of t(t 1) 2. It has been observed that a design satisfying condition (i) and (ii) is nearly optimal. The minimum number of replications required for such a design is t 2 or t according as t is even or odd. Departure from exact optimality of an approximate optimal design has been investigated numerically for some cases and found to be negligible for AR(1) and MA(1) models. These designs also perform satisfactorily under one realistic nonstationary errors-in-variable model. For an incomplete block design, the optimality requirements are that the design should be a BIBD satisfying condition (i) to (iii), and in addition, the design obtained by deleting 543

8 the edge plots of each block should also be a BIBD. Such a BIB design has been termed as Equineighbourhood BIB design by Cheng(1983). Results of Gill and Shukla(1985a) were generalized to nonbinary block designs case by Kunert(1986) who also investigated the optimality of designs under specific optimality criteria. Some methods of construction of efficient equineighbourhood incomplete block designs having block size 3 have been given by Jacroux (1998) These all studies relate to experimental situations where the experimenter is interested in making all possible paired comparisons. However, for the experimental situations where experimenter is interested in making test treatment-control treatment comparisons. Das(1991) studied the universally optimality aspects of block designs when the errors follow a heteroscedastic and correlated variance - covariance structure. The first order autocorrelation model and nearest neighborhood model follow as a particular case of this model. The model assumed by Das(1991) is y = τ + D β + µ 1 + e, + () () where E e = 0, D e = V = A I. j Under this set up, the Binary Block Designs of type P were defined as the block designs which are binary and for which off diagonal elements of C d matrix are equal. It was also shown that a binary block design of type P, d*, whenever it exists, is universally optimal over D ( v,b,k, V). Further, for proper block designs, a new class of designs known as Position Permuted Design (PPD) were introduced. For a PPD every pair of treatments ( i,i' ) appears in all the ( s,t) th positions of the block for i i' = i,..., v and s t = i,...,k. It has been proved that a PPD, whenever exists, is binary block design of type P and hence is Universally optimal over D v,b,k,v. ( ) A BIB design with k = 2, is always a PPD and therefore a binary variance balanced design of type P and hence universally optimal in D v,b,2,v. ( ) k For k > 2, every treatment pair must occur in all the positions of the block. 2 Example: Consider the BIB design ( 7 7,,3,3,1 ) Now according to the condition for a PPD, the final design is

9 . Similarly the results have also been obtained for non-proper block designs and also particular cases of V. The above design is universally optimal over D ( 7,21,3,V ) 3.2 Two-Dimensional Designs In many cases experimental units are arranged in two-dimensional array where correlation may extend along the rows and columns. GLS analysis may be considered as an alternative to the usual analysis incorporating row and column effects. Matern(1972), Berenblut and Web(1974) and Duby et al.(1977) studied the performance of various twodimensional designs when observations are assumed to be correlated. Kiefer and Wynn(1981) (see appendix) proposed NN balanced Latin squares for OLS analysis when observations follow first-order spatial moving average model. Martin(1982) considered design and analysis under stationary torus lattice process. He proved the optimality of NN balanced torus designs for first - order autonormal model. Martin (1986) discussed the optimality ordering of planer designs, for both OLS and GLS estimation under various correlation structures. He showed that for long range correlation structures ordering is invariant under the optimality criteria whereas for short range correlation structures the order changes with optimality criterion. He further showed that efficiency gain of GLS estimation over OLS estimation is only marginal for long range correlation structures. Construction and enumeration of two-dimensional NN balanced designs has been given by Freeman (1979a, b, 1981) and Bailey(1984). Gill and Shukla(1985b) studied the optimality of two-dimensional designs under autonormal and doubly-geometric correlation models. NN balanced designs, in which every two distinct treatments occur equally often as NN in rows and columns, were proved to be optimal for the first-order autonormal model. Martin and Eccleston (2001) derived some effective updating formulae for useful approximation to A-optimality asa part of a general algorithmic technique and hence obtained optimal and near optimal designs for dependent observations. References Atkinson,A.C.(1969). The use of residuals as a concomitant variable. Biometrika, 56, Bailey,R.A.(1984). Quasi-complete Latin Squares: construction and randomization. J.R.Statist.Soc., B46, Barlett,M.S.(1938). The approximate recovery of information from field experiments with large blocks. J.Agric.Sci., 28, Barlett,M.S.(1978). Nearest neighbour models in the analysis of field experiments. J.R.Statist.Soc., B40, Berenblut,I.I. and Web,G.I.(1974). Experimental designs in the presence of autocorrelated errors. Biometrika, 61, Besag,J.(1974). Spatial interaction and statistical analysis of lattice systems (with discussion). J.R.Statist.Soc., B26, Besag,J.(1978). Discussion of paper by M.S.Bartlett. J.R.Statist.Soc., B40,

10 Besag,J. and Kempton,R.(1986). Spatial methods in the analysis of field experiments using neighbouring plots. Biometrics, 42, Bhaumik,D.R. (1990). Optimal incomplete block designs for comparing treatments with a control under the nearest neighbour correlation model. Utilitas Mathematica, Cheng,C.S.(1983). Construction of optimal balanced incomplete block designs for correlated observations. Ann. Statist., 11, Cox,D.R.(1951). Some systematic experimental designs. Biometrika, 38, Das,P.(1991). Studies on optimality of designs. Unpublished Ph.D. Thesis, IARI, New Delhi. Draper,N.R. and Farragi,D.(1985). Role of the Papadakis estimator in one and two dimensional field trials. Biometrika, 72, Duby,C., Guyon,X. and Prum,B.(1977). The precision of different experimental designs for a random field. Biometrika, 64, Freeman,G.H.(1979a). Some two-dimensional designs balanced for nearest neighbours. J.R.Statist.Soc., B41, Freeman,G.H.(1979b). Complete Latin squares and related experimental designs. J.R.Statist.Soc., B41, Freeman,G.H.(1981). Further results on quasi-complete Latin squares. J.R.Statist.Soc., B43, Gill,P.S.(1986). Nearest Neighbour Balanced Designs for Comparative Experiments: Optimality and Analysis. Ph.D. Thesis, I.I.T., Kanpur. Gill,P.S. and Shukla,G.K.(1985a). Efficiency of nearest neighbour balanced block designs for correlated observations. Biometrika, 72, Gill,P.S. and Shukla,G.K.(1985b). Experimental designs and their efficiencies for spatially correlated observations in two dimensions. Commun.Statist: Theor. Meth., 14, Green,P., Jennison,C. and Seheult,A.H.(1983). Discussion of paper by G.N.Wilkinson et al. J.R.Statist.Soc., B45, Green,P., Jennison,C. and Seheult,A.H.(1985). Analysis of field experiments by least squares smoothing. J.R.Statist.Soc., B47, Jacroux,M.(1998). On the construction of efficient equineighbourhood incomplete block designs having block size 3. Sankhya B, 60(3), Kempton,R.A.(1984). Spatial Methods in Field Experiments. Report of papers presented at a Biometric Society Workshop held at the University of Durham, U.K. Kempton,R.A. and Howes,C.W.(1981). The use of neighbouring plot values in the analysis of variety trials. Appl. Statist., 30, Kiefer,J.(1960). Optimum experimental designs. Froc. Pourth Berk. Symp., 2, Kiefer,J. and Wynn,H.P.(1981). Optimum balanced block and latin square designs for correlated observations. Ann. Statist., 9, Kunert,J.(1986). Some optimal block designs for correlated errors. Submitted to Biometrika. Kunert,J. and Martin,R.J.(1986a). On the optimality of finite Williams II(a) designs. Submitted to Commun.Statist. Kunert,J. and Martin,R.J.(1986b). Some results on the optimality properties of finite Williams' designs. Submitted to Commun.Statist. 546

11 Martin,R.J.(1982). Some aspects of experimental designs and analysis when errors are correlated. Biometrika, 69, Martin,R.J.(1986). On the design of experiments under spatial correlation. Biometrika, 73, Martin,R.J. and Eccleston,J.A.(2001). Optimal and near-optimal designs for dependent observations. Statist. Appl., 3, Matern,B.(1972). Performance of various designs of field experiments when applied in random fields. 3 rd Conference Advisory Group Of Forest Statisticians: Institute National De La Recherche Agronomique, Paris Papakadis,J.S.(1937). Methode statisque pour des experiences sur. Champ. Bull Inst. Amel. Plantes a Salonique, 23. Papakadis,J.S.(1940). Comparison de differentes methodes d' experimentation polytechnique. Revesta Argentina de Agronomia, 7, Papakadis,J.S.(1954). Ecologia de los cultives 2, Buenos Aires. Papakadis,J.S.(1970). Agricultural Research. Buenos Aires. Papakadis,J.S.(1984). Advances in yhe analysis of field experiments. Poceedings of the Academy of Athens, 59, Pearce,S.C.(1978). The control of environmental variation in some West Indian maize experiments. Trop. Agric. (Trinidad), 55, Pearce,S.C.(1980). Randomized blocks and some alternatives: A study in tropical conditions. Trop. Agric. (Trinidad), 57, Pearce,S.C. and Moore,C.S.(1976). Reduction of experimental error in perennial crops, using adjustment by neighbouring plots. Expl. Agric., 12, Quenouille,M.H.(1949). Problems in plane sampling. Ann. Math. Statist., 20, Shukla,G.K. and Gill,P.S.(1986). Nearest neighbour designs for comparative experiments: optimality and analysis. Proceedings of the Symposium on Optimization, Design of Experiments and Graph Theory. Indian Institute of Technology, Bombay during December 15-17, pp Street,D.J. and Street,A.P.(1985). Design with partial neighbour balance. J.Statist.Plan.Inf., 12, Whittle,P.(1954). On stationary process in the plane. Biometrika, 41, Wilkinson,G.N.(1984). Nearest Neighbour methodology for design and analysis of field experiments. Proc. 12 th int. Biom. Conf., Tokyo, Wilkinson,G.N., Eckert,S.R., Hancock,T.W. and Mayo,O.(1983). Nearest Neighbour(NN) analysis of field experiments(with discussion). J.R.Statist.Soc., Williams,R.M.(1952). Experimental designs for serially correlated observations. Biometrika, 39, Wynn,H.P.(1978). Discussion of paper by S.M.Bartlett. J.R.Statist.Soc., B40, 161. APPENDIX Williams Type Iia and Iib designs for AR(1) model for t=4 IIa: 2(1,2,3,4) (2,3,1,4) (3,1,4,2) IIb: 1(1,2,3,4) (4,1,3,2) (2,4,1,3) (3,4,2,1) 547

12 Optimal block designs (Gill and Shukla, 1986) for t=4 (2,1,0,3); (2,3,1,0); (2,0,3,1) (3,1,2,0); (0,3,2,1); (3,2,0,1) NN balanced Latin Squares (Kiefer and Wynn, 1981) ) for t=3 and 4 t= t=4 and ρ 1 = ρ Pair of 4 4 balanced Latin Squares (balanced for diagonals) Type I designs (Gill, 1986) for 6 treatments 1(1,2,0,3,5,4,)4; 2(2,3,1,4,0,5)5; 3(3,4,2,5,1,0)0; Type II designs (Gill, 1986) for 6 treatments 2(2,1,0,3)3; 2(2,3,1,0)0; 2(2,0,3,1)1; 3(3,1,2,0)0; 3(3,2,0,1)1; 0(0,3,2,1)1; 548