Efficiency of Neighbouring Designs for First Order Correlated Models

The comparison of efficiency of Complete and Incomplete Nearest Neighbour Balanced Block Designs over regular block design using average variance, generalized variance and min-max variance with the error term  given in the NNBD model follows using first order correlated models. It is observed that, H R and D R show increasing efficiency values for direct and neighbour effects (left and right) for MA(1) models. The A R and G R show neither increasing nor decreasing efficiency values are observed for direct and neighbouring effects for AR(1) and MA(1) models. In the case of ARMA(1,1) model, neither increasing nor decreasing efficiency values have been observed for average variance and generalized variance. The E R shows decreasing efficiency values with  in the interval 0.1 to 0.4 for direct and neighbouring effects for AR(1), MA(1) and


Introduction
The assumptions in the classical (Fisherian) block model are that the response on a plot to a particular treatment does not affect the response on the neighbouring plots and the fertility associated with plots in a block is constant.

IASSL
ISSN-2424- 6271 15 However, in many fields of agricultural research, like horticultural and agroforestry experiments, the treatment applied to one experimental plot in a block may affect the response on the neighbouring plots if the blocks are linear with no guard areas between the plots. If the treatments are varieties, neighbour effects may be caused by differences in height, root vigor, or germination date especially on small plots, which are used in plant breeding experiments. Treatments such as fertilizer, irrigation, or pesticide may spread to adjacent plots causing neighbour effects. Such experiments exhibit neighbour effects, because the effect of having no treatment as a neighbour is different from the neighbour effects of any treatment. Competition or interference between neighbouring units in field experiments can contribute to variability in experimental results and lead to substantial losses in efficiency. In case of block design setup, if each block is a single line of plots and blocks are well separated, extra parameters are needed for the effect of left and right neighbours. An alternative is to have border plots on both ends of every block. Each border plot receives an experimental treatment, but it is not used for measuring the response variable. These border plots do not add too much to the cost of one-dimensional experiments. The estimates of treatment differences may therefore deviate because of interference from neighbouring units. Neighbour balanced block designs, where in the allocation of treatments is such that every treatment occurs equally often with every other treatment as neighbours, are used for modeling and controlling interference effects between neighbouring plots. Azais et al.  (1997) observed that the performance of NNBD is quite satisfactory for the remaining models. Druilhet (1999) studied optimality of circular neighbour balanced block designs obtained by Azais et al. (1993). Bailey (2003) has given some designs for studying one-sided neighbour effects. These neighbour balanced block designs have been developed under the assumption that the observations within a block are uncorrelated. In situations where the correlation structure among the observations within a block is known, may be from the data of past similar experiments, it may be advantageous to use this information in designing an experiment and analyzing the data so as to make more precise inference about treatment effects (Gill and Shukla, 1985). Kunert et al. (2003) considered two related models for interference and have shown that optimal designs for one R

Model Structures
The designs considered here are assumed to be in linear blocks, with neighbour effects only in the direction of the blocks (say left-neighbour or right-neighbour or both). Because the effect of having no treatment differs from the neighbor effects of any treatment, designs with border plots have been considered, which is, designs with one point added at each end of each block. The border plots receive treatments but are not used for measuring the response variables. The plots, which are not on the borders, are inner plots. The length of a block is the number of its inner plots. It is further assumed that all the designs are circular, that is the treatment on border plots is same as the treatment on the inner plot at the other end of the block.  Gill and Shukla, (1985). If the errors within a block follow an ARMA(1,1) model then  Santharam & Ponnuswamy (1997). The NN correlation structure, the  is a matrix with diagonal entries as 1 and off-diagonal entries as  .
Model (2.1) can be rewritten in the matrix notation as follows The joint information matrix for estimating the direct and neighbour (left and right) effects under correlated observations estimated by generalized least squares is obtained as follows: The above v v 3 3  information matrix   C for estimating the direct effects and neighbour effects of treatments in a block design setting is symmetric, nonnegative definite with row and column sums equal to zero. The information matrix for estimating the direct effects of treatments from (2.3) is as follows: where and Similarly, the information matrix for estimating the left neighbour effect of treatments   l C and right neighbour effect of treatments    C can be obtained.

Construction of Design
This series of design bas been investigated under the correlated error structure. It is seen that the design turns out to be pair-wise uniform with 1   and also variance balanced for estimating direct   1 V and neighbour effects 

Comparison of Measures of Efficiency of NNBD
In this section, we study the behaviour of some estimators of  and 2   . The nearest neighbour balanced block design and regular block design data sets were generated with the following true parameters: The estimation of 2   based on nearest neighbour balanced block design and regular block design were compared using the following three measures.

Average Variance Comparison
Consider the measure of harmonic means will also be considered as an index of efficiency.

Generalised Variance Comparison
Another way to compare regular block design and nearest neighbour balanced block design is the ratio: This gives a better comparison of regular block design and nearest neighbour balanced block design.

Min-max Variance Comparison
This closeness is measured by the ratio of the smallest nonzero eigen-value to the largest eigen value of the information matrix. Note that this ratio independent of 2   . For comparing nearest neighbour balanced block design and regular block design, we take the ratio

Results and Conclusion
We have compared the efficiencies of NNBD using average variance, generalized variance and min-max variance when the errors follow first order correlated models.