GENERALIZATION OF NEAREST NEIGHBOUR TREATMENTS

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1 International Journal of Advances in Engineering & Technology, Jan 04 GENERALIZATION OF NEAREST NEIGHBOUR TREATMENTS USING EUCLIDEAN GEOMETRY Rani S, Laxmi R R Assistant Professor, Department of Statistics Hindu Girls College, Sonipat Haryana, India Professor, Department of Statistics, M D U Rohtak, Haryana, India ABSTRACT Neighbour design is important to know that the competition effects between neighbouring plots that is the response on a given plot is affected by the treatment applied on neighbouring plots as well as by the treatments applied to that plot In this paper Neighbour design is constructed for OS series using Euclidean Geometry with parameters v=s, b=s(s+), r=s+, k=s, λ=, whether s is a prime number or power of a prime number Here two-sided (left & right) neighbours for every treatment upto q=(s-)-th order for neighbour design is obtained And it is observed that for these neighbour treatments property of circularity holds for the complete design as well as within the each set of s series It is further observed that neighbour treatments of qth - order follow the property of circularity of the same order KEY WORDS: Galois field, BIBD, Euclidean geometry, Neighbour Design, left neighbours, Right neighbours, Circularity I INTRODUCTION Neighbour balanced designs are more useful to remove the neighbour effects in experiments because these designs are a tool for local control in biometrics, agriculture, horticulture and forestry These designs are, therefore, useful for the cases where the performance of a treatment is affected by the treatments applied to its neighboring plots Rees (967) introduced neighbour designs in serology and constructed these designs in complete blocks as well as incomplete blocks Some suitable designs have also been given by Lawless (97), Hwang (973), Dey and Chakravarty (977), Azais et al (993), Iqbal et al (009) Laxmi and Rani (009) obtained the patterns of neighbour treatments of first - order neighbours for every treatment of neighbour designs of the OS series considering two sided (left &right) border plots Laxmi and Parmita (00) suggested a method of finding left neighbours of a treatment in a neighbour design for OS series without constructing the actual design Laxmi and Parmita (0) further suggested a method of finding right neighbours for the OS series Rani and Laxmi observed left and right (two-sided) first - order neighbours of every treatment for OS series using Euclidean Geometry and has shown that these neighbours follow the property of circularity of first-order Rani et al (03) observed left and right (two-sided) second-order neighbours of every treatment for OS series and has shown that these neighbours follow the property of second-order circularity The objective of this paper is given a systematic procedure for finding out two-sided neighbours for neighbour design with parameters v= s, b= s(s+), r= s+, k=s, λ= for different values of s even without constructed the actual design The designs of OS series can be obtained by first forming a finite -dimensional EG(, s), where s is either a prime number or power of a prime number, by using the elements of GF(s), treating the points as treatments, all possible lines as blocks, and then the 784 Vol 6, Issue 6, pp

2 International Journal of Advances in Engineering & Technology, Jan 04 points on a line as the contents of the block corresponding to the line This method is considered for s being a prime number or power of a prime number by taking s=3 and s=4 With the resulted BIBD neighbour design is constructed by using the border plots In bordered designs only the neighbours of treatments in interior plots are of interest Thus a border plot has no neighbours but can be a neighbour of an interior plot and all interior plots have two nearest neighbours II NEIGHBOUR DESIGN OF OS SERIES USING EUCLIDEAN GEOMETRY Orthogonal series for Balanced Incomplete Block Design (BIBD) with parameters v=s, b=s(s+), r=s+, k=s, λ= and v = b = s +s+, r = k = s+, λ= were introduced by Yates (936) The first series was named as OS series and the second series was named as OS series Construction of neighbour design of OS series using Euclidean geometry one may refer to Rani and Laxmi ( ) The first order left and right (two-sided) neighbours for every treatment of OS series have been discussed by Rani and Laxmi ( ) Further, second-order left and right (two-sided) neighbours for every treatment of OS series have been discussed by Rani and Meena (03) This paper interests to find out left and right (two-sided) neighbours upto q th order for neighbour design of OS series where q=,,, s- Let us consider the neighbours of a treatment for the neighbour design for different value of s () Neighbours of a treatment When s=3 The parameters of OS series when s=3 will becomes: v=9, b=, r=4, k=3, λ= and neighbours of first and second- order for a treatment can be easily obtained using the method discussed by Rani and Laxmi ( ) and Rani and Meena (03) Authors considered the third- ordered neighbours of every treatment for the neighbour design and found that these third-ordered neighbours are the treatments themselves Further, higher ordered neighbours that is fourth -ordered, fifth- ordered and so on is the repetition of first order and second- order neighbour treatments respectively It implies that neighbours for a treatment upto second-order [(s-)-th] can be obtained after that the repetition of treatments is there Rani and Laxmi ( ) has found and given a list of neighbours of first-order for every treatment of neighbour design for OS series when s=3 Rani and Meena (03) has given a list of neighbours of second- order neighbours for every treatment for the same design for s=3 Here these left and right (two-sided) neighbours () of first- order, second-order are further summarized in Table Table Two-sided neighbours upto q-th order when s=3 Treatment Series in st -order st -order Other nd -order nd - order other no i which i left right left right lies i s 7,8,9 4,5,6 3, 4,5,6 7,8,9,3,3 3, 3,, 4 s+ i s,,3 7,8,9 6,5 7,8,9,,3 5,6 5 4,6 6,4 6 5,6 6,5 7 s+ i s 4,5,6,,3 9,8,,3 4,5,6 8,9 8 7,9 9,7 9 8,9 9,8 It is further observed that these neighbours have a systematic pattern which is summarized in the form of series in Table Table Two- sided neighbours in form of series upto q-th order when s=3 Treatment Series in st -order st -order Other nd -order nd - order other 785 Vol 6, Issue 6, pp

3 International Journal of Advances in Engineering & Technology, Jan 04 no i S s+ s+ s s+ s+ s which i lies Common right nbhr series i s s+ i s s+ i s i-,i+ s+ i s s+ i s i-,i+ s+ i s i s s+ i s i-,i+ s+ i s i s i-,i+ s+ i s s+ i s i s i-,i+ i s s+ i s i-,i+ () Neighbours of a treatment When s=4 The parameters of the OS series thus becomes: v=6, b=0, r=5, k=4, λ= Neighbours of first- order and second -order for a treatment can be easily obtained using the method discussed by Rani and Laxmi ( ) and Rani and Meena (03) Authors considered the third- ordered and fourth- ordered neighbours of treatments for the neighbour design and found that the fourth- ordered neighbours are the treatments themselves Further, higher ordered neighbours ie fifth- ordered and so on is the repetition of first- order, second- order and third- ordered neighbour treatments respectively It implies that neighbours for a treatment of third- order (s-) can be obtained after that the repetition of neighbour treatments takes place Rani and Laxmi has found and given a list of neighbours of first order for every treatment for neighbour design of OS series when s=4 Rani and Meena (03) has given a list of neighbours of second- order for every treatment for the same design for s=4 Neighbours of third- order for a treatment can be easily obtained by observing the treatment as two plot away in left direction and two plot away in right direction for left and right neighbors respectively ie using circularity property as suggested by Rani and Laxmi ( )and Rani and Meena (03) It is further observed that these neighbours () have a systematic pattern which is summarized in the Table : Table Two- sided neighbours in form of series upto q-th order when s=4 oln no treatment Series in which st -order st- order common st - order other no i i lies right i s 3s+ i s s+ i s i-,i+ 3 S s+ s+ i s i s s+ i 3s i-,i+ s+ s+3 s s+ s+ i 3s s+ i s 3s+ i s i-,i+ s+ s+3 3s 3s+ 3s+ 3s+3 s 3s+ i s s+ i 3s i s i-,i+ Table contd nd -order nd -order nd order 3 rd -order 3 rd -order common 3 rd -order other right Other 786 Vol 6, Issue 6, pp

4 International Journal of Advances in Engineering & Technology, Jan 04 s+ i 3s s+ i 3s i-,i+ s+ i s 3s+ i s i-3,i+3 3s+ i s 3s+ i s i-,i+ s+ i 3s i s i-3,i+3 i s i s i-,i+ 3s+ i s s+ i s i-3,i+3 s+ i s s+ i s i-,i+ i s s+ i 3s i-3,i+3 (3) Neighbours of a treatment When s=5 The parameters of the OS series thus becomes: v=5, b=30, r=6, k=5, λ= Here authors considered the neighbours of first- order, second- order and higher-ordered for the neighbour design and found that the fifth- ordered neighbours are the treatments themselves Further, higher ordered neighbours that are sixth- ordered and so on, is the repetition of first- order, second- order upto fourth- ordered neighbour treatments respectively It implies that neighbours for a treatment upto fourth- order (s-) can be obtained Left and right neighbours of fourth- order for a treatment can be further obtained by observing the treatments as three-plot away in left and right direction of that treatment ie using property of circularity of fourth order Further neighbours () have a systematic pattern which is summarized in the Table 3: Table 3 Two- sided neighbours in form of series upto q-th order when s=5 Coln no Treatment Series in st -order st -order st -order nd -order nd -order no i which i lies common other common left nbhr right nbhr series series i s 4s+ i s s+ i s i-,i+ 3s+ i 4s s+ i 3s 3 4 s s+ s+ i s i s s+ i 3s i-,i+ 4s+ i s 3s+ i 4s s+ s+3 s+4 s s+ s+ i 3s s+ i s 3s+ i 4s i-,i+ i s 4s+ i s s+ s+3 s+4 3s 3s+ 3s+ i 4s s+ i 3s 4s+ i s i-,i+ s+ i s i s 3s+ 3s+3 3s+4 4s 4s+ 4s+ 4s+3 4s+4 s 4s+ i s 3s+ i 4s i s i-,i+ s+ i 3s s+ i s Table contd nd -order other 3 rd -order 3 rd -order 3 rd -order other 4 th -order 4 th -order 4 th - order other 787 Vol 6, Issue 6, pp

5 International Journal of Advances in Engineering & Technology, Jan 04 i-,i+ s+ i 3s 3s+ i 4s i-3,i+3 s+ i s 4s+ i s i-4,i+4 i-,i+ 3s+ i 4s 4s+ i s i-3,i+3 s+ i 3s i s i-4,i+4 i-,i+ 4s+ i s i s i-3,i+3 3s+ i 4s s+ i s i-4,i+4 i-,i+ i s s+ i s i-3,i+3 4s+ i s s+ i 3s i-4,i+4 i-,i+ s+ i s s+ i 3s i-3,i+3 i s 3s+ i 4s i-4,i+4 (4) Neighbours of OS series for any value of s The parameters of the OS series are : v=s, b=s(s+), r=s+, k=s, λ= for which both - sided neighbours are to be found whether s is either a prime number or power of a prime number (i)consider the treatment number i where i=,,,s (ii) Then find the series in which the treatment number i lies The series is defined in such a way that the sequence of first s treatments ie ( to s ) of the design form the first series, the sequence of next s treatments ie (s+ to s) form the second series and the sequence of next s treatments ie (s+ to3s) form the third series and so on so the last series consists of the extreme last treatment from s(s-)-th to s -th treatment Thus, there are s series upto the treatment number s The (s+)-th series of treatment numbers s + to s +s reduces to to s with modv So the (s+)-th series is again the first series of the design It again holds true for the next (s+)-th and so on series which proves that the design is circular (iii) Then find out the neighbour series and neighbour series for any treatment number: Let the treatment number i lies in the j th series where (j=,,,s) then (j-)-th series ie the previous series is the first- order neighbour series and (j+)-th series ie the next series is the firstorder neighbour series (j-)-th series is the second-order left neighbour series and (j+)-th series is the second-order right neighbour series For example: let the treatment number i lies in the j-th series where (j=,,s) then (j-)-th series is the one series away in left direction is the second-order neighbour series and (j+)-th series is the one series away in right direction is the second order neighbour series In the similar fashion, q-th left and right neighbour series can be observed where q=,,s- As there are s+ replications of each treatment there will be s+ left neighbors and s+ right neighbors making s+ in total number of neighbours of any order for a treatment (iv) Two more neighbours other than these two common series can be find by the concept of neighbour means successor or predecessor: These should be (i-q)-th and (i+q)-th neighbour treatments for the q-th order neighbours of treatment number i, where q=,,s- These two neighbour treatments follow the property of circularity of q-th order within the series in which series the treatment number i lies Hence the systematic way of finding left neighbours and right neighbours () of q-th order of any treatment is summarized in the Table 3 Table 3Two- sided neighbours in form of series upto q-th order for any value of s Coln no treatment no i s Series in which i lies st -order common left st -order st - order other nd -order common left i s s(s-)+ i s s+ i s i-,i+ s(s-)+ i s(s-) 788 Vol 6, Issue 6, pp

6 International Journal of Advances in Engineering & Technology, Jan 04 s+ s+ s s+ s+ 3s s(s-)+ s(s-)+ s(s-) s(s-)+ s(s-)+ s s+ i s i s s+ i 3s i-,i+ s(s-)+ i s s+ i 3s s+ i s 3s+ i 4s i-,i+ i s s(s-)+ i s(s-) s(s-3)+ i s(s-) s(s-)+ i s i-,i+ s(s-4)+ i s(s-3) s(s-)+ i s s(s-)+ i s(s-) i s i-,i+ s(s-3)+ i s(s-) Table contd nd -order nd -order q th -order q th -order common q th -order other right other series s+ i 3s i-,i+ s+ i s s(s-)+ i s i-q, i+q 3s+ i 4s i-,i+ s+ i 3s i s i-q, i+q 4s+ i 5s i-,i+ 3s+ i 4s s+ i s i-q, i+q i s i-,i+ s(s-)+ i s s(s-3)+ i s(s-) i-q, i+q s+ i s i-,i+ i s s(s-)+ i s(s-) i-q, i+q After observing the s-th order two-sided neighbours it was found that these s-th ordered neighbours are the treatments themselves Further, higher-ordered neighbours that is (s+)-th, (s+)-th and so on, is the repetition of first-order, second-order upto the (s-)-th ordered neighbour treatments respectively It implies that neighbours for a treatment upto (s-)-th ie highest - order can be obtained after that the repetition takes place It was noted from the above table that the second-order left neighbours and right neighbours of every treatment are same when s=4 This may be due to the circularity of second-order as shown by Rani and Meena (03) It was further observed that for s=8, fourth-ordered left and right neighbours are same, for every treatment It was found true for each even value of s (for which the incomplete block neighbour design can be constructed) of OS series So it is inferred that whenever s is an even number there (s/)-th order two-sided (left & right) neighbours of a treatment shall be same due to circularity of the design 789 Vol 6, Issue 6, pp

7 International Journal of Advances in Engineering & Technology, Jan 04 III CONCLUSION In case of OS series where v=s, b=s(s+), r=s+, k=s, λ= either s is a prime number or power of a prime number neighbour treatments are same either using the method of MOLS discussed by Laxmi et al ( ) and Laxmi et al ( ) or Euclidean Geometry It is further observed that for a given value of s, the neighbour treatments can be find out from -th order to (s-)-th order for treatment number from to s and these neighbour treatments follows the property of circularity of the same order for the neighbour treatments between the complete design as well as within each set of s series ACKNOWLEDGEMENT I wish to thank Dr R R Laxmi for encouraging this research work at the department of Statistics, M D University, Rohtak I am also grateful to a referee for very useful comments concerning the paper REFERENCES [] Azais, J M, Bailey, RA and Monod, H, (993) A catalogue of efficient neighbour design with border plots Biometrics,49, 5-6 [] Dey, A and Chakravarty, R (977) On the construction of some classes of neighbour designs Journal of Indian Society of Agricultural Statistics, 9, [3] Hwang, F K (973) Constructions for some classes of neighbor designs Annals of Statistics, (4), [4] Iqbal, I, Tahir, M H and Ghazali, S S A (009) Circular neighbor-balanced designs using cyclic shifts Science in China Series A: Mathematics, 5(0), [5] Laxmi, R R and Parmita (00) Pattern of left neighbours for neighbour balanced incomplete block designs J of statistical sciences Vol, number, 9-04 [6] Laxmi, R R and Parmita (0) Pattern of right neighbours for neighbour balanced incomplete block designs J of statistical sciences Vol3(), [7] Laxmi, R R and Rani, S (009) Construction of Incomplete Block Designs For Two Sided Neighbour Effects Using MOLS J Indian Soc Stat Opers Res, Vol XXX, No-4,7-9 [8] Laxmi, R R, Rani, S and Parmita (accepted) : Two-sided first-order neighbour for OS series of BIBD (journal of agri bio research) [9] Laxmi, R R, Rani, S and Parmita (accepted): Two- Sided Neighbours of an Order for Neighbour Design of OS Series (journal of agri bio research) [0] Lawless, J F (97) A note on certain types of BIBD balanced for residual effects Annals of Mathematical Statistics, 4, [] Rani, S and Meena Kumari (accepted) : Second-Order Neighbour Effects using Euclidean Geometry Source :Review of research journal, Vol 3(), Nov03 [] Rees, H D, (967) Some designs of use in serology Biometrics, 3, [3] Yates (936): Incomplete randomized blocks Ann Eugen, 7, Vol 6, Issue 6, pp