Construction of Some New Three Associate Class Pbib Designs with Two Replicates

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1 Iteratioal Joural of Academic Research i Progressive Educatio ad Develomet Costructio of Some New Three Associate Class Pbib Desigs with Two Relicates Silver J.Key Rambaei School of Sciece, Deartmet of Mathematics ad Comuter Sciece, Chekoilel Uiversity College, P.O. Box 11,Eldoret-Keya key.silver.jetoo@gmail.com Chirchir Edwi Kikemoi School of Sciece, Deartmet of Mathematics ad Comuter Sciece, Chekoilel Uiversity College, P.O. Box 11,Eldoret-Keya Tum Isaac Kikosgei School of Sciece, Deartmet of Mathematics ad Comuter Sciece, Chekoilel Uiversity College, P.O. Box 11,Eldoret-Keya Abstract Some ew series of three associate Partially Balaced Icomlete Block Desigs with miimal blocks ad two relicates have bee costructed by itroducig aother blak diagoal etry to triagular associatio scheme. The geeralizatio of arameters for ay eve ositive iteger greater tha or equal to eight have also bee give. Secific desig has also bee costructed to illustrate the results umerically. Keywords: Partially Balaced Icomlete Block, Three associate classes, suerimose. Itroductio By chagig the arragemet of treatmets or omittig some blocks ad or treatmets, we obtai desigs that may be a class of ew desigs. Usig this techique Bose ad Nair (1939) itroduced some PBIB desigs. Atiqullah (1958) established that cosiderig PBIB desigs based o triagular associatio scheme with,,, r =, ad, the ecessary coditio for existece of these PBIB desigs is the existece of symmetrical triagular PBIB desigs with,, ad. Other methods were give by Shrikade (1960, 1965) ad Chag et al (1965) based o the existece of certai BIB desigs ad cosiderig the dual of a BIB desig as omittig certai blocks from the BIB desig. Joh (1966) showed that triagular associatio scheme ca be described by reresetig the treatmets by ordered airs (x, y) with 1 x < y q ad the 188

2 Iteratioal Joural of Academic Research i Progressive Educatio ad Develomet geeralized (Joh, 1966) for the case that, (q > 3). Arya ad Narai (1981) discussed a ew associatio scheme called trucated triagular (TT) with five associate classes whe with a eve ositive iteger 8, ad used to costruct artial diallel crosses. Chig-Shui et al (1984) came u with a geeral ad simle method of costructio based o the relatio of triagular ad l tye of PBIB desig to the lie grah theory. A costructio of triagular desigs with ested rows ad colums is give by Agrawal ad Prasad (1984). Siha ad Saei (004) gave a series of triagular (teary) desigs with ested row ad colums. Recetly, Garg et al (May, 011) costructed some ew series of triagular ad four associate PBIB desigs with two relicatios by usig dualizatio techique. Costructio of Desigs The desig is costructed from two squared arrays of rows ad colums ( is eve ositive iteger greater tha or equal to eight) with both diagoal etries ij (i=j ad i + j = +1) i array havig o treatmets allocated to as illustrated i figure Figure1. Arragemet of treatmet etries i a squared array i ad j are itegers (1 i, j ) such that; Each row ad colum of the square array has treatmet etries. The treatmet etries are allocated i the array by followig two subsequet stes 1. The iitial set of v treatmet etries are first filled o oe side of the diagoal etries.. The secod set of v treatmet etries are allocated i such a way that give ay two etries ad i j are allocated to treatmet x say, if ad oly if the subscrits i + i = j + j. That is; ay two etries allocated to the same treatmet are characterized by the sum ad equivalece of subscrits i ad j of the two etries ad i j such that: Every treatmet i the array aears twice, 189

3 Iteratioal Joural of Academic Research i Progressive Educatio ad Develomet A air of treatmets exist i both the i th row ad j th colum (i, j = 1,,... ). The secod array is foud by trasosig the first array ad the suerimosig it o the first so as to obtai a array cotaiig two treatmets i each cell as i Figure Figure. Arragemet of treatmet etries i the suerimosed square array Results Takig each row or colum of figure to costitute a block, we obtai a desig with the followig arameters distict blocks yieldig Associatio scheme Two treatmets are said to be: i. First associates if they both occur i the same row ad colum. ii. Secod associates if they both occur i the same row or the same colum but ot both. iii. Third associates if they either occur i the same row or i the same colum. Givig rise to the followig associatio arameters ( 10) 1 3 4( 4)

4 Iteratioal Joural of Academic Research i Progressive Educatio ad Develomet ( 4) 0 ( 10) ik ( 4) 0 ( 10) ( 4) ( 6) ( ) ( 6) ( 8) ( 18) ( 8).1 Illustratio Takig =8 we obtai three associate class PBIB desig with the blocks Block 1: (1,, 3, 4, 5, 6, 7,,, 19,, ) Block : (1, 6, 7, 8, 9, 10, 11,,, 0, 3, ) Block 3: (, 5, 8, 11,,,,, 18,,, 3) Block 4: (3, 4, 9, 10,,,,, 18, 19, 0, ) With the arameters ad the associatio matrixes give by:

5 Iteratioal Joural of Academic Research i Progressive Educatio ad Develomet Coclusio I this aer we have costructed some ew series of three associate class PBIB desigs with miimal ossible blocks ad two relicatios from a eve squared array greater tha or equal to eight by leavig both the diagoal etries blak ad suerimosig the squared array by its trasose. The restrictio of the umber of relicatios to two ad the miimal umber of blocks costructed i this aer is desirable i large exerimets to miimize cost. Refereces Agrawal. H ad Prasad J, 1984, Costructio of artially balaced icomlete block desigs with ested rows ad colums. Biom. J, Arya A. S ad Narai Prem, 1981, Trucated triagular associatio scheme ad related artial diallel crosses Sakhya: Idia joural of statistics B, Pt 1, Atiqulla M, 1958, O cofiguratio ad o-isomohisim of some icomlete block desigs Idia joural of statistics, 0 series 3, 4, 7-8 Bose R.C, Nair K.R, 1939, Partially Balaced Icomlete Block desigs Sakhya 4, Chag Li- Cheg, Liu Chagwe ad Liu Wa Ru, 1965, icomlete block desigs with triagular arameters for k 10 ad v 10 Scietia siica, Chig-Shui Cheg, Costace G.M, Hedayat A.S, 1984, A uified method for costructig PBIB desigs based o triagular ad L-schemes J.R.statist.soc 46, 1, -37. Garg D. K, Jhaji H.S ad Mishra G, May, 011, Costructio of Some New Triagular ad Four Associate Class PBIB Desigs with Two Relicates Iteratioal Joural of Mathematical Scieces ad Alicatios 1,, Kishore S. Saei K, 004, Some series of block desigs with ested rows ad colums. Australasia joural of combiatorics. 9, Joh, P. W. M, 1966, A extesio of the triagular associatio scheme to three associate classes. J. Roy. Statist. Soc B, 8, Shrikhade, S. S, 1960, Relatios betwee certai icomlete block desigs, I: Cotributios to Probability ad Statistics. I. Olki (ed.). Staford, CA: Staford Uiversity Press, Shrikhade S. S, 1965, O a class of artially balaced icomlete block desigs, A. Math. Statist 36,