Internatonal Journal of Theoretcal & Appled Scences 6(1): 50-54(2014) ISSN No. (Prnt): 0975-1718 ISSN No. (Onlne): 2249-3247 Generalzed Incomplete Trojan-Type Desgns wth Unequal Cell Szes Cn Varghese, Seema Jagg and Eldho Varghese Indan Agrcultural Statstcs Research Insttute Lbrary Avenue, New Delh, INDIA (Correspondng author Cn Varghese) (Receved 15 December 2013, Accepted 21 March, 2014) ABSTRACT: Generalzed Incomplete Trojan-Type Desgns are row-column desgns n whch each cell, correspondng to the ntersecton of row and column, contans more than one treatment and the rows are ncomplete. In some expermental stuatons, t s not possble for the expermenter to have the row-column ntersectons of equal sze. Generalzed ncomplete Trojan-type desgns wth unequal cell szes are to be obtaned for such stuatons. In ths paper, a method of constructon of a seres of generalzed ncomplete Trojan-type desgn wth unequal cell szes has been developed. Key Words: Row-column desgn, Trojan-type desgn, Sem-Latn square, Unequal cell sze I. INTRODUCTION Row-column desgns (RCDs) are used n expermental stuatons when the heterogenety n the expermental materal s due to two cross classfed sources. Most commonly, the row-column desgns have only one unt correspondng to the ntersecton of each row and column. However, there may be nstances when the number of treatments s substantally large and a more general class of row-column desgns s requred wheren there s more than one unt n each row-column ntersecton. For example, consder a sensory tral to make a comparatve ratng of 12 food products that are presented to 3 panel members. The constrant here s that each member can assess a maxmum of 4 products n a sesson, more than whch may cause assessor fatgue. The ratng may vary from member to member and sesson to sesson and hence the panel members and sessons can be treated as row as well as column classfcatons. There are 3 rows and 3 columns, but the number of products to be rated s 12 and hence each row-column ntersecton has to be further dvded nto 4 sub-sessons or cells. Sem-Latn square, a generalzed row-column desgn wth n rows and n columns wheren the ntersecton of each row and each column contans a cell of k subdvons resultng n nk cells each row and each column, s approprate for such an expermental stuaton. Thus, each row and each column s complete. Darby and Glbert [1] descrbed Trojan squares based on sets of mutually orthogonal supermposed Latn squares as a specal class of sem-latn squares. These desgns are shown to be maxmally effcent desgns for par-wse treatment comparsons n the plots-wthnblocks stratum by Baley [2]. Some combnatoral propertes of sem-latn squares and related desgns were dscussed by Preece and Freeman [3]. Baley [4, 2] obtaned a range of sem- Latn and Trojan square desgns and showed that the Trojan squares are the optmal choce of sem-latn squares for par-wse comparsons of treatment means. Incomplete Trojan squares obtaned by omttng a sngle row from a complete Trojan square have been dscussed by Edmondson [5], whch are of consderable practcal utlty, when there s a lmtaton of resources. Edmondson [6] constructed generalzed ncomplete Trojan square desgns based on a set of cyclc generators. Some methods of constructon of sem-latn squares were gven by Bedford and Whtaker [7]. Further, Baley and Monod [8] gave some effcent sem-latn rectangles useful for plant dsease experments. Subsequently, Dharmalngam [9] used Trojan square desgn to obtan partal trallel crosses. Jagg et al. [10] defned Generalzed Incomplete Trojan-Type Desgns n whch each cell, correspondng to the ntersecton of row and column, contans more than one treatment and the rows are ncomplete. They have developed a method of constructng these desgns. These desgns are avalable for any number of treatments 6 wth flexblty n choosng the cell sze dependng on the expermental resources avalable. Many tmes, t s not possble for the expermenter to have the row-column ntersectons of equal sze. Generalzed ncomplete Trojan-type desgns wth unequal cell szes are sutable for such stuatons. Here, a method of constructon of a seres of generalzed ncomplete Trojan-type desgn wth dfferent cell szes has been developed.
II. GENERALIZED INCOMPLETE TROJAN- TYPE DESIGNS Defnton (Jagg et al., [10]): Consder an array wth m rows and n columns n whch each ntersecton of row and column s dvded nto k sub-unts formng a total of mnk sub-unts n all. A Generalzed Incomplete Trojan-Type desgn s an arrangement of v treatments n mnk sub-unts, such that each treatment occurs n every column α ( 2) tmes, n every row β tmes (β = 0 or 1) and on every sub-unt γ tmes (γ = 0 or 1). A general method of constructng generalzed ncomplete Trojan-Type desgns wth unequal cell szes s explaned n the subsequent secton. III. METHOD OF CONSTRUCTION OF GENERALIZED INCOMPLETE TROJAN-TYPE DESIGN Let there be v treatments (v > 5) denoted by 1,2,, v. Arrange the frst s treatments 5 s (v-1) n the frst row of the desgn. Varghese, Jagg and Varghese 51 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 1 v 4 5 6 7 8 1 2 v 6 7 8 1 2 3 4 v 7 8 1 2 3 4 5 v 8 1 2 3 4 5 6 Group these s treatments nto n groups (n 2) of szes n k, = 1, 2,, n, ( k = s ) such that at least one or = 1 all of the groups are of dfferent sze. Each group of treatments belongs to a cell n the frst row of the desgn. Now develop (v-1) more rows of the desgn cyclcally, column-wse, by addng 1 to the treatment n the prevous row (reduced mod v). Thus, we get a generalzed ncomplete Trojan-type desgn wth cell szes k for v treatments n m = v rows of sze s and n columns of sze vk. There are several ways to form the n cells n the frst row and hence one can obtan a number of desgns wth dfferent set of parameters for a gven number of treatments, v. Example 3.1: Let there be v = 8 treatments. Further let s = 7 and n = 2. Takng k 1 = 5 and k 2 = 2, the generalzed ncomplete Trojan-type desgn s obtaned n 8 rows and 2 columns as: The nformaton matrx C = - 0.6 - - - - 0.6 The nformaton matrx of treatment effects and the varances pertanng to estmated elementary treatment contrasts under a four-way classfed model were computed by wrtng a SAS code. The elementary treatment contrasts are estmated wth three types of varances, vz., 0.3231, 0.3685 and 0.3885. The average varance s calculated as 0.3646.
Varghese, Jagg and Varghese 52 For 8 treatments, takng s = 6, n = 2, k 1 = 4 and k 2 = 2, the followng desgn s obtaned: 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 v 4 5 6 7 8 1 v 5 6 7 8 1 2 v 6 7 8 1 2 3 v 7 8 1 2 3 4 v 8 1 2 3 4 5 The nformaton matrx C = - 0.5-0.3-0.3-0.5 The varances are 0.3976, 857, 0.5405 and 0.5714 and average varance s 884. Agan, one can have a generalzed ncomplete Trojan-type desgn n 8 rows and 2 columns havng two dfferent cell szes k 1 = 4 and k 2 = 3 as gven below: 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 1 v 4 5 6 7 8 1 2 v 6 7 8 1 2 3 4 v 7 8 1 2 3 4 5 v 8 1 2 3 4 5 6
Varghese, Jagg and Varghese 53 The nformaton matrx C = - 0.8-0.3-0.3-0.8 The varances pertanng to estmated elementary treatment contrasts are 0.3221, 0.3744, 268 and 489 and the average varance s 0.3851. Another arrangement of 8 treatments s gven below n 8 rows and 3 columns havng three dfferent cell szes k 1 = k 2 = 2 and k 3 = 3: 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 1 v 4 5 6 7 8 1 2 v 6 7 8 1 2 3 4 v 7 8 1 2 3 4 5 v 8 1 2 3 4 5 6 The nformaton matrx C = - 0.3-0.3 The varances are 055, 0.5972, 0.7247 and 0.7660 gvng an average varance of 0.6030.A generalzed ncomplete Trojan-type desgn n 8 rows and 2 columns havng cell szes k 1 = 3 and k 2 = 2 can be obtaned for 8 treatments as: 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 v 4 5 6 7 8 v 5 6 7 8 1 v 6 7 8 1 2 v 7 8 1 2 3 v 8 1 2 3 4
Varghese, Jagg and Varghese 54 The nformaton matrx C = - 0.3 There are 4 types of varances, vz., 0.5355, 0.7506, 0.9072 and 0.9558 and the average varance had turned out to be 0.7632. It can be seen that for a fxed total row sze, f the cells are formed such that there are less number of cells wth bgger cell szes, the varance s less. The more the number of cells, the more s the varance for a gven row sze. REFERENCES [1]. L.A. Darby and N. Glbert, The Trojan Square. Euphytca, 7, 183-188(1958). [2]. R.A. Baley, Effcent sem-latn squares. Statstca Snca, 2, 413-43(1992). [3]. D.A. Preece and G.H. Freeman, Sem-Latn squares and related desgns. J. R. Statst. Soc., B45, 267-277(1983). [4]. R.A. Baley, Sem-Latn squares. J. Statst. Plan. Inf., 18, 299 312(1988). [5]. R.N. Edmondson, Trojan square and ncomplete Trojan square desgn for crop research. J. Agrc. Scences, 131, 135-142(1998). [6]. R.N. Edmondson, Generalsed ncomplete Trojan desgns. Bometrka, 89(4), 877-891(2002). [7]. D. Bedford and R.M. Whtaker, A new constructon for effcent sem Latn squares. J. Statst. Plan. Inf., 98, 287-292(2001). [8]. R.A. Baley and H. Monod, Effcent sem-latn rectangles: Desgns for plant dsease experments. Scand. J. Statst., 28, 257-270(2001). [9]. M. Dharmalngam, Constructon of partal trallel crosses based on Trojan square desgn. J. Appled Statst., 29(5), 675-702(2002). [10]. S. Jagg, C. Varghese, E. Varghese and V.K. Sharma, Generalzed ncomplete Trojan-type desgns. Statstcs and Probablty Letters, 80, 706-710 (2010).