Response odels nd ini.l designs for mixtures of n of m ites useful for intercropping nd oer investigtions BY W. T. FEDERER B.iomet:r.ics lln.it:, Cornell lln.ivers.it:y, It:hc, NY 14853, ll.s.a. AND D. RAGHA VARAO.Deprt:ment: of St:t:.ist:.ics, Temple lln.ivers.it:y, Ph.i.ldelph.i, PA 19122, ll. S.A. BU-890-M-B October 1986 SUMMARY Consider e sitution where n of m cultivrs re grown togeer in mixture such s t found in intercropping investigtions. Response model equtions for n=2 re formulted in mnner kin to t found for dillel crossing experiment in genetics. This sitution vries considerbly from dillel crossing in t n ~ 2 nd yields my or my not be vilble for ech member of e mixture. Response model equtions were formulted nd en miniml tretment designs were obtined to derive lest squres solutions for e prmeters of e model. This ws done for bo cses, i.e., when individul member yields were vilble nd when ey were not. Vrinces of estimble contrsts re lso given. Applictions to oer res of investigtion wi exmples re given. Some key words: Blnced incomplete block; N-blends; Generl mixing bility; Specific mixing bility. * In e Technicl Report Series of e Bio~etrics Unit, Cornell University.
-2-1. INTRODUCTION Intercropping investigtions involve e growing of two or more cultivrs on e sme re of lnd, where cultivr my be line, vriety, nd/or species. It is centuries-old prctice in tropicl griculture, nd to some extent in temperte zone griculture. Agriculturl, biologicl, nd sttisticl investigtions hve tended to ignore e problems of reserch in is re due to e complexities involved in modeling nd interpreting such experiments. Consider n investigtion involving m cultivrs, sy {l,,m}. One cn form v non-empty sets s 1,,Sv of em cultivrs. These sets my ll be of equl or unequl sizes nd ey my be of ny crdinlity 1,2,,m. The clss of sets so formed is clled e tretment design of e m cultivrs nd ese v tretments will be used in n pproprite experiment design like completely rndomized design, rndomized complete block design, n incomplete block design, etc. If e set S hs single element i, en S = {i} is clled e i sole cultivr or uni-blend. If S {i,j}, en e tretments is clled e hi-blend of cultivrs i nd j. If S ~ {i,j,k} en e tretment S is clled e tri-blend of cultivrs i, j, nd k. clled en-blend of cultivrs i 1,,in. If S= {i 1,,in}' e tretment S is en For simplicity, let us consider e experiment design used wi e v tretments to be n orogonl design. The dt cn be collected eier for ech individul cultivr of e tretment S ssocited wi n experi mentl unit, or e dt cn be collected for e whole experimentl unit to which e tretment S is pplied. Let S= {il,,in}. In e former cse, let yi.(s ) denote e men yield of e - J tretments for j=l,,n. In e ltter cse, i. cultivr used in e J let Ys denote e men
-3- yield of e tretment S. Note t in Ys only e totl of n cultivrs is vilble nd e individul cultivr yields re not. If more complicted experiment designs re used for e v tretments, one my replce yi.(s ) nd Ys by e estimted djusted tretment effects J pproprite to e experiment design used. Thus wiout loss of generlity we ssume e response vrible yi (S ) or Ys for furer nlyses j discussed in is pper. In Section 2 we develop model to interpret yij(s) nd give miniml designs which will enble us to estimte ll e prmetric contrsts of interest. In Section 3, n nlysis of such miniml designs will be presented. In Sections 4 nd 5 similr pproch will be used for Ys. In e concluding section we will give some pplictions. 2. MODEL AND MINIMAL DESIGNS WITH RESPONSE VARIABLE Yi.(S ) J When we consider e response of cultivr i. used in e tretment ] s= {il,,in}' it my be ffected by e following components: (i) its reltive performnce s monoculture or uni-blend, nd such n effect will be denoted by ~ + ~~ ; 1j (ii) Its effect becuse of its use in mixtures, nd such n effect will be clled generl mixing bility. It will be denoted by 6i.; J (iii) its bility to respond well (or poorly) becuse of e presence of ech of e oer n-1 cultivrs in t blend. We denote such effects by y, (. ) fork l,,n; k~j nd we cll em e first order specific 1j 1k mixing bilities of e ij cultivr wi e ik cultivr. One cn esily note t Yij(ik) ~ Yik(ij) nd us e first order mixing bilities re not e usul first order interction of e cultivrs -ij nd ik;
-4- (iv) its bility to respond well (or poorly) becuse of e presence of ech distinct pir of e oer n-1 cultivrs in t blend. We denote such effects by ri (i. ) fork,~= l,,n; k~j, ~*j, k<~, nd ey cn j k' 1 ~ be clled second order specific mixing bilities of e ij cultivr wi ik nd i~ cultivrs; nd (v) Continuing in is fshion, it depends on e ird,,(n-l) order specific mixing bilities of e i. J cultivrs. cultivr wi e oer As n illustrtion, it cn be noted t + where E( ) is e expected vlue of e rndom vrible in e preneses. In n experiment involving m cultivrs, it is possible for ech cultivr to hve monoculture effect, generl mixing bility, specific mixing bilities of first,, nd (m-1) order. In prctice, n ex perimenter my be interested in drwing inferences on t most t order specific mixing bilities for given t (t S m-1). By using uni-blends, hi-blends,, (t+l)-blends, one cn mke such inferences. When using mixtures wi different number of cultivrs, e problems of plnt density per hectre for e~ch cultivr in e mixture nd of unequl error vrinces rise. It becomes bsolutely necessry to use uni-blends if e experimenter is interested in drwing conclusions bout generl mixing bilities. If uni-blend~ re not used in e experiment, ~t. nd 6i. J J effeits will be completelj confoundedl nd one cn drw inferences only bout ~. 1, J In is pper, we restrict our ttention to
-5- blends using e sme number of cultivrs, t is, e tretment sets S (X hving e sme crdinlity. The effects cn be reprmeterized, if necessry, nd e following conditions or restrictions my be imposed on e prmeters: m I 't... o, i=1 1 = 0 ' for given i, j 1 ~i, m I j 1 2 = 0, m 2 r.(j. ) = 0, for given i nd j 1,,jt_ 1, j =1 1 1',Jt t For ny k, e number of k order specific combining bility prm eters on e i cultivr, yi('.. j )is (m-1) choose k, nd e number J 1' ' k of restrictions is (m-1) choose (k-1). Thus, e number of independent k d. fi b. i b 'li h i 1.. or er spec1 c com 1n ng 1 ty prmeters on t e cu t1vr 1s ( m-1 ) - ( m-1 ) = ( m-1)! (m-2k) k k-1 k!(m-k)! nd hence e totl number of k order specific combining bility prmeters is (m-1)! ( m) m k! (m-k)! (m-2k) = k. (m-2k)
-6- In view of is, to drw inferences bout~. nd t most t order specific 1 mixing bilities of e cultivrs, e number of responses needed is 1 + (m-1) + ( ~) (m-2) + ( ~) (m-4) + + (:) (m-2t) ( t:1 ) (t+1), for e men, ~is (cultivr effects in mixture), nd first to t order specific mixing bilities, respectively. Thus using m cultivrs, one cn drw inferences for t most [m/2-1] order specific mixing bilities where[ ] is e gretest integer function becuse m - 2t > 0. On e oer hnd, if n experimenter is interested in estimting t order specific mixing bilities, e number of cultivrs to be used in e experiment m should stisfy m > 2t. The required number (t+1) times m choose (t+1) of responses cn be obtined by using v equl to m choose (t+1) tretment sets S, where e sets S S form n 1' ' v irreducible blnced incomplete block design of ll combintions of (t+1) cultivrs selected from e m cultivrs nd en noting e response on ech of e (t+1) cultivrs in e mixture tretment sets S. Such designs re miniml designs where miniml tretment design is one in which e number of responses for e cultivrs in mixtures is smllest for estimting e prmetric contrsts of interest. A solution of e norml equtions for e unknown prmeters for estimting e required contrsts cn be esily obtined, nd will be given for t = 2 in e next section. 3. A SOLUTION OF PARAMETERS FOR TREATMENT DESIGNS OF SECTION 2 Consider n experiment wi m cultivrs in which e experimenter is - interested in drwing inferences up to second order specific mixing
-7- bilities. As noted in e lst section, e miniml tretment design consists of v equls m choose 3 tretments s 1,,Sv is mde up of ll triples of e m cultivrs nd obtining responses for ech of e 3 cultivrs in ech mixture. However, n experimenter my be interested in using blends of n cultivrs, n~3. The tretments cn be lid out in n pproprite experiment design. Let yi.(s ) be e men response of e J ij cultivr used in e mixture S from n orogonl experiment design. One djusts yi.(s ) for blocks if non-orogonl experiment design is J used. We give e results here using yi.(s ) from n orogonl ex J periment design for generl n nd specilize e results for n=3. As consequence of e conditions or restrictions imposed on e prmeters, e following is solution for e prmeters which cn be used to estimte e contrsts of ~is' ri(j)s, nd yi(j,k)s which re of interest to e experimenter: i i - y - ( n-1) 2 Y. ( S )] ies jts 1 ' ' nd where r i(jk).. ~ [ 2 y i,, j ' kes y = 2 i, yi(s ) I i( s ) nd s = ( m-3) _ 2(m-4) + (m-5) n-3 n-4 n-5
-8- When n 3 e bove simplify to: Yi(j) "" (m-l~(m-3) [(m-3)! Yi(S ) - 2 t. Yi_(s )] u;i,j SN ;ies,jts.... < nd Yi(j,k) (m-2~(m-3) [<m- 3)(m-4)! Y i(s ) u;i,j,kes - (m-4)! y - (m-4) I y u i jes kts i(su) u i kes jts i(su) ' ' u' ' ' ' ( u + 2! yi(s )] ies j kts ' ' ' where is e overll men of e mens. If e mens of e cultivr blends re bsed on r replictions nd if 2 is e error vrince for e responses in e originl orogonl design i"i', vr(yi(j) - yi(j')) 2 2 /{{m-3)r}, vr(yi(j,k) - yi(j,k')) 2{m-4)2/{(m-3)r}, nd ( j t k) " ( j 'k.. )
-9- Expressions for e solutions of prmeters cn be similrly obtined for oer designs of is type, when e interest is in drwing inferences bout higher order specific mixing bilities. An nlysis of vrince cn be obtined by stndrd meods. 4. MODEL AND MINIMAL DESIGNS WITH RESPONSE VARIABLE Ys Subject to e terminology introduced in Section 1, if Ys is e men of e mixture tretments = {i i } bsed on e orogonl 1' ' n experiment design used, n n n n n n n c ~* + L ~i + L A.. + L Ai i i +. + j=1 j j,k 1;j<k 1 j 1 k j,k.~=1;j<k<~ j k ~ A il, i, ' n where A.. = y, (. ) 1j1k 1j 1k + yik(1'j.)' y i ( i1 '. 'i 1) n n- For convenience, we my put A.. z A i ' A 1k1j i i = >..i i i = Ai i i j 1k - 1 j k, j,_ k k j,_ A... = Ai. i Ai. i, etc. The prmeter A. i behves like n 1k1,t 1j, 1j k.t 1k j 1j k interction term for e ij nd ik cultivrs; but in essence it is e sum of combining bilities of ij cultivr wi ik nd t- of ik cultivr wi i.. A similr interprettion cn be given for e oer A prmeters. J In
-10- e sme vein s in Section 2, we my cll A,. s e first order 1j 1k specific mixing bility of ij. nd ik cultivrs, A ijiki,t s e second order specific mixing bility of i., ik nd i 1 cultivrs,, Ai. s J ~ 1', 1 n en order specific mixing bility of i 1,,in cultivrs. As in Section 2, e effects cn be reprmeterized, if necessry, nd e following conditions or restrictions my be imposed on e prmeters: m L "[i = o i=l m L i 1 k m I A... = 0, for given i. nd ik' ij.~ik' i, ij., i 8 ik' i =1 1j1k1,t J ~ ~.t m L i =1 1 For ny k, e number of k order specific combining bility pr- - meters A is m choose (k+l) nd e number of restrictions imposed 1l,tk+1
-11- is m choose k. Thus e number of independent k order specific combining bility prmeters is ( m ) _ (m)... (m) (m-2k-l) k+l k k (k+l) If e experimenter is interested in drwing inferences bout e ~i nd t most t order specific mixing bilities of e cultivrs using mixtures of e sme size wi one response for ech mixture, e number of responses needed is 1 + (m-l) + (m 1 ) (m-3) + (m) (m-5) + + (m) (m-2t-l) ( m) 2 2 3 t t+l - t+l Thus using m cultivrs one cn drw inferences on t most [(m-l)/2] order specific mixing bilities. On e oer hnd, if n experimenter is inter ested in estimting t order specific mixing bilities, e number of cultivrs to be used in e experiment m should stisfy m > 2t+l. Note t in Section 2, m > 2t, wheres here we need m > 2t+l. The required number m choose (t+l) of responses cn be obtined by using v equls m choose (t+l) tretments S, where e sets s 1,,S form n irreducible v blnced imcomplete block design of ll combintions of (t+l) cultivrs selected from e m cultivrs nd en noting e response on ech of e v tretments S. Such designs re miniml designs for estimting ll e prmetric contrsts of interest. A solution of e norml equtions for e unknown prmeters to estimte e required contrsts cn be esily obtined nd will be given for t = 2 in e next section. 5. A SOLUTION FOR PARAMETERS FOR THE TREATMENT DESIGNS OF SECTION 4 Consider n experiment wi m cultivrs on which e experimenter is interested in drwing inferences up to second order specific mixfng bilities s noted in Section 4; e tretment desi-gn consists of v equls
-12- m(m-1)(m-2)/6 tretments s 1,,Sv consisting of ll triples of em cultivrs nd noting e responses on e tretment mixtures. The tretments cn be lid out in n pproprite experiment design. Let y 5 be e men response of e S 0: tretment mixture when e experiment design is n orogonl one. One my replce Ys by djusting it for block effects 0: if non-orogonl experiment design is used. We give e results here using Ys. 0: It cn be esily verified t e following is solution of e prmeters which cn be used to estimte e contrsts of interest of ~. s, 1 ll n-tuples of e m cultivrs: s, when n(~3) cultivrs re used in ech S consisting of 0: 0: ~. = { - (m-1) } (m-2) 1 ~ Ys - n-1 Y 1 n-1 ;H;S 0: where s is given previously, y=i:ys/(:) 0: 0: nd u -(m-3) - n-3 3 (m-4) + 1 (m-5) _ l (m-6) n-4 2 n-5 2 n-6
-13- When n=3, e bove reduces to A.. { ~ y - (m-2)y- (m-3)(i, + i.)}/(m-4) 1 J i J'ES s 1 J ' ' t.. k = 1] ~ y.. k s s ;1,], - y - where y 6 EyS /m(m-l)(m-2) is e overll men of mens. If 2 is e vrince of e experimentl unit totl response in blends of cultivrs nd if e replictions re used for ech blend in e orogonl design i :F i' vr(aij- tij') = 2(m-3) 2 /{r(m-2)(m-4)}, i ± j r A 2 2 {(m-5) 2 + 2(m-S)2 4 } vr(aijk- ijk') = r (m-3) 2 (m-3)2(m-4) 2 + (m-3) 2 (m-4) 2 kfl:k'. Expressions for e solutions of prmeters cn be similrly obtined for oer designs of is type, when e interest is in drwing inferences bout higher order specific mixing bilities. An nlysis of vrince cn be obtined by stndrd meods. 6. APPLICATIONS AND CONCLUDING REMARKS Severl experiments of e type described bove hve been conducted nd e bove sttisticl nlyses re considered to be pproprite for em. Three of ese re described below. In e first one e experimenter, Steven Kuffk, Cornell University, ws interested in biomss production of six species or cultivrs in mixtures of ree. There were 20
-14- such mixtures. His interest centered on generl, first order, nd second order mixing effects. The experiment design ws rndomized complete block design. In second experiment he used blnced incomplete block design. The models nd nlyses discussed here re not confined to intercropping experiments only. For second ppliction, preference rtings for eight soft drinks in mixtures (group~) of four using doubly blnced incomplete block tretment design of 14 blocks hs been conducted by Rghvro nd Wiley (1986). Their interest centered on first order specific mixing competing effects of one soft drink on noer on preference rtings by individul users of soft drinks. The results of is pper re lso useful in studying e effects of one question on noer in survey design using e block totl procedure of Rghvro nd Federer (1979). Mny oer pplictions could be mde. Unequl numbers of cultivrs or items in mixture my lso be used. D. B. Hll, described procedures for such situtions in 1975 Cornell University Msters Thesis entitled "Miniml designs to estimte hi-specific mixing bility." Note t solutions for ~i = 6i + ~~were obtined here. To obtin solutions for 6. nd ~*i individully it is necessry to 1 include uni-blends or monocultures. REFERENCES Rghvro, D. nd Federer, ~. T. (1979). Block totl response s n lterntive to e rndomized response meod in surveys. J. Royl S~~2s~. Soc., Ser2es B, 41(1), 40-45. Rghvro, D. nd Wiley, J. B. (1986). Testing competing effects mong soft drink brnds. In S~~is~icl Design: Theory nd Prc~ice. Proceed2ngs of Conference in Honor of ft'l~er I. Federer. Eds. C.E. McCulloch, S.J. Schwger, G. Csell, nd S.R. Serle, pp. 161-176. Cornell University Press, Ic, New York.