Analyzing genotype-by-environment interaction using curvilinear regression

Transcription

1 357 Analyzing Scintia GE Agricola using curvilinar rgrssion Analyzing gnotyp-by-nvironmnt intraction using curvilinar rgrssion Dulc Gamito Santinhos Prira 1 * Paulo Canas Rodrigus 24 wona Mjza 3 Stanislaw Mjza 3 João Tiago Mxia 2 1 Univrsidad d Évora/Escola d Ciências Tcnologia CMA-UE Dpto. d Matmática R. Romão Ramalho Évora Portugal. 2 FCT/UNL Cntro d Matmática Aplicaçõs Caparica Portugal. 3 Poznan Univrsity of Lif Scincs Dpt. of Mathmatical and Statistical Mthods ul. Wojska Polskigo Pozna Poland. 4 SLA Campus Lisboa Laurat ntrnational Univrsitis Estrada da Corria Lisboa Portugal. *Corrsponding author <dgsp@uvora.pt> Editd by: Thomas Kumk Rcivd Jun Accptd May ABSTRACT: n th contxt of multi-nvironmnt trials whr a sris of xprimnts is conductd across diffrnt nvironmntal conditions th analysis of th structur of gnotyp-bynvironmnt intraction is an important topic. This papr prsnts a gnralization of th joint rgrssion analysis for th cass whr th rspons (.g. yild) is not linar across nvironmnts and can b writtn as a scond (or highr) ordr polynomial or anothr non-linar function. Aftr idntifying th common form rgrssion function for all gnotyps w propos a slction procdur basd on th adaptation of two tsts: (i) a tst for paralllism of rgrssion curvs; and (ii) a tst of coincidnc for thos rgrssions. Whn th hypothsis of paralllism is rjctd subgroups of gnotyps whr th rsponss ar paralll (or coincidnt) should b idntifid. Th us of th Schffé multipl comparison mthod for rgrssion cofficints in scond-ordr polynomials allows to group th gnotyps in two typs of groups: on with upward-facing concavity (i.. potntial yild growth) and th othr with downward-facing concavity (i.. th yild approachs saturation). Thortical rsults for gnotyp comparison and gnotyp slction ar illustratd with an xampl of yild from a non-orthogonal sris of xprimnts with wintr ry (Scalcral L.). W hav dltd 10 % of that data at random to show that our mtorology is fully applicabl to incomplt data sts oftn obsrvd in multi-nvironmnt trials. Kywords: Schffé multipl comparison mthod joint rgrssion analysis tst for paralllism tst of coincidnc ntroduction Farmrs and scintists aim to idntify suprior prforming gnotyps across a wid rang of nvironmntal conditions. Hr by nvironmnts w man combinations of locations and yars. Th main sourc of diffrncs btwn gnotyps in thir yild stability is th fact that th gnotyp and nvironmnt ffcts ar not additiv i.. gnotyp-by-nvironmnt intraction (GE) is prsnt in th data. This intraction can b du to contrasting drought strss lvls wintr low tmpratur strss abiotic strsss growing cycl duration availability of nutrints tc. Th GE can b xprssd ithr as crossovrs whn two diffrnt gnotyps chang in rank ordr of prformanc whn valuatd in diffrnt nvironmnts or inconsistnt rsponss of som gnotyps across nvironmnts without changs in rank ordr. Th study and undrstanding of ths intractions is a major challng for brdrs and agronomic rsarchrs attmpting to improv complx traits (.g. yild) across nvironmntal conditions. Various tchniqus hav bn usd to analyz th intraction in gnral and GE in particular. Radrs intrstd in thos mthods ar rfrrd to.g. Aastvit and Mjza (1992); Annicchiarico (2002); Gauch (1992); Kang and Gauch (1996); Romagosa t al. (2009). Rgrssion is on of th most popular and most applicabl mthods usd for infrnc about gnotyp comparison and slction in th contxt of multi-nvironmntal xprimnts. n rgrssion analysis two sts of variabls ar usd th first charactrizing gnotyps and th scond nvironmnts. Th so-calld adjustd mans (or som othr gnotypic charactristic) for gnotyps usually constitut obsrvations of th dpndnt variabl. n our illustration w tak th original obsrvations of th phnotypic variabl as a ralization of ach dpndnt variabl. Th indpndnt variabl is dfind by nvironmntal indxs which rprsnt a masurmnt of productivity. Although Finlay and Wilkinson (1963) dfind ths nvironmntal indxs as th avrag ovr all nvironmnts for vry gnotyp in this study w comput thm with an itrativ zigzag algorithm (Mxia t al. 1999; Prira and Mxia 2010) which lads to th bst linar unbiasd stimators of th joint rgrssion paramtrs. n joint rgrssion analysis (JRA) aftr slcting th variabl of intrst (.g. yild) th joint rgrssion modl adjusts a linar rgrssion pr gnotyp across all nvironmnts ( 2011; Rodrigus t al. 2011) on a synthtic variabl masuring productivity th nvironmntal indx. Svral variants of JRA hav bn proposd. Th on in which w will b intrstd hr was first proposd by Gusmão (1985) who showd that th prcision in analyzing sris of randomizd block xprimnts was incrasd by considring nvironmnt indxs for individual blocks instad of only on nvironmntal indx pr nvironmnt. Mxia t al. (1999) introducd th L 2 nvironmntal indxs obtaind by minimizing th sum of sums of squars of rsiduals both in ordr to th cofficints of th rgrssions and to th nvironmntal indxs. Hr a gnralization of th joint rgrssion analysis is prsntd for cass whr th rspons (.g. th yild) is not linar across nvironmnts and can b writtn as a scond (or highr) ordr polynomial or

2 358 Analyzing GE using curvilinar rgrssion anothr non-linar function. n our considrations w shall start with th stimation of th rgrssion functions (linar or curvilinar) indpndntly for all gnotyps. n th scond stp th hypothsis of paralllism of rgrssions for all gnotyps is tstd. n gnral it is xpctd that th hypothsis of paralll rgrssion lins will b rjctd bcaus of th prsnc of GE in th data. f w rjct th paralllism of rgrssion curvs th nxt stp to invstigat GE is to try to find subgroups of gnotyps with similar rsponss to diffrnt nvironmntal conditions. Th gnotyps can thn b dividd into groups basd on th Schffé multipl comparison mthod for rgrssion cofficints i.. th gnotyps with similar bhavior will b groupd togthr and th calculations prformd for ach group sparatly. Th mthodology will b illustratd by using a data st for wintr ry (Scalcral L.) from multi-nvironmnt trials carrid out in th yars 1997 and 1998 in Słupia Wilka Poland (52 o 13 N 17 o 13 E). Matrials and Mthods Th zigzag algorithm For convninc lt us considr data arrangd in a two-way array with rows and b columns. Suppos is a continuous rspons variat (.g. yild) for gnotyp n block j if prsnt. Th joint rgrssion modl discussd hr is an xtnsion of that of Finlay and Wilkinson (1963) whr th nvironmntal indxs ar computd for ach block instad of for ach nvironmnt. Assuming that th yild vctors ar indpndnt normal homoscdastic and that gnotyp s prsnt in block j (or rplicat) th joint rgrssion modl can b writtn as: a i + b i c j + ij (i1... ; j 1... b) (1) with a i th intrcpt and b i th slop for gnotyp i c j th block nvironmntal indx and ij th rsiduals. Ths nvironmntal indxs rprsnt th avrags ovr a block/suprblock and can b sn as a (spatial) masur of productivity. Gusmão ( ) showd that th prcision in analyzing sris of randomizd block xprimnts was improvd considring nvironmntal indxs for individual blocks of only on nvironmntal indx pr xprimnt instad of on nvironmntal indx pr nvironmnt. This proposal rsults in K xprimnts ach with b blocks i.. Kb supporting points pr rgrssion instad of only K such points usd by th classical Finlay-Wilkinson joint rgrssion modl (Finlay and Wilkinson 1963). To stimat th modl paramtrs w wish to minimiz (2) whr p ij is th wight of gnotyp n block j. f th gnotyp is absnt w tak p ij 0. Whn th gnotyp occurs w tak pij pj j 1 2 b. Ths wights may diffr from block to block to xprss diffrncs in th rprsntativnss of th blocks. f thr ar svral blocks in th sam location thir wights will b th sam. n th illustration prsntd in this papr w us 1 and 0 for th wights bcaus no information was availabl about th rlvanc of th importanc of blocks. Th zigzag algorithm (Prira and Mxia 2010) is usd to minimiz th loss function (2) itrativly with rspct to a i to b i and to th nvironmntal indx x j. For th complt cas (i.. all th gnotyps ar prsnt in ach nvironmnt) th avrag yild pr block can b a good initial valu for sarching th nvironmntal indxs (Gusmão 1985). Whn incomplt blocks ar usd on may tak th avrag yilds for th corrsponding suprblock as th initial valus. n th worst cas any initial valus may b takn sinc th computation tim dos not incras much. W assum that th yild vctors hav componnts normally and indpndntly distributd so that th zigzag algorithm will lad to maximum liklihood stimators and nabl us to mak infrncs whil comparing gnotyps. Th zigzag algorithm may b dscribd as follows: (i) (ii) (iii) (iv) (v) Calculat th initial valus for th nvironmntal indxs x b 0 which rang within th intrval [a 0 b 0 ] whr a 0 Min{ x x 0b } and b Max{ x... x } ; b Minimiz th function and ; To minimiz functions: and obtain j 1 2 b minimiz th b to obtain th nw vctor x 0 of nw nvironmntal indxs; Standardiz th vctor of nvironmntal indxs to kp th rang unchangd. With a 0 Min{ x 01 x 0b} b 0 Max{ x 01 x 0b} tak b0 a0 x1j a0 + ( x 0i a 0) b b 0 a ; to obtain th vctor x 0 1 th nw nvironmntal indxs. Rpat stps (ii) to (iv) until succssiv sums of sums of squars of wightd rsiduals diffr by lss than a fixd valu. At th nd of ach itration a standardization of th adjustd nvironmntal indxs is carrid out so that th rang dos not chang from itration to itration. Th procdur is carrid out until th goal function stabilizs.

3 359 Analyzing GE using curvilinar rgrssion Th nvironmntal indxs adjustd in this way ar calld L 2 nvironmntal indxs bcaus th L 2 norm was usd. Th dscribd zigzag algorithm is a vrsion of th itrativ algorithms xisting in th litratur; s for xampl Digby (1979); Gabril and Zamir (1979) and Ng and Williams (2001). Prira and Mxia (2010) provd th convrgnc of th zigzag algorithm and that th adjustd paramtrs could b sn as maximum liklihood stimators. Th altrnativ algorithms only considrd th numrical adjustmnt. Tst for coincidnc Considring as bfor th phnotypic obsrvation for gnotyp n block j j 1 b; i 1 and X j th nvironmntal indx for nvironmnt j can b writtn as b i0 m i b i1 z 1 b i2 z 2 )... b it z t. (6) Th null hypothsis to tst th paralllism of rgrssion functions can b writtn as H 0 : b ik b ck i 1 ; k 1 t (7) whr b ck dnots th common k th rgrssion cofficint qual for all gnotyps. Lt us considr now th cas whn som of th obsrvations ar missing. Thn lt n i ( b) b th numbr of nvironmnts in which th gnotyp s obsrvd and N ni. i 1 Classical rgrssion tchniqus ar usd to stimat th paramtrs by th last squars mthod indpndntly for ach gnotyp. Thn w hav b i0 + b i1 z 1j b it z tj + ij (3) whr b ik (k 0 1 t) ar th t+1 unknown rgrssion cofficints z kj (f k (x j )) ar known functions of th nvironmntal indxs x j ij ar indpndnt and idntically distributd random variabls following normal distribution with E( ij ) 0 for all i j and Aftr idntifying th rgrssion functions for all gnotyps a tst of coincidnc for rgrssions can b usd to chck whthr th yild rsponss for gnotyps ar similar with rspct to nvironmntal indxs. This tst can b prformd in two stags (Klinbaum t al. 2008; Williams 1967): (i) a tst for paralllism of rgrssion curvs; and (ii) a tst of coincidnc for thos rgrssions. Equation (3) can b rwrittn as m i + b i1 (z 1j z 1 ) + b i2 (z 2j z 2 ) b it (z tj z t ) + ij j 1... b; i 1... (5) Aftr cntring th obsrvations w hav (4) whr b i is th vctor of stimators of rgrssion paramtrs for th gnotyp i Z i is an ( ni t) matrix of cntrd valus of xplanatory variabls Y i [Y i1 Y i2... Y ] i Th SS has n t 1 dgrs in i of frdom. Tabl 1 givs th analysis of varianc to tst th paralllism of rgrssion functions. Th statistic F C undr H 0 follows an F cntral distribution with ( 1) t and N t dgrs of frdom. Aftr rjcting th hypothsis that all intrcpts ar qual w can tst th hypothsis of th form: H 01 : b 10 b b 0. (8) To tst H 01 it is ncssary to calculat th stimator of common rgrssion cofficints b C (undr hypothsis H 0 ). Ths stimators can b obtaind by solving normal quations of th form Z Zb C Z Y (9) whr Tabl 1 Analysis of varianc for paralllism of th rgrssion lins. Sourc of variation d. f. Sum of squars Man squar F-ratio SSRC RC t Combind Rgrssion t SS R C b C ZY Btwn rgrssions ( 1)t SS bzy -SS C R C i 1 C SSC ( 1 ) t F C C Combind Rsiduals N t SS ' YY bzy i 1 i 1 SS N t Total within gnotyps N SS ' Y YY i 1

4 360 Analyzing GE using curvilinar rgrssion ' By SS C YY b CZY YY bczy w dnot th i 1 comon sum of squars for dviation from rgrssion with ν N t dgrs of frdom. Lt Y b th vctor of all N obsrvations corrsponding to th vctor of t rgrssion paramtrs and an N t matrix Z ~ of obsrvd valus of Z. Using th standard rgrssion tchniqu w obtain th sum of squars for th rsidualss E Y Y bz Y. Th ovrall analysis of varianc to tst th hypothss H 0 and H 01 is prsntd in Tabl 2. Th statistic F undr H 01 follows an F-distribution with 1 and N t dgrs of frdom. Plant matrials To illustrat th dscribd mthodology w us th yild data from a wintr ry (Scalcral L.) xprimnt obtaind in multi-nvironmnt trials carrid out at Słupia Wilka (Poland) in 1997 and n ach dsign thr wr four suprblocks with four blocks of four plots ach. Each gnotyp occurrd in on plot pr suprblock. Th data st usd in this illustration is a subst with fiv gnotyps (CHD_296 RAH_596 RAH_697 RAH_797 and URSUS) and 32 blocks. From this twoway tabl with 32 rows and fiv columns 10 % of th data valus wr dltd to simulat th possibility of having missing valus du to psts animals or othr likly factors. Th rmoval of missing valus was prformd so as to produc an idntical numbr of missing clls pr gnotyp. Th summary of th two-way data is prsntd in Tabl 3. Rsults and Discussion Aftr applying th zigzag algorithm to th nonorthogonal sris of xprimnts dscribd in th prvious sction th nvironmntal indxs ar obtaind and usd as indpndnt variabl for th rgrssion curvs. Th rspons function (i.. yild) with rspct to nvironmntal indx was stimatd by svral functions and th adjustd cofficint of dtrmination R 2 obtaind. Tabl 4 shows th adjustd R 2 for all th considrd functions and all fiv gnotyps. Although all th adjustd R 2 ar high and vry similar w hav dcidd to us quadratic rgrssion to xprss th rsponss of gnotyps bcaus this modl is that for all gnotyps it givs th bst fit of th data as can b sn from Tabl 4. Th adjustd rgrssion cofficints of th quadratic modl R 2 and p-valus arprsntd in Tabl 5. Figur 1 rprsnts th adjustd quadratic rgrssions for th fiv wintr ry gnotyps in study. Aftr rjcting th hypothsis that th rgrssion is not a quadratic curv for ach of th gnotyps (p < Tabl 5) w ar ld to tst whthr th rgrssion curvs ar paralll for all fiv gnotyps (cf. Tabl 1). Th hypothsis that th rgrssion lins ar paralll is rjctd (Tabl 6) as xpctd aftr analyzing Figur 1 and from a prliminary analysis whr GE was found in this data. n th nxt stp of invstigating GE w trid to find subgroups of gnotyps whr rsponss ar paralll (or coincidnt). By simpl inspction of th cofficints (Tabl 5) w can considr two groups of gnotyps: (1) CHD_296; RAH_797 and URSUS; (2) RAH_596 and RAH_697. n this cas this is in accordanc with th prliminary analysis prsntd in Figur 1. To quantify th diffrncs w can us multipl comparison mthods such as Schffé (Schffé 1959; Millr 1991). Whn using th Schffé mthod rprsnting by f 1 a r g th 1 a quantil of th cntral F distribution with r and g dgrs of frdom and s 2 b im i 1... m 01 2 th varianc of th rgrssion cofficint b im th pairs of quadratic rgrssion cofficints which satisfy th condition m ar diffrnt at th significanc lvl a. Sinc th rgrssion cofficints b 1 and b 2 do not diffr among nvironmnts (p < 0.05) w prsnt in Tabl 7 only th rsults for th Schffé multipl comparison mthod of th b 0 cofficints. Tabl 3 Dscriptiv statistics for th fiv gnotyps and th nvironmntal indx. Gnotyps n i Man Std. Dv. Min. valu Max. valu CHD_ RAH_ RAH_ RAH_ URSUS Environmntal indx Tabl 2 Ovrall analysis of varianc to tst th coincidnc of th rgrssion lins. Sourc of variation d. f. Sum of squars Man squar F-ratio Ovrall rgrssion t SS b ZY Btwn intrcpts 1 SS SS E - SSC R SSR t R SS 1 F Btwn rgrssions ( 1)t SS C C F C Rsidual-combind N t SS Total within gnotyps N 1 YY

5 361 Analyzing GE using curvilinar rgrssion Tabl 4 Adjustd R 2 valus for svral rspons functions. Gnotyp Function CHD_296 RAH_596 RAH_697 RAH_797 URSUS Linar Logarithmic nvrs Quadratic Compound Powr S-Curv Growth Exponntial Logistic Tabl 5 Rgrssion cofficints cofficint of dtrmination and p-valu for th fiv gnotyps. Rgrssion cofficints Gnotyp R 2 p-valu b 0 b 1 b 2 CHD_ <0.001 RAH_ <0.001 RAH_ <0.001 RAH_ <0.001 URSUS <0.001 Tabl 6 ANOVA for paralllism of all fiv quadratic rgrssion lins. Sourc of variation d.f. Sum of squars Man squar F-ratio p-valu Combind rgrssion Diffrnc of rgrssions < Combind rsiduals Total within groups Figur 1 Adjustd quadratic rgrssions for th fiv wintr ry gnotyps in study. Th abscissa corrsponds to th yild and th ordinat to th nvironmntal indx. Th dots rprsnt th gnotyps and th solid lins th adjustd quadratic rgrssions.

6 362 Analyzing GE using curvilinar rgrssion Th groups obtaind with th Schffé mthod at significanc lvl 1 % ar th sam as thos givn by th simpl inspction of th cofficints or by analysis of Figur 1. Th multipl comparison mthod of Schffé mad it possibl to divid th gnotyps into two groups: on group with upward-facing concavity (i.. potntial yild growth) and othr with downward-facing concavity (i.. th yild approachs saturation). nspcting th cofficints spcially b 2 it is possibl to s th form of th yild curvs. f b 2 is positiv th curv will b convx othrwis concav. Tabl 8 shows th common rgrssion cofficints for all gnotyps togthr and ach of th two groups obtaind using th Schffé multipl comparison mthod whil Tabl 9 givs th ANOVA to tst th paralllism of rgrssion lins for ach of th two groups of gnotyps. Th hypothss of paralllism btwn th quadratic rgrssions wr rjctd for th first group of gnotyps (Tabl 9). Howvr w do not rjct th sam hypothsis for th scond group. Hnc in this cas w can go on stp furthr and tst whthr th rgrssions in th scond group ar coincidnt (hypothsis H 01 ). From th ANOVA prsntd in Tabl 10 w rjct that hypothsis i.. although th quadratic rgrssion lins ar paralll thy ar distinct. Thrfor th adjustd rgrssion functions for th gnotyps RAH_596 and RAH_697 can b writtn as: Yˆ x 0.066x 2 RAH _ ; Yˆ x 0.066x 2 RAH _ Tabl 7 Schffé multipl comparison tsts of th b 0 cofficints. b b i0 l0 RAH_596 RAH_697 RAH_797 URSUS CHD_ ** 8.949** 2.145* NS RAH_ NS ** 9.985** RAH_ ** ** RAH_ NS *Significant at th 0.05 probability lvl; **Significant at th probability lvl; NS not significant at th 0.05 probability lvl. Tabl 8 Common rgrssion cofficints cofficint of dtrmination and p-valus for all gnotyps togthr and ach of th two groups (cntrd data). Rgrssion cofficints Groups of gnotyps R 2 p-valu b 0 b 1 b 2 All <0.001 Group <0.001 Group <0.001 Tabl 9 ANOVA to tst th paralllism of rgrssion lins for ach of th two groups of gnotyps. Group Sourc of variation d.f. Sum of squars Mansquar F-ratio p-valu 1 Combind Rgrssion Diffrnc of Rgrssions < Combind Rsiduals Total within groups Combind Rgrssion Diffrnc of Rgrssions NS 0.98 Combind Rsiduals Total within groups NS not significant at th 0.05 probability lvl. Tabl 10 ANOVA to tst th coincidnc of rgrssion functions in th scond group (RAH_596 and RAH_697). Sourc of variation d.f. Sum of squars Man squar F-ratio p-valu Ovrall rgrssion Btwn intrcpts * Btwn rgrssions NS 0.98 Rsidual-combind Total-within gnotyps *Significant at th probability lvl; NS not significant at th 0.05 probability lvl.

7 363 Analyzing GE using curvilinar rgrssion whr th b 1 and b 2 ar th common rgrssion cofficints from Tabl 8 and th b 0 cofficints ar calculatd according to xprssion (6). W point out that w did not find groups of gnotyps with idntical rgrssions; this may b du to th high lvl of GE. All that w can say is that gnotyps RAH_596 and RAH_697 hav yilds paralll with that of th scond gnotyp a littl highr. Conclusions Th hypothsis of paralllism of rgrssion curvs was rjctd which is natural in multi-nvironmnt trials with intraction btwn gnotyp and nvironmnt. Th main diffrnc in th two subgroups of gnotyps whr th rsponss ar paralll is that on group had upward-facing concavity (i.. potntial yild growth) and th othr had downward-facing concavity (i.. th yild approachs saturation) which can hlp brdrs in thir gnotyp slction. Th approach proposd in this papr is gnral and applicabl to any sris of xprimnts conductd in multi-nvironmnt trials or simply to th cas of two-way classifid data. Acknowldgmnts Dulc G. Prira is a mmbr of th CMA-UE a rsarch cntr financd by th Scinc and Tchnology Foundation Portugal. Th work was partially supportd by Ministry of Scinc and Highr Education Grant N N and by th Fundação para a Ciência a Tcnologia (Portugus Foundation for Scinc and Tchnology) through PEst-OE/MAT/U0297/2011 (CMA). W thank th anonymous rfrs and th Associatd Editor for thir suggstions which gratly improvd th papr. Rfrncs Aastvit A.H.; Mjza S A slctd bibliography on statistical mthods for th analysis of gnotyp nvironmnt intraction. Biultyn Ocny Odmian 25: Annicchiarico P Gnotyp Environmnt ntractions: Challngs and Opportunitis for Plant Brding and Cultivar Rcommndations. Food and Agricultural Organization Rom taly. (FAO Plant Production and Protction Papr 174). Digby P.G.N Modifid joint rgrssion-analysis for incomplt varity nvironmnt data. Journal of Agricultural Scinc 93: Finlay K.W.; Wilkinson G.N Analysis of adaptation in a plant-brding programm. Australian Journal of Agricultural Rsarch 14: Gabril K.R.; Zamir S Lowr rank approximation of matrics by last-squars with any choic of wights. Tchnomtrics 21: Gauch H.G Statistical Analysis of Rgional Yild Trials: AMM Analysis of Factorial Dsigns. Elsvir Amstrdam Nthrlands. Gusmão L An adquat dsign for rgrssion analysis of yild trials. Thortical and Applid Gntics 71: Gusmão L nadquacy of blocking in cultivar yild trials. Thortical and Applid Gntics 72: Kang M.S.; Gauch H.G. ds Gnotyp-by-nvironmnt intraction. CRC Prss Boca Ratón FL USA. Klinbaum D.G.; Kuppr L.L.; Nizam A.; Mullr K.E Applid Rgrssion Analysis and Othr Multivariabl Mthods. 4d. Thompson Highr Education Blmont CA USA. Mxia J.T.; Prira D.G.; Bata J L 2 nvironmntal indxs. Biomtrical Lttrs 36: Millr R.G Simultanous Statistical nfrnc. Springr Nw York NY USA. Ng M.P.; Williams E.R Joint-rgrssion analysis for incomplt two-way tabls. Australian & Nw Zaland Journal of Statistics 43: Prira D.G.; Mxia J.T Comparing doubl minimization and zig-zag algorithms in joint rgrssion analysis: th complt cas. Journal of Statistical Computation and Simulation 80: Prira D.G.; Rodrigus P.C.; Mjza S.; Mxia J.T A comparison btwn joint rgrssion analysis and AMM: a cas study with barly. Journal of Statistical Computation and Simulation 82: Rodrigus P.C.; Prira D.; Mxia J.T A comparison btwn JRA and AMM: th robustnss with incrasing amounts of missing data. Scintia Agricola 68: Romagosa.; van Euwijk F.A.; Thomas W.T.B Statistical analyss of gnotyp by nvironmnt data. v.3. n: Phohns J.; Nuz F.; Carna M.J. ds. Handbook of Plant Brding. Elsvir Nw York NY USA. Schffé H Th Analysis of Varianc. Wily Nw York NY USA. Williams E.J Rgrssion Analysis. Wily Nw York NY USA.