A Coffcnt of Dtrmnaton (R ) for Lnar Mxd Modls Hans-Ptr Ppho 1,* 1 Bostatstcs Unt, Insttut of Crop Scnc, Unvrsty of Hohnhm, 70 593 Stuttgart, Grmany *mal: ppho@un-hohnhm.d SUMMARY. Extnsons of lnar modls ar vry commonly usd n th analyss of bologcal data. Whras goodnss of ft masurs such as th coffcnt of dtrmnaton (R ) or th adjustd R ar wll stablshd for lnar modls, t s not obvous how such masurs should b dfnd for gnralzd lnar and mxd modls. Thr ar by now svral proposals but no consnsus has yt mrgd as to th bst unfd approach n ths sttngs. In partcular, t s an opn quston how to bst account for htroscdastcty and for covaranc among obsrvatons nducd by random ffcts. Ths papr proposs a nw approach that addrsss ths ssu and s unvrsally applcabl. It s xmplfd usng thr bologcal xampls. KEY WORDS: Gnralzd lnar modls; Gnralzd lnar mxd modls; Lnar mxd modls; Total varanc; Goodnss-of-ft; Smvarogram. 1. Introducton Th coffcnt of dtrmnaton, also dnotd as R, s prhaps th most popular masur of goodnss of ft for lnar modls (LM) (Drapr and Smth, 1998). It s dfnd as th proporton of th corrctd sum of squars that s xpland by th modl. LM can b 1
xtndd n two mportant ways. Th frst s to allow for random ffcts, ladng to lnar mxd modls (LMM) (Sarl t al., 199), and th scond s to allow for othr dstrbutons than th normal, ladng to gnralzd lnar modls (GLM) (Nldr and McCullagh, 1989). Both xtnsons can b combnd, amountng to gnralzd lnar mxd modls (GLMM) (McCulloch t al., 005; L t al., 006; Stroup, 013). Usrs of GLMM procdurs hav a kn ntrst n masurng goodnss of ft n a smlar fashon as avalabl for LM va R. Svral proposals hav bn mad for R masurs, som targtng GLM (Zhang, 017), othrs targtng LMM (Edwards t al., 008; Lu t al., 008; Dmdnko t al., 01), and yt othrs covrng GLMM (Nagakawa and Schlzth, 013; Jagr t al., 017; Nakagawa t al., 017; Stoffl t al., 017). Som of th proposals ar applcabl only to somwhat spcalzd sttngs and spcfc lnar prdctors (.g., wth ndpndnt random ffcts wth constant varanc). Although a lot of progrss has bn mad rcntly, thr dos not as yt sm to b a gnrally agrd procdur that s broadly applcabl to any GLM, LMM, or GLMM, ncludng modls wth complx varanc-covaranc structur such as thos ndd,.g., wth rpatd masurs (Jnnrch and Schluchtr, 1986) or spatal data (Zhang, 00). For an R masur for GLMM to fnd broad usag and b appalng to data analysts, t nds to hav a smpl dfnton n trms of varanc xpland that s asy to undrstand and communcat. Morovr, such a masur should rduc to th usual R n cas of LM. In th prsnc of random ffcts, th masur should also account for any covaranc and htrognty of varanc among obsrvatons. Th purpos of ths papr s to propos such a masur and llustrat ts proprts and usag. Th ky nw da compard to prvous proposals s that wth corrlatd data th total varanc s bst assssd basd on th varanc of parws dffrncs rathr than th margnal varanc of th obsrvd data, whch s commonly usd. Th papr s organzd as follows. To st th stag, w brfly rvst th dfnton of R for LM. In th nxt scton an R masur for LMM s proposd. Ths s
subsquntly xtndd to th varanc xpland by random ffcts and to GLMMs. Thr xampls ar usd to llustrat th mthod.. Th coffcnt of dtrmnaton for LM Th LM s gvn by y X, (1) whr y s a rspons vctor of lngth n, wth fxd ffcts vctor and assocatd dsgn matrx X, and N0, V ~ I n s th rsdual rror vctor. It s assumd that th fxd ffcts comprs a common ntrcpt. In ordr to assss th xplanatory powr of th rmanng ffcts n, all of ths ffcts ar droppd, yldng th null modl y 1 n, () whr 1 n s a vctor of ons and s th ntrcpt. Th rsdual rror undr ths null modl s dstrbutd as N0, V ~ 0 I n 0. Th coffcnt of dtrmnaton can b dfnd as V, V 0, (3) V0 whr V quantfs th total varanc mpld by th varanc-covaranc structur V, and V V V V, 0 0 s th varanc xpland by th ffcts addd n th full modl 3
rlatv to th null modl. For LM w us V,, and V V and hnc 0 0 V, 0 0. (4) 0 1 0 0 If th varanc componnts n (4) ar stmatd by maxmum lklhood (ML), w obtan th ordnary coffcnt of dtrmnaton, R, for LM. If th varancs ar stmatd by rsdual maxmum lklhood (REML), th adjustd R rsults. 3. Masurng total varanc n LMM Th LMM can b dfnd as (Sarl t al., 199) y X Zu, (5) whr y, and X ar as dfnd for LM, u s a random ffcts vctor u N0, G dsgn matrx Z and s a rsdual rror vctor N0, R assumd to b mutually ndpndnt such that y NX, V ~ wth ~. Random vctors u and ar ~ wth V ZGZ T R. (6) Our coffcnt of dtrmnaton for fxd ffcts n LMM wll hav th sam form as th on usd for LM (quaton 4), now usng th varanc-covaranc structur V for th full modl 4
n quaton (5) and V 0 for th null modl, n whch X n (5) s rplacd by 1. Th dfnton of V should allow for possbl htrognty of varanc,.., htrognty of th dagonal lmnts n V, and for covaranc btwn obsrvatons,.., non-zro offdagonal lmnts n V. Morovr, th dfnton should b such that th masur rducs to n th common R for LM whn random ffcts ar droppd and R I n. Fnally, w rqur th masur to b addtv,.., V V V, (7) 1 1 V bcaus ths allows componnts of xpland varanc to b dcomposd n a natural way. Two masurs of total varanc ar consdrd. Th frst s basd on th margnal varanc (mv). For an ndvdual rspons varabl y ths s gvn by mvy v, (8) whr v j s th j-th lmnt of V. Ths margnal varanc may b avragd across obsrvatons to yld th avrag margnal varanc (AMV) n 1 AMV V mvy 1 tracv. (9) n n 1 Th trac of a varanc-covaranc matrx s a common masur of total varanc n multvarat analyss. Th major downsd of ths crtron s that t dos not account for covarancs v j j (Mustonn, 1997; Johnson and Wchrn, 00, p.139). Thr ar svral altrnatv masurs of total varanc commonly usd n multvarat analyss that 5
allow accountng for covaranc, such as th dtrmnant of V, also known as gnralzd varanc n ths contxt (Wlks, 193; Goodman, 1968; Sbr, 1984; Stank, 1990; Mostonn, 1997), or ts standardzd form, gvn by th postv n-th root of th gnralzd varanc, V V 1/ n, whch s dnotd as standardzd gnralzd varanc (SnGupta, 1987) and rducs to for LM. Ths crtron s a nonlnar functon of th lmnts of V, howvr, and as such fals to mt th addtvty rqurmnt (7). Morovr, n dgnrat cass wth vry larg corrlatons th gnralzd varanc may bcom zro (Mostonn, 1997; Johnson and Wchrn, 00, p.130). Also, whn V s dagonal, dagonal lmnts, whras th arthmtc man, as obtand by n V 1/ s th gomtrc man of th AMV V, sms mor ntutvly appalng. In addton thr ar othr rlatd masurs of total or "xpland" varanc usd to dfn multvarat tst statstcs (Plla's trac, Hotllng-Lawly trac, Wlk's lambda, Roy's largst root; s,.g., Johnson and Wchrn, 00, p.139), but all of ths ar also nonlnar functons of th lmnts of th varanc-covaranc matrcs nvolvd. For ths rasons, w do not consdr such altrnatv masurs. Instad, w propos a scond masur that s also a lnar functon of th lmnts of V. Th man motvaton for ths nw masur s to account for th covaranc among obsrvatons n an asly ntrprtabl way. Rathr than focusng on th margnal varanc, w may consdr th varanc of a dffrnc btwn obsrvatons. Arguably, ths provds a natural way to account for covaranc. For two ndvdual rspons varabls y and y j j, th "smvaranc" (sv) s dfnd as half th varanc of y y : j sv 1 1 y y j vary y j v v jj vj,. (10) 6
Th smvaranc (or smvarogram) s a masur of varanc commonly usd for spatally corrlatd data (Isaacs and Srvastava, 1991). If thr s a postv covaranc v j smvaranc s rducd n comparson wth th avrag of th two margnal varancs v and v jj. Convrsly, f th covaranc s ngatv, th smvaranc s ncrasd compard to th avrag margnal varanc. Ths naturally accounts for th ntrplay btwn varancs v and covarancs v j n govrnng th ovrall varanc btwn obsrvatons. Th avrag smvaranc (), avragd across all pars of obsrvatons n th data vctor y, s th scond proposd masur of total varanc: j, th V n 1 n 1 y, y trac sv j VP, (11) n n 1 n 1 1 j1 whr P I n n 1 1 V whn V 1 T AMV n n. It s radly vrfd that V R I n (.., for an LM) as rqurd. 4. Total varanc xmplfd for a balancd on-way modl wth random group ffcts To llustrat and compar th proprts of AMV V and V and to s how covaranc s accountd for by th lattr, consdr th on-way random-ffcts modl for a groups and m obsrvatons pr group, gvn by I u y 1 1, (1) n a m 7
whr u N0, rprsnts a random group ffcts, and th rror N0, rprsnts ~ I a u ~ I n n am random dvatons from th rspctv group mans. Th varanc of y s V a m u m I J I. (13) Undr ths modl thr s a postv covaranc btwn obsrvatons wthn th sam group, whras obsrvatons from dffrnt groups ar ndpndnt. Th parws corrlaton among obsrvatons wthn th sam group, th ntra-class corrlaton, quals and approachs unty whn. For ths modl w fnd AMV V and u 1 V am a 1m. It may b argud that V 1 u u s th mor manngful masur of total varanc hr bcaus th margnal varanc of only occurs for u obsrvatons btwn groups but not wthn groups, whr th varanc amounts to only. Among th n n 1/ pars of obsrvatons, m 1/ hnc hav varanc sv, y y j am pars occur wthn groups and btwn thm. Th othr a 1m / btwn groups and hav varanc sv y y j, u y y j a pars occur btwn thm, whch s qual to mv,. Avragng ths varancs across all pars ylds V u, confrmng that ths s ndd a manngful masur of total varanc. Also not that n th xtrm cas of havng only on group 1 a, w stll fnd AMV V, although for all pars of obsrvatons th only rlvant varanc componnt s, whch s wll rflctd by th fact that V n ths lmtng cas. Ths xampl llustrats that V u not only accounts for th covarancs among obsrvatons but also for th dsgn of th study, gvn hr by th AMV numbr of groups, a, and th group sz, m. By contrast, V cannot account for ths structur, as t s focussd ntrly on th dagonal of V. u 8
1/ n m1 W not n passng that th standardzd gnralzd varanc V m 1/ m u (Sarl t al., 199, p. 443) s obvously nonlnar n th varanc paramtrs, hnc volatng th addtvty rqurmnt (7), and dos not hav th sam ntutv ntrprtaton as and V do. AMV V 5. A coffcnt of dtrmnaton for random ffcts n LMM W can also dfn a coffcnt of dtrmnaton for random ffcts u as V 0 ZGZ T u. (14) Th varanc xpland jontly by th fxd and random ffcts can b assssd by R V u u 1. (15) 0 Furthrmor, on account of th addtvty rqurmnt for V as pr quaton (7), w can comput partal coffcnts of dtrmnaton (Edwards t al., 008) by parttonng th random ffcts as Zu Z u Z u... and th assocatd varanc as 1 1 T T T T ZGZ Z G Z Z G Z... and thn usng Z G Z to assss th xpland varanc at 1 1 1 th lvl of ndvdual random ffcts u. 9
6. Extnson to GLMM A GLMM has th lnar prdctor X Zu (16) wth fxd and random ffcts as dfnd for LMM n quaton (5). Th obsrvaton vctor y s assumd to hav condtonal xpctaton 1 y g E, (17) whr g. s a lnk functon. It may b assumd that th condtonal dstrbuton of y gvn s from an xponntal famly, n whch cas th modl can b stmatd,.g., by full ML usng adaptv Gaussan quadratur (Pnhro and Bats, 1995) or by th Laplac mthod (Wolfngr, 1993). An altrnatv s to only mak an assumpton about th varanc functon var y and allow ovrdsprson rlatv to paramtrc dstrbutons n th xponntal famly. Such modls can b fttd by psudo-lklhood (Wolfngr and O'Connll, 1993) or pnalzd quas-lklhood (Brslow and Clayton, 1993). Anothr opton to allow for ovrdsprson s to nclud a random unt ffct among th random ffcts u on th lnar prdctor scal. Th rsdual varanc occurs on th obsrvd scal ("R-sd"), whch s not th lnar prdctor scal xcpt whn an dntty lnk s usd, whras th varanc du to random ffcts occurs on th lnar prdctor scal ("G-sd") (Stroup, 013). Ths maks t dffcult to 10
assss th total varanc. A furthr complcaton s that varanc on th obsrvd scal dpnds on th condtonal man va th lnar prdctor (16), whch n turn dpnds on th random ffcts and thus th ovrall varanc structur. Whn consdrng a null modl wth fxd ffcts rducd to 1 n, th condtonal man structur, gvn th random ffcts u, s altrd. In ordr to prsrv as much as possbl of that condtonal man structur so that rsdual varanc on th obsrvd scal s modlld proprly, t s suggstd hr to gnrally add a random unt ffct f wth zro man and var f R f n th lnar prdctor (Nakagawa and Schlzth, 013): X Zu f. (18) Th man purpos of th random unt ffct f s to captur any unxpland varanc on th lnar prdctor scal. Also not that th random unt ffct accounts for ovrdsprson, and ovrdsprson s clarly to b xpctd f mportant prdctors ar omttd from th fxd ffcts structur X. In ordr to prsrv addtvty, t s ncssary to assss th total varanc on th lnar prdctor scal. To ths nd, w furthr ntroduc an auxlary random rsdual vctor h T h 1, h,... and consdr th xtndd lnar prdctor ~ h and th condtonal varanc var var ~ 1 g, askng whch varanc-covaranc structur varh 1 g ~ vary R h lads to (Foully t al., 1987). Not that ths auxllary random ffct h s not to b confusd wth th random unt ffct f ncludd n th lnar prdctor (18). Th auxllary rsdual varanc varanc-covaranc matrx R h s thn usd along wth th unt varanc R f to dfn a 11
~ T ~ V ZGZ R (19) wth ~ R R f R h on th lnar prdctor scal, whch s usd n plac of V to assss th total varanc for GLMMs. For th spcal cas of a bnomal dstrbuton and a logt lnk, a logstc dstrbuton may b assumd for h, whch has varanc varh / 3 assum a standard normal dstrbuton for 1 1997). Ths rsults ar xact,.., var g. Smlarly, wth a probt lnk w may h, whch has varanc var 1 ~ vary h (Kn and Engl,. For othr dstrbutons and lnks, no such xact rsults ar avalabl, so w tak rcours to an approxmaton basd on a Taylor srs xpanson (Foully t al., 1987; Nakagawa t al., 017). It s assumd hr that var 1/ A 1/ RA y, (0) whr A s a dagonal matrx wth dagonal lmnts qual to valuatons of th varanc functon at th man and R s an unknown matrx. Ths varanc structur allows for ovrdsprson, and stmaton n cas of ovrdsprson rqurs psudo-lklhood mthods to b usd (Wolfngr and O'Connll, 1993). Not that th LMM s a spcal cas of ths modl wth dntty lnk and A I. For GLMMs wth condtonal rror dstrbutons n th xponntal famly w hav R In. It may b assumd that var 1/ h R W 1/ RW, (1) h 1
whr W s a dagonal matrx wth functons of th man on th dagonal. Expandng g 1 ~ n a Taylor srs about th man of th lnar prdctor, w fnd that to frst ordr var 1 g ~ D 1/ 1/ * W RW D *, () whr D 1 ~ ~ * dag g ~. Comparng coffcnts btwn (0) and () ylds A and hnc W D ~ W D ~ D 1 ~ A D 1 ~. Ths approach s an xtnson of a mthod proposd by Foully t al. (1987) for Posson data and by Bnnwtz t al. (014) for ovrdsprsd bnomal and Posson data. For xampl, wth ovrdsprsd Posson data wth 1 vary and log-lnk, w fnd var probt lnk w hav h 1 m h. For ovrdsprsd bnomal data wth var, whr m s th bnomal sampl sz of th -th obsrvaton, s th -th bnomal probablty and. s th standard normal probablty dnsty (Bnnwtz t al., 014). 7. Exampls Exampl 1: W consdr th data by Potthoff and Roy (1964) from an orthodontc study wth 11 grls and 16 boys. At ags 8, 10, 1 and 14 yars, th dstanc (mm) from th cntr of th ptutary to th ptrygomaxllary fssu was masurd for ach chld. Ths datast has also bn usd by othr authors to llustrat thr proposd R masurs (Zhng, 000; Edwards t al., 008; Orln and Edwards, 008). W hr ft th sam modls as n Edwards t al. (008), usng thr nomnclatur to dntfy modls so our rsults can b drctly compard to thr Tabl II. Th fxd-ffcts parts of th modl ar dnotd as Modls I to III. Modl I 13
comprss just an ntrcpt and a lnar ag ffct, Modl II has a gndr-spcfc ntrcpt n addton and Modl III has an ntracton of gndr and lnar ag addd compard to Modl II. Th data clarly has a rpatd-masurs varanc-covaranc structur that nds to b modlld. Th varanc-covaranc structurs consdrd by Edwards t al. (008) ar: 1 = random ntrcpt for ndvduals, = random ntrcpt and slop wth unstructurd varanc-covaranc matrx, and 3 = wth htrognty of rsdual varanc btwn grls and boys. Th rsdual ffcts n ths modls ar assumd to b ndpndnt wth th sam varanc at ach tm pont. For comparson, w addd th followng varanc-covaranc structurs: 0 = ndpndnt rsdual wth constant varanc (corrspondng to an LM), 4 = unstructurd rsdual varanc-covaranc, 5 = unstructurd rsdual varanc-covaranc wth htrognty btwn grls and boys. Rsults ar shown n Tabl 1 for th Akak Informaton Crtron (AIC) basd on REML and stmats of basd on both REML and ML. Covaranc modl 3 fttd bst accordng to AIC wth all thr fxd-ffcts modls (I, II, and III). For all varanc-covaranc modls, th stmats of ndcat that ncluson of a gndr man ffct s mportant, whras th ntracton of gndr wth lnar tm provds only margnal addtonal mprovmnt. Th stmats of for both ML and REML ar rathr lowr than th R valus rportd by Edwards t al. (008; Tabl II) basd on Waldtyp F-statstcs wth dffrnt approxmatons of th dnomnator dgrs of frdom. Comparson wth th stmats of for covaranc modl 0 for ML and REML, corrspondng to ordnary and adjustd R for LM, rspctvly, suggsts that th R -valus n Edwards t al. (008) may ovrstat th goodnss of ft. Estmats of largr than thos of AMV tnd to b slghtly, so accountng for th covaranc among obsrvatons dos mak a dffrnc, albt not a vry larg on n ths xampl. - Tabl 1 about hr - 14
Exampl : Nakagawa and Schlzth (013) consdr data on btl larva sampld from 1 populatons. Wthn ach populaton, larva wr obtand from two mcrohabtats, subjctd to two dffrnt dtary tratmnts, and dstngushd as mal and fmal. Sxd pupa wr rard n contanrs, ach holdng ght anmals from th sam populaton. Thr ar thr rsponss: () body lngth (Gaussan dstrbuton), frquncy of two mal colour morphs (bnary dstrbuton), and () th numbr of ggs lad by ach fmal (Posson dstrbuton). Th modls fttd by Nakagawa and Schlzth (013) had ndpndnt homoscdastc random ffcts for populaton and contanr and fxd ffcts for tratmnt, habtat. For body lngth, th modl also comprsd a fxd man ffct for sx. Ths modls wr fttd hr usng REML for body lngth and th Laplac mthod (Wolfngr, 1993) for th morph data and gg counts. For morph frquncy w assumd a logt lnk and stmatd th coffcnt of dtrmnaton accountng for bnary varanc on th lnar prdctor scal by sttng var / 3 h and cov, 0 h j, th varanc of th logstc dstrbuton. W h j stmatd th prdctd Posson man xp basd on th lnar prdctor (18) and, 1 followng Foully t al. (1987), st var h and cov, 0 h j data and gg counts, w fttd ndpndnt homoscdastc random ffct for unts ( R f I n f h j. For both morph ) on th lnar prdctor scal n ordr to captur any unxpland varanc not takn up by th othr random ffcts. Varanc stmats for all trats agr wth thos of Nakagawa and Schlzth (013) to thr dcmal placs (Tabl ). For body lngth, most of th varanc s xpland by random ffcts. Th stmat of s rlatvly clos to th condtonal R ( R ) of 74% u GLMM (c) rportd by Nakagawa and Schlzth (013), and thr margnal R ( R GLMM (m) ) of 39% s 15
clos to th stmat of. For th bnary morph data, th varancs for populaton and contanr ar slghtly largr for th full modl than th modl wth fxd ffcts droppd, suggstng that fxd ffcts hav no xplanatory powr for ths trat. Consquntly, s stmatd to b nar zro. By way of comparson, Nakagawa and Schlzth (013) rport a margnal R for th fxd ffcts of 8%, whch s somwhat unxpctd gvn th smallr stmatd varancs undr th null modl. Ths lkly rsults from th fact that thy only us th ft of th full modl to comput R, usng th fxd-ffcts stmats thmslvs to stmat th varanc xpland by ths. By contrast, our approach uss both th fts of th full and null modls and asssss xpland varanc basd on stmats of V and V 0. Th gg counts hav largr stmatd coffcnts of dtrmnaton (Tabl 3), but agan ths ar smallr than thos rportd n Nakagawa and Schlzth (013). - Tabl about hr - Exampl 3: Zhang (017) usd data from a study of nstng horssho crabs (Agrst, 1996) to llustrat dffrnt dfntons of R for GLM. A total of 173 crabs wr assssd for colors (C), spn condtons (SC), carapac wdth (CW), and wght (W), ach wth a mal crab attachd to hr n hr nst. To nvstgat th ffct of ths factors on th numbr of satllts,.., any othr mals rdng nar a fmal crab, a GLM wth log-lnk and Posson dstrbuton was fttd by Zhang (017). Hr, w ft a GLMM wth ndpndnt homoscadastc random ffct for unts ( R f I n ) n th lnar prdctor so that th unxpland varanc undr th rducd modls can b capturd by that ffct and thus allocatd to th modl's varanc. Th modls ar fttd by both th Laplac mthod (Wolfngr, 1993) and adaptv Gaussan quadratur (Pnhro and Bats, 1995). Th 1 auxllary varanc was st to var h and cov, 0 f h j h j as n Exampl. W 16
fttd th sam fxd ffcts modls as Zhang (017), so our rsults may b drctly compard to altrnatv R masurs n Tabl 1 of that papr. Bcaus n ths cas V ~ s dagonal, and AMV concd. On avrag, stmats of n Tabl 3 ar slghtly largr than th R masurs proposd by Zhang (017), but gv a smlar rankng among modls. W also fttd a GLM by psudo-lklhood and rsdual psudo-lklhood (Wolfngr and O'Connll, 1993), usng a Taylor srs xpanson around (18) (Tabl 4). Coffcnts of dtrmnaton basd on psudo lklhood ar slghtly lowr n magntud compard to thos n Tabl 3. Rsdual psudo-lklhood (akn to REML) ylds smallr coffcnts of dtrmnaton than psudo-lklhood (akn to ML) as xpctd. Th rsdual psudolklhood mthod pcks th modl wth th sngl covarat W as th bst, whras th othr mthods, all of whch us ML, slct mor complx modls. Ths outcom llustrats that us of REML-lk mthods for stmatng varanc componnts n GLMM lads to coffcnts of dtrmnaton that bhav lk adjustd R for LM. Zhang (017) also found th sam bst modl usng an adjustd R masur for GLM. - Tabls 3 and 4 about hr - 8. Dscusson Thr ar svral proposals of R masurs for GLM, LMM and GLMM n th ltratur, and most of thm shar th dsrabl proprty to rduc to th common R n cas of LM (Camron and Wndmjr, 1996). Ths may b consdrd a ncssary but not a suffcnt condton, howvr. Thr ar svral proposals, mostly targtng GLM, that ar functons of 17
th maxmzd lklhood of quas lklhood undr th full and null modls, most notabl among thm th lklhood rato statstc (Maddala, 1983; Cox and Snll, 1983; Mag, 1990; Naglkrk, 1991; Zhng, 000) and th Kullback-Lblr dvrgnc (Camron and Wndmjr, 1997). Whl ths masurs crtanly hav thr mrts, thy ar mor dffcult to communcat to rsarch scntsts prmarly famlar wth ordnary last squars for LM and th concpt of "xpland varanc". Thr ar svral masurs, manly proposd for LMM, that mak us of quadratc forms of y. For xampl, Bus (1973), Kramr (005) and Dmdnko t al. (01) assss th unxpland varanc for fxd ffcts basd on th wghtd rsdual sum of squars, SS W T 1 y X V y X. Whl ths masur dos account for covarancs among obsrvatons, t dos so va 1 V rathr than th varanc V, whch may b dffcult to xplan to non-statstcans. Also, t s not mmdatly obvous how xactly SS W s rlatd to th varanc of th data V. Hr, I am not rfrng to th mathmatcal rlatonshp, whch s obvous, but for an ntutv xplanaton that s asly graspd by a rsarch scntst. Fnally, th approach only works for LMM but not for GLMM. Edwards t al. (008) and Jagr t al. (017) proposd R masurs that ar motvatd by th fact that for LM th F-statstc for comparng th full and rducd modls can b wrttn as a functon of R and vc vrsa. Ths fact s usd to dfn R masurs for LMM and GLMM by analogy basd on th Wald-typ F-statstc for th sam typ of comparson. A practcal dffculty wth ths approach s that rsdual dgrs of frdom nd to b dtrmnd, and thr ar svral approxmatons n us, ladng to dffrnt valus of R [s Tabl II n Edwards t al. (008) for a lucd xampl]. Also, Wald-typ F-statstcs nvolv 1 V, as dos SS W. A furthr complcaton s that dffrnt mxd modl packags comput F-statstcs 18
dffrntly, so wth complx varanc-covaranc modls thr may b dffrncs btwn packags (Ppho and Edmondson, 018). Furthrmor, whras th R for LM can ndd b wrttn as a functon of an F-statstc, that rprsntaton dos not lnd tslf so wll to communcat th ntrprtaton n trms of xpland varanc. For that purpos t s prfrabl to xprss th R as n quaton (3) or n trms of sums of squars. But such an analogous rprsntaton dos not sm to b forthcomng whn a Wald-typ F-statstc s usd to dfn R for LMM and GLMM. Nxt, thr ar svral proposals for R masurs that xclusvly oprat on unwghtd sums of squars of dvatons btwn obsrvd data and fttd valus (.g., Vonsh t al., 1996; Lu t al., 008) or th unwghtd sum of squars of fttd valus for X (Nakagawa and Schlzth, 013; Nakagawa t al., 017). Ths masurs do not mak xplct us of V, othr than n th gnralzd last squars stmator of. Hnc, t may b sad that ths masurs do not account for covarancs among obsrvatons or htrognty of varanc. Ths papr has proposd a nw mthod to assss th coffcnt of dtrmnaton that s gnrally applcabl to any GLM, LMM or GLMM, rgardlss of th varanc-covaranc structur usd, and rducs to R and adjustd R for LM. Th proposd coffcnt of dtrmnaton s assssd on th lnar prdctor scal, allowng an addtv dcomposton of total varanc n cas of random ffcts, whch would not b forthcomng wth approachs that targt varanc on th obsrvd scal, such as th rcnt proposal by Zhang (017) for GLM. Whn varanc componnts ar stmatd by ML (approxmat or xact), th coffcnt of dtrmnaton bhavs lk th ordnary R for LM, whras wth REML-lk mthods of stmaton (Ppho t al., 018), th coffcnt of dtrmnaton works lk th adjustd R for LM. Two masurs of total varanc wr consdrd that concd for ndpndnt data. Th frst, AMV V, uss only th margnal varanc,.., th dagonal 19
lmnts of V. Smlar approachs hav bn usd by many authors proposng R masurs for LMM and GLMM, prmarly whn G has a smpl varanc-componnts structur (Snjdrs and Boskr, 1999; Nakagawa and Schlzth, 013; Nakagawa t al., 017). A major lmtaton of ths masur, howvr, s that t cannot account for th covaranc among obsrvatons. A ky fatur of th approach basd on th scond masur, V, whch s novl, s that covaranc among obsrvatons s takn nto account by assssng total varanc n trms f th man varanc of a dffrnc among obsrvatons. Ths da has roots n gostatstcs whr smvarograms ar usd routnly to dscrb spatal varanc and covaranc (Isaacs and Srvastava, 1991), and t also bars som rlaton to xprmntal dsgn, whr ffcncy may b assssd n trms of th avrag parws varanc of a dffrnc among tratmnts (Buno Flho and Glmour, 003; Wllams and Ppho, 015), and to mthods usd for th stmaton of hrtablty n plant brdng xprmnts. Not that th hrtablty has an ntrprtaton of R for th rgrsson of phnotyps on gnotyps (Falconr and Mackay, 1996). In fact, whn y s a random vctor of adjustd gnotyp mans basd on an analyss of an ndvdual xprmnt or of a srs of xprmnts, R s th assocatd varanc-covaranc matrx of adjustd mans, I n u G s th gntc varanccovaranc matrx, and X 1n, thn u s quvalnt to th broad-sns hrtablty dfnd n q. (19) n (Ppho and Möhrng, 007). All analyss wr mplmntd usng th MIXED and GLIMMIX procdurs of SAS. Ths procdurs wr usd to ft th full and null LMM, producng stmats of V, V 0 and R. For GLMM, output from GLIMMIX was post-procssd to obtan stmats of V ~ ~, V 0 and R ~. Ths outputs wr thn submttd to a macro that computs stmats of, Th full cod for th thr xampls s provdd wth th Supportng Informaton. u and u. 0
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Tabl 1 AMV Coffcnts of dtmnaton (, ) of LMM for rpatd-masurs dntal data (Exampl 1). Varanccovaranc modl Fxd ffcts modl $ AIC (REML) REML AMV ML AMV 0 I 511 0.49 & 0.49 & 0.56 # 0.56 # II 487 0.398 & 0.398 & 0.410 # 0.410 # III 486 0.406 & 0.406 & 0.43 # 0.43 # 1 I 451 0.54 0.49 0.6 0.57 II 44 0.391 0.388 0.41 0.410 III 438 0.40 0.399 0.46 0.43 I 451 0.353 0.349 0.363 0.360 II 443 0.471 0.469 0.49 0.491 III 441 0.481 0.479 0.505 0.503 3 I 43 (bst) 0.330 0.37 0.339 0.336 II 45 (bst) 0.448 0.446 0.468 0.467 III 4 (bst) 0.463 0.461 0.486 0.484 4 I 455 0.383 0.381 0.393 0.391 II 449 0.495 0.494 0.516 0.515 III 445 0.506 0.505 0.58 0.57 5 I 45 0.39 0.34 0.41 0.35 II 446 0.385 0.38 0.406 0.403 III 441 0.401 0.398 0.49 0.46 0 = ndpndnt rsdual wth constant varanc (corrspondng to an LM), 1 = random ntrcpt for ndvduals, = random ntrcpt and slop wth unstructurd varanc-covaranc matrx, 3 = wth htrognty of rsdual varanc btwn grls and boys, 4 = unstructurd rsdual varanc-covaranc, 5 = 4 = unstructurd rsdual varanc-covaranc wth htrognty btwn grls and boys. $ I = lnar ag ffct only, II = lnar ag ffct and gndr man ffct, III = II wth lnar ag-by-gndr ntracton addd & Equvalnt to R for LM # Equvalnt to adjustd R for LM 6
Tabl Coffcnts of dtrmnaton (, u, u ) and varanc componnt stmats (obtand by Laplac mthod) for btl data (Exampl ). Paramtr Body lngth (Gaussan) $ Trats Morph data (bnary) $ Egg count (Posson) $ Varanc componnts & Null modl Full modl Null modl Full modl Null modl Full modl Populaton 1.181 1.379 0.946 1.110 0.303 0.304 Contanr.06 0.35 0.000 0.006 0.01 0.03 Unts 1.4 1.197 0.000 0.000 0.171 0.100 Coffcnt of AMV AMV AMV dtrmnaton 0.4009 0.3906 0.0377 0.040 0.009 0.0091 u 0.3330 0.3498 0.467 0.635 0.0474 0.051 0.7339 0.7405 0.089 0.33 0.0566 0.0603 u All fxd ffcts droppd from full modl. & All stmats agr closly wth thos rportd n Nakagawa and Schlzth (013) up to th thrd dcmal plac. $ An LMM was fttd to th Gaussan data, whras GLMM wr fttd to th bnary and Posson data wth logt-lnk and log-lnk, rspctvly. For th bnary modl, th var / cov, j. For th auxllary varanc was st to h 3 and h h j 0 1 Posson modl th auxllary varanc was st to var cov j. h and, 0 h h j 7
Tabl 3 Coffcnt of dtrmnaton and stmats of unt varanc f componnt for GLMM fttd to crab count data (Posson GLMM wth log-lnk) by th Laplac mthod and by Gaussan quadratur (Exampl 3). Covarat Estmaton mthod modl $ Laplac Gaussan quadratur AMV Unt AMV Unt varanc varanc (C,SC,CW,W) 0.03 0.96 0.1 0.98 (C,CW,W) 0.01 0.96 0.10 0.99 (C,CW) 0.181 0.99 0.190 1.01 (C,W) 0.06 0.96 0.13 0.98 (CW,W) 0.184 0.98 0.19 1.01 C 0.055 1.14 0.059 1.18 SC 0.03 1.17 0.03 1.1 CW 0.16 1.01 0.170 1.04 W 0.186 0.98 0.193 1.01 1.1 1.5 Total varanc was assssd on th lnar prdctor scal basd on R ~. Th lnar prdctor had a random unt ffct to captur unxpland varanc. Th auxllary 1 varanc was st to varh and covh, h j 0 j. $ C = colour, SC = spn condton, CW = carapac wdth, W = wght 8
Tabl 4 Coffcnt of dtrmnaton and stmats of unt varanc f for GLMM fttd to crab count data (Posson GLMM wth log-lnk) by th (rsdual) psudo-lklhood mthod (Exampl 3). Covarat modl $ Estmaton mthod Rsdual psudolklhood Psudo-lklhood & AMV Unt AMV Unt varanc varanc (C,SC,CW,W) 0.097 0.83 0.175 0.75 (C,CW,W) 0.118 0.81 0.173 0.75 (C,CW) 0.111 0.8 0.156 0.77 (C,W) 0.133 0.80 0.177 0.75 (CW,W) 0.136 0.79 0.158 0.76 C 0.010 0.91 0.046 0.86 SC 0.003 0.9 0.07 0.88 CW 0.18 0.80 0.139 0.78 W 0.148 0.78 0.160 0.76 0.9 0.91 Total varanc was assssd on th lnar prdctor scal basd on R ~. Th lnar prdctor had a random unt ffct to captur unxpland varanc. Th auxllary 1 varanc was st to varh and covh, h j 0 j. $ C = colour, SC = spn condton, CW = carapac wdth, W = wght Taylor srs xpanson around X Zu f Opton RSPL n GLIMMIX procdur of SAS & Opton MSPL n GLIMMIX procdur of SAS 9