Page 1. Before-After Control-Impact (BACI) Power Analysis For Several Related Populations (With Unknown Variance Matrix) Richard A.

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1 Pag Bfor-Afr Corol-Impac (BACI) Powr Aalysis For Svral Rlad Populaios (Wih Ukow Variac Marix) Richard A. Hirichs Spmbr 0, 00 Cava: This xprimal dsig ool is a idalizd powr aalysis buil upo svral simplifyig assumpios (Tabl ). For a spcific xprim, a mor accura porrayal of powr may rquir chagig hs assumpios ad h udrlyig quaios. This aalysis should b rad as a rough guid o powr. Iroducio Currly hr ar may warshd projcs udrway i h Columbia Basi o drmi h survival ffcs of various maagm acios o salmo survival. For xampl, hr ar a sris of isivly moiord warshds (IMWs) big sablishd for h purpos of br udrsadig how salmo rspod o diffr approachs o rsor habia. Wh hs projcs ar ru as xprims, i is possibl o idify h ffcivss of rsoraio ad ohr maagm acios. Th powr aalysis prsd hr was moivad by h d o dsig hs xprims i such a way ha hy hav a good chac o dc sigifica survival chags i salmo wh hy occur. Usig his ool ca giv a xprimr a rough ida of h umbr of yars o ru a xprim wha saisical powr ca b xpcd basd o diffr assumpios abou variac-covariac srucurs for survival amog h salmo populaios sudid. Th framwork for h aalysis giv hr, alhough dvlopd wih salmo i mid, fis io h framwork of h Bfor-Afr-Corol-Impac (BACI) xprim. Such BACI-yp xprims fid applicaio byod Columbia Rivr salmo survival (Osbrg ad Schmi 996). A a priori powr aalysis is dvlopd for a BACI-yp xprim aimd a simaig a commo chag i survival for svral populaios. Th xprim icluds a Bfor priod whr all populaios rciv o ram followd by a Afr

2 Pag priod whr oly h ram populaios rciv ram. This is a gralizaio of h BACI-yp xprim whr h corol populaio ad impac populaio ar sampld o im bfor ad o im afr h ram (Gr 979, Osbrg ad Schmi 996). Th assumpios for h aalysis ar giv i Tabl. I is assumd ha i h absc of ram, all populaios hav a commo ma log survival. Bcaus his assumpio ad ohrs may o hold i pracic, his aalysis should b rad as a rough guid. Usually, xprims ar dsigd so ha hy achiv a powr of 0.80 or grar. Th mai goal of his work is o dmosra h probabiliy of dcig a ffc o survival wh svral rlad populaios wih a commo ma survival ar usd i a BACI-yp xprim ad h variac marix mus b simad. This goal is accomplishd by dscribig h xprim i a saisically rigorous way, sig up h liklihood fucio, dvlopig maximum liklihood simaors, h usig Mo Carlo simulaio o sima powr. Powr is h probabiliy of rjcig h ull hypohsis of o ram ffc. Bcaus horical formulas wr o availabl for h sima of h ram ffc wh variac was ukow, Mo Carlo simulaios wr usd o sima powr isad of a formula. A A A3 A4 A5 A6 A7 A8 Tabl. Assumpios usd i h powr aalysis. Th obsrvaios of log(survival) follow a mulivaria ormal disribuio. Thr is o srial dpdc i log(survival). All populaios hav a commo ma log(survival) bfor ram. Afr ram, h corol populaios coiu o hav h sam commo ma as xhibid i h Bfor yars, ad h ram populaios also hav a commo ma, bu shifd by a cosa amou (ram ffc) ha is h sam for all ram populaios. Th masurm rrors i log(survival) follow a mulivaria ormal disribuio ad h rrors ar idpd of h rror du o acual yar-o-yar viromal variabiliy. Th simaor of h ram ffc is a maximum liklihood sima. Th variac-covariac marix rprsig h rror i log(survival) is o kow ad mus b simad. Th variac-covariac marix aks h form of a iraclass covariac marix wih qual variacs ad qual covariacs. Ths ar assumpios for a idalizd xprim. For a spcific applicaio, a mor accura xprimal dsig may rquir chagig hs assumpios ad h udrlyig quaios. Thrfor his aalysis should b rad as a rough guid o powr. Th wbsi coais a wb-basd ool ha implms his powr aalysis wih h addd assumpios ha, i h daa graig modl, variacs i log(survival) ar qual for all populaios ad h corrlaios i log(survival) ar qual for ach populaio pair. This is h iraclass covariac srucur sudid by R.A. Fishr (95). Th cod for implmig his powr aalysis,

3 Pag 3 which may b foud i Appdix A, was implmd i R, a sysm for saisical compuaio ad graphics (Vrabls al. 00). Mhods To coduc h powr aalysis, a modl was formulad ad maximum liklihood simaors wr drivd (Mood al. 974). Ths simaors wr h usd as h basis for sig h ull hypohsis of o ram ffc usig Mo Carlo simulaio. Powr was h calculad as h probabiliy ha h ull hypohsis is rjcd. Th modl. I was assumd ha ma log(survival) bfor ram was h sam for ach populaio ad qual o. Afr ram, h ma log(survival) of h ram populaios shifs by h amou for h ram populaios whil h corol populaios coiu o hav a ma log(survival) of. I was also assumd ha yar-o-yar variabiliy i log(survival) ad masurm rror followd a mulivaria ormal disribuio wih variac Σ Σ y Σm, whr Σ y is h variaccovariac marix ha dscribs yar-o-yar variabiliy i h absc of masurm rror, ad Σm rprss h variac-covariac marix of h masurm rror. I his implmaio of h BACI xprim, h variac-covariac marix, Σ, was assumd o b ukow so ha i mus b simad alog wih ohr wo modl paramrs ( ad ). Maximum liklihood simaors. To driv maximum liklihood simaors, a xprssio for h liklihood fucio is dd. For h modl dscribd abov, h logliklihood fucio is l ( θ, Σ) C ( / )l Σ (/ ) ( x 0θ) Σ ( x 0 θ)) () θ) Σ ( x ) ( / ) ( x θ ; whr θ ; Σ is h ukow variac-covariac marix wih qual variacs ad qual covariacs; C is a cosa ha dos o dpd o h paramrs; is h umbr of yars prior o ram; is h oal umbr of yars of h xprim; x is a k-vcor of obsrvd survivals i yar ; k is h umbr of populaios (ram + corol) usd i h xprim; is a k-vcor of s; is a k -vcor of k 0s followd by k s, whr k rprss h umbr of corol populaios ad k rprss h

4 Pag 4 umbr of ram populaios. Th vcor x is arragd so ha h k corol populaios prcd h k ram populaios. Maximum liklihood simas for, ad Σ ar sough. Wh Σ is simad all populaios hav h sam variac ad all pairs of populaios hav h sam covariac. This givs ris o h iraclass covariac marix srucur sudid by Fishr (95) whr all diagoal ris ar qual ad all off-diagoal ris ar qual. Usig maximum liklihood hory, simaig quaios for h Bfor ma, ram ffc, ad h covariac marix ar dvlopd. I h cas of h bfor ma ad ram ffc, maximizig h liklihood fucio is quival o a solvig h gralizd las squars problm of miimizig x 0 Σ 0 x 0 SS ; x 0 Σ x () whr x rprss h k-vcor of sampl mas of log(survival) i h Bfor priod, ad x rprss h k-vcor of sampl mas of log(survival) i h Afr priod. This gralizd sum of squars may b wri i h familiar form SS ; y Bθ Ω y Bθ (3) 0 Σ 0 whr y x x, B, ad Ω. I his form, h 0 Σ gralizd las squars soluio, h calld h Aik simaor (Prss 005), is kow o b

5 Pag 5 y Ω B B Ω B θ T T ˆ. (4) Afr cosidrabl marix algbra, w may wri ) ( ) ( ˆ ˆ ˆ Σ x x Σ x Σ x θ μ ; (5) whr x is a k-vcor rprsig populaio-spcific sampl mas ovr h ir duraio of h xprim. Also wll kow is h codiioal variac of h sima of θ (giv ha h variac-covariac marix): ˆ var Σ Σ B Ω B θ T. (6) Nx, h simaig quaio for h variac covariac marix is drivd. To do his, h parial drivaivs of h liklihood fucio ar calculad wih rspc o h ivrs of h covariac marix ( Σ ), ad s o zro. Th variac marix has h form of a iraclass covariac marix. I his cas, h ivrs of h variac-covariac marix also has a iraclass covariac marix srucur ad may b wri as I Σ b b (a ). (7)

6 Pag 6 No ha all of h diagoal ris of h ivrs covariac marix ar qual o h scalar quaiy a, ad all of h off-diagoal ris ar qual o h scalar quaiy b. I his spcial cas, h log-liklihood fucio may b wri as l( Σ ) C ( / )l Σ (/ ) z Σ z. (8) whr z x 0θ wh ad z x θ wh. Th followig formulas ar usd for calculaig h parial drivaivs of his logliklihood fucio l Σ a k l Σ ad k( k ), b (9) whr is h commo variac rm i h variac marix ad is h commo covariac valu. Also usd ar h formulas zσ a z zz ad zσ z (0) zz zz b Armd wih hs quaios, i is show ha l( Σ a ) ( / ) k (/ ) zz () ad

7 Pag 7 l( Σ b ) ( / ) k( k ) (/ ) z z zz. () Sig hs wo parial drivaivs qual o zro yilds h simaig quaios ˆ zz k (3) ad z z z z ˆ. k( k ) (4) Th maximum liklihood simaor for h iraclass covariac marix is hrfor qual o Σˆ ( ˆ ˆ ) I. (5) ˆ MLE umrical algorihm. Armd wih h simaig quaios a algorihm o solv hm for h MLEs is ow drivd. A iraiv procdur is usd bcaus h maximum liklihood sima of θ dpds o h maximum liklihood sima of Σ i quaio (5). Th algorihm is basd o a procdur calld iraivly rwighd las squars or IRLS, which is a spcial cas of iraiv simaig quaios (IEE), wih kow covrgc propris (Jiag al. 007). I pracic, h mhod will covrg quickly if h umbr of obsrvaios is sufficily grar ha h umbr of simad paramrs. Th oal umbr of simad modl paramrs is always 4: a corol ma, a ram ffc, a commo variac, ad a commo covariac paramr.

8 Pag 8 Th iraiv procdur bgis by sig h iiial sima of h variac (0) marix, call i ˆΣ, qual o h idiy marix. Th x sp is o mak a iiial (0) sima of h θ vcor. This is accomplishd by usig ˆΣ i plac of Σ i quaio (6) (0) (0) ad solvig for ˆθ. Th sima ˆθ is h usd i quaios (3)-(5) o g a () updad sima of h variac marix, ˆΣ. This ir procdur is rpad wih h mos rc updas of h paramrs uil h liklihood fucio fails o dcras by som spcifid olrac. Saisical powr calculaios. Saisical powr is simad wih a Mo Carlo procdur. Nomial valus of h modl paramrs ar spcifid, ad h Mo Carlo rplicaios of h maximum liklihood simaors ar cosrucd. Th s saisic usd i his powr aalysis is Tˆ ˆ / s( ˆ), (6) whr s (ˆ ) is h squar roo of h variac of ˆ obaid by subsiuig Σˆ for Σ i quaio (6). This saisic is kow o hav a sud s -disribuio i h cas of a liar rgrssio wih uiform variac ad ucorrlad rrors, which is a spcial cas of h modl cosidrd hr. A horical disribuio for h s saisic was o assumd. Isad, Mo Carlo simulaio was usd o driv a sima of is disribuio udr h ull hypohsis of o ram ffc. This was h usd o sima powr. A alraiv o his s saisic would b basd o h liklihood raio (Mood al. 974), which would b compard o a chi-squar disribuio wih o dgr of frdom. Saisical powr was simad by h followig 4-sp procdur: Sp. Firs obai Nsim rplicaios of h s saisic (dfid i quaio 6) wh h ru ram ffc is. Ths rplicaios ar xprssd as Tˆ, T,, T Nsim,whr ˆ ˆ / ( ˆ Ti i s i ) whr i is h ih rplicaio of h ram ffc sima, ad ( ˆ s i ) is calculad by subsiuig Σ ˆ i for Σ i quaio (6). Sp. Us hs rplicaios of h s saisic o build a s of rplicaios of h s saisic udr h ull hypohsis. To do his, dfi h rplicaios udr h ull hypohsis as ˆ T / s( ˆ ) i i

9 Pag 9 Sp 3. Assumig ha h probabiliy of a Typ I rror is s o h valu, h criical valu is h simad as h avrag of h valus q a / ad q, a / whr qa / rprss h / quail of h rplicaios of h ram ffc sima i Sp ad q rprss h / quail. a / Sp 4. Powr is simad as h fracio of h Mo Carlo rplicaios of h s saisic (grad i Sp ) whos absolu valu xds h criical valu calculad i Sp 3. I Sp, a shor cu was usd: Nsim rplicaios from h ull disribuio of h s saisic wr grad by subracig ˆ / s( i ) from rplicaios grad i Sp. This shorcu was possibl bcaus if x is a radom sampl of log survivals durig a sigl yar from h Afr priod i Sp wih ru ram ffc of, h x is a radom sampl durig a sigl yar of h Afr priod wh h ull hypohsis is assumd (ru ram ffc of zro). I ohr words, h saisic ˆ is a locaio saisic, for which icrasig log(survival) of rad populaios durig h Afr priod by icrass h saisic ˆ islf by. Wh a locaio saisic is usd, h -lik saisic i quaio (6) is kow o b a appropria s saisic (Efro ad Tibshirai 993). Th sadard rror of ˆ was simad as h squar roo of h sampl variac of h Nsim rplicaios of ˆ : Nsim s ( ˆ) ( ˆ ˆ )/( Nsim ). i i (7) whr ˆ was h sampl ma of h Nsim rplicaios of ˆ. Th Mo Carlo sima of h coffici of variaio was simad as CV ( ˆ) s ( ˆ) /. (8) Exampl. Cosidr h cas whr h umbr of ram populaios quals h umbr of corol populaios ( k k ); h umbr of bfor yars quals h umbr of afr yars 0or, alraivly, 5; h graig modl

10 Pag 0 uss a commo variac of.0 ad a commo covariac of 0.5; masurm rror is zro; ad h probabiliy of a yp I rror is L h ram ffc vary from 0 o log() ad compar h powr obaid wh h variac marix is kow o h powr obaid wh h variac marix is simad. Th rsuls of his xrcis may b foud i Tabl. Noic ha h powr calculaios i h cas wh variac is rad as kow ad i h cas wh i is rad as ukow ar clos, v wh h oal yars of xprimaio dcrass from =0 o =0. Noic also how powr dclis as h oal umbr of yars of xprimaio dclis.

11 Pag Tabl. Saisical powr udr wo diffr simaio assumpios: ukow variac, variac ukow ad havig iraclass covariac marix srucur. I his xrcis, k k ad simulaios ar ru wih variac of.0 ad a corrlaio of 0.5 ad a masurm rror of zro. Th probabiliy of a yp I rror was s o Th ru ram ffc () varis from 0 o log(). A oal of 0,000 Mo Carlo simulaios wr usd wh was simad. = =0 = =5 dla kow simad kow simad

12 Pag Ackowldgms This work was suppord by Bovill Powr Admiisraio corac # Thaks o Charli Pauls, Rishi Sharma, ad Tracy Hillma for hir valuabl rviws. Thaks o Bria Maschhoff for implmig his aalysis as a wb ool a

13 Pag 3 Rfrcs Dwyr, P.S Applicaios of marix drivaivs i mulivaria aalysis. Joural of h Amrica Saisical Associaio 6: Efro, B. ad R.J. Tibshirai A iroducio o h boosrap. Moographs o Saisics ad Applid Probabiliy 57. Chapma & Hall/CRC, Nw York, Nw York. Fishr, R.A. 95. Saisical Mhods for Rsarch Workrs. Olivr ad Boyd, Ediburgh, Scolad. Gr, R.H Samplig dsig ad saisical mhods for viromal biologiss. Wily ad Sos, Nw York, Nw York. Jiag, J. Lua, Y., ad Y. Wag Iraiv simaig quaios: liar covrgc ad asympoic propris. Th Aals of Saisics 35: Mood, A.M, Graybill, F.A., ad D.C. Bos Iroducio o h hory of saisics, Third Ediio. McGraw-Hill, Nw York, Nw York. Osbrg, C.W. ad R.J. Schmid Dcig cological impacs causd by huma aciviis. I Dcig Ecological Impacs: Cocps ad Applicaios i Coasal Habias, R.J Schmi ad C.W. Osbrg, Ediors. Acadmic Prss, Nw York, Nw York. Vrabls, W.N., Smih, D.M., ad R Dvlopm Cor Tam. 00. A Iroducio o R. Nos o R: A Programmig Evirom for Daa Aalysis ad Graphics Vrsio.. ( ). hp://

14 Pag 4 Appdix A. R cod usd o calcula saisical powr for h BACIyp xprim wh h variac-covariac marix is simad #Program o sima powr of a baci xprim # wh h variac-covariac marix is ukow. Variac is simad # alog wih h ram ffc ad ad h corol populaio ma # Th simad variac-covariac marix ca has h # form of a iraclass covariac marix #Baci cod usig Mo Carlo simas of powr #Nsim umbr of Mo Carlo simulaios #s is variacs (assumd qual for all populaios) #rho is corrlaio bw ach pair of populaios # umbr of bfor yars # umbr of afr yars #k umbr of corol populaios #k umbr of ram populaios #m masurm rror #alpha prob. yp I rror (rjcig ull hypohsis wh ru) #dla -- ru ram ffc rprsig diffrc i aural log survival l(sram/scorol) #h iraclass covariac marix srucur is assumd. library(mass) baci<-fucio(nsim=000,s=,rho=.9,=5,=5,k=,k=,m=log(.0),alpha=0.05,dla=log(.50)){ k<-k+k SIG<-marix(srho,col=k,row=k) diag(sig)<-s+mm INVSIG<-solv(SIG) dlas<-rp(na,nsim) ss<-rp(na,nsim) #Do Mo Carlo simulaios o g rplicaios of dla ad s for(ii i :Nsim){ brs<-baci.simas(s=s,rho=rho,=,=,k=k,k=k,m=m,alpha=alpha,dla=dla) if(!is.ull(brs)){ dlas[ii]<-brs$par[] ss[ii]<-g.s(brs$sig,=,=,k=k) s<-sqr(var(dlas,a.rm=t)) #g criical valu of disribuio udr ull hypohsis cri<-quail(x=(dlas-dla)/ss,probs=c(alpha/,-alpha/),a.rm=t) cri<-ma(abs(cri)) powrx<-abs(dlas/ss)>cri powr<-ma(powrx,a.rm=t) good<-sum(!is.a(dlas/ss)) rur(lis(nsim=nsim,good=good,s=s,rho=rho,=,=,k=k,k=k,m=m, alpha=alpha,dla=dla,s=s,cv=s/dla,powr=powr)) #oupus #good is h umbr of simulaios ha rsul i a valid sima #s sadard rror #cv coffici of variaio #powr probabiliy of rjcig h ull hypohsis of o ffc

15 Pag 5 baci.simas<-fucio(s=,rho=.9,=5,=5,k=,k=,m=log(.0),alpha=0.05,dla=log(.50)){ <-+ k<-k+k par<-c(log(.0),dla) SIG<-marix(srho,col=k,row=k) diag(sig)<- s+mm xma<-mvrorm(=,mu=rp(par[],k),sigma=sig) xma<-mvrorm(=,mu=c(rp(par[],k),rp(par[]+par[],k)),sigma=sig) xma<-cbid((xma),(xma)) rs<-myopim(xma=xma,s=s,rho=rho,m=m,=,k=k) rur(rs) #Ira uil maximum liklihood simas ar obaid #solvig h simaig quaios which wr #drmid by sig h parial drivaivs of h #liklihood fucio o zro. myopim<-fucio(xma,s,rho,m,,k){ k<-dim(xma)[] <-dim(xma)[] #bgi wih OLS rgrssio simas of ha paramrs SIG<-diag(,k) par<-g.pars(xma,sig,,k) SIG<-g.SIG(par,xma,,k) #chck codiio umbr of SIG c<-kappa(sig) if((/c)<=.-5){ warig("sig is compuaioally sigular i myopim") rur(null) lf<-lf(par=par,x=xma,=,k=k,sig) ol<-.-5 rr<-.ol(abs(lf)+ol) ir<-0 #look for rlaiv liklihood fucio covrgc whil(rr>ol(abs(lf)+ol)){ par<-g.pars(xma,sig,,k) SIG<-g.SIG(par,xma,,k) #chck codiio umbr of SIG c<-kappa(sig) if((/c)<=.-5){ warig("sig is compuaioally sigular i myopim") rur(null) lf<-lf(par=par,x=xma,=,k=k,sig) rr<-abs(lf-lf) lf<-lf ir<-ir+ if(ir>000){ warig("oo may iraios i myopim") rur(null) rur(lis(par=par,sig=sig)) #Us simaig quaios o solv for paramr valus #Rurs corol ma (mu) ad ram ffc (dla) g.pars<-fucio(xma,sig,,k){ <-dim(xma)[] <--

16 Pag 6 k<-dim(xma)[] E<-rp(,k) E<-c(rp(,k),rp(0,k-k)) E<-c(rp(0,k),rp(,k-k)) xbar<-apply(xma[,(+):],c(),ma) xbar<-apply(xma,c(),ma) INVSIG<-solv(SIG) dla<-((e)%%invsig%%xbar)((e)%%invsig%%e)- (E)%%INVSIG%%xbar((E)%%INVSIG%%E) d<-((e)%%invsig%%e)((e)%%invsig%%e)-(/)((e)%%invsig%%e)^ dla<-dla/d mu<-(e)%%invsig%%xbar-((e)%%invsig%%e)dla d<-(e)%%invsig%%e mu<-mu/d rur(c(mu,dla)) #log-liklihood fucio lf<-fucio(par,x,,k,sig){ INVSIG<-solv(SIG) <-dim(x)[] k<-dim(x)[] z<-x lik<--k.5log(pi)-.5log(d(sig)) for(ii i :){ z[,ii]<-x[,ii]-rp(par[],k) lik<-lik-.5(z[,ii])%%invsig%%z[,ii] for(ii i (+):){ z[,ii]<-x[,ii]-c(rp(par[],k),rp(par[]+par[],k-k)) lik<-lik-.5(z[,ii])%%invsig%%z[,ii] rur(lik) #G h sima variac-covariac marix #This is basd o h simaig quaios #variac is ukow ad has h #form of a iraclass covariac marix g.sig<-fucio(par,x,,k){ iform<- <-dim(x)[] k<-dim(x)[] z<-x SIG<-marix(0,col=k,row=k) for(ii i :){ z[,ii]<-x[,ii]-rp(par[],k) SIG<-SIG+z[,ii]%%(z[,ii])/ for(ii i (+):){ z[,ii]<-x[,ii]-c(rp(par[],k),rp(par[]+par[],k-k)) SIG<-SIG+z[,ii]%%(z[,ii])/ if(iform==){ s<-ma(diag(sig)) s<-(sum(sig)-sum(diag(sig)))/(kk-k) SIG<-marix(s,col=k,row=k) diag(sig)<-s

17 Pag 7 rur(sig) #Rur s sima basd o SIG #This is a horical formula #drivd i h papr g.s<-fucio(sig,=5,=5,k=){ k<-dim(sig)[] k<-k-k INVSIG<-solv(SIG) <-rp(,k) s<-(+)()%%invsig%% <-c(rp(,k),rp(0,k)) <-c(rp(0,k),rp(,k)) d<-()%%invsig%%+()%%invsig%% d<-d()%%invsig%%-(()%%invsig%%)^ s<-sqr(s/d) rur(s)