Page 1. Before-After Control-Impact (BACI) Power Analysis For Several Related Populations. Richard A. Hinrichsen. March 3, 2010

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1 Page Before-Afer Corol-Impac BACI Power Aalysis For Several Relaed Populaios Richard A. Hirichse March 3, Cavea: This eperimeal desig ool is for a idealized power aalysis buil upo several simplifyig assumpios Table. For a specific eperime, a more accurae porrayal of power may require chagig hese assumpios ad he uderlyig equaios. For eample, assumig ha variace will be esimaed isead of kow. Therefore his aalysis should be reaed as a rough guide o power. Iroducio Currely here are may waershed projecs uderway i he Columbia Basi o deermie he survival effecs of various maageme acios o salmo survival. For eample, here are a series of iesively moiored waersheds IMWs beig esablished for he purpose of beer udersadig how salmo respod o approaches o resore habia. Whe hese projecs are ru as eperimes, i is possible o ideify he effeciveess of resoraio ad oher maageme acios. This aalysis was moivaed by he eed o desig hese eperimes i such a way ha hey have a high likelihood o deec sigifica survival chages i salmo whe hey occur. Usig his aalysis ca give a eperimeer a rough idea of he umber of years o ru a eperime ad deermie wha saisical power ca be epeced based o wha is kow abou he variace-covariace srucures for survival amog he salmo populaios sudied. The framework for he aalysis give here, alhough developed wih salmo i mid, fis io he framework of he Before-Afer-Corol-Impac BACI eperime. Such BACI-ype eperimes fid applicaio beyod Columbia River salmo survival Oseberg ad Schmi 996. A a priori power aalysis is developed for a BACI-ype eperime aimed a esimaig a chage i survival for several populaios. The eperime icludes a Before period where all populaios receive o reame followed by a Afer period where oly he reame populaios receive reame. This is a geeralizaio of he BACI-ype eperime where he corol populaio ad impac populaio are sampled oe ime before ad oe ime afer he reame Gree 979, Oseberg ad Schmi 996. The assumpios for he aalysis are give i Table. I is assumed ha i he absece of reame, all populaios have a commo mea log survival. Because his assumpio ad ohers may o hold i pracice, his aalysis should be reaed as a rough guide.

2 Page The mai goal of his work is o demosrae he probabiliy of deecig a effec o survival whe several relaed populaios wih a commo mea survival are used i a BACI-ype eperime. This goal is accomplished by describig he eperime i a saisically rigorous way, seig up he likelihood fucio, ad he usig maimum likelihood heory o esimae power. Power is he probabiliy of rejecig he ull hypohesis of o reame effec. Table. Assumpios used i power aalysis. A The observaios of logsurvival follow a mulivariae ormal disribuio. A There is o serial depedece. A3 All populaios have a commo mea logsurvival before reame. A4 Afer reame, he corol populaios coiue o have he same commo mea as ehibied i he Before years, ad he reame populaios also have a commo mea, bu shifed by a amou he effec size ha is he same for all reame populaios. A5 The measureme errors i logsurvival follow a mulivariae ormal disribuio ad he errors are idepede of he error due o acual year-o-year eviromeal variabiliy. A6 The esimaor of he reame effec is a maimum likelihood esimae. A7 The variace-covariace mari represeig he error i logsurvival is kow. These are assumpios for a idealized eperime. For a specific applicaio, a more accurae eperimeal desig may require chagig hese assumpios ad he uderlyig equaios. Therefore his aalysis should be reaed as a rough guide o power. The websie coais a web-based ool ha implemes his power aalysis wih he added assumpios ha he variaces i logsurvival are equal for all populaios ad he correlaios i logsurvival are equal for each pair of populaios. The R-code for implemeig his power aalysis may be foud i Appedi A. Mehods To derive he esimaor ad is variace for he power aalysis, maimum likelihood is used Mood e al I is assumed ha mea logsurvival before reame is he same for each populaio ad is equal o µ. Afer reame, he mea logsurvival of he reame populaios shifs by he amou δ o he value of µ for he reame populaios while he corol populaios coiue o have a mea logsurvival of µ. I is also assumed ha year-o-year variabiliy i logsurvival ad measureme error follow a mulivariae ormal disribuio. Thus he oal variace mari is y + m, where y is he variace-covariace mari ha describes yearo-year variabiliy i he absece of measureme error, ad m represes he variacecovariace mari of he measureme error. Uder hese assumpios, he log-likelihood fucio may be wrie as

3 Page 3 [ ] [ ] / / + + C l where l is he log-likelihood fucio wih vecor argume [ ] µ µ wih eries represeig he corol ad reame meas, respecively; is he umber of years prior o reame; is he oal umber of years of he eperime; is a k-vecor of observed survivals i year, k is he umber of populaios reame + corol used i he eperime; is a k-vecor of s; is a k -vecor of s, where k represes he umber of corol populaios; is a k -vecor of s, where k represes he umber of reame populaios. The vecor is arraged so ha he k corol populaios precede he k reame populaios. To obai he maimum likelihood esimae of he reame effec, he firs parial derivaives are arraged i a vecor called he gradie ad se o zero. The gradie vecor is give by [ ] [ ] [ ] l where represes he sample mea of logsurvival for he Before period ad represes he sample mea of logsurvival observaio i he Afer period. To esimae he reame effec, his equaio is se o zero ad solved for he maimum likelihood esimae ˆ. The maimum likelihood esimae of he reame effec is [ ]ˆ ˆ δ. This yields ˆ δ 3 where represes he vecor of sample meas over he eire duraio of he eperime.

4 Page 4 To deermie he variace of he esimaors, he iformaio mari is calculaed. By maimum likelihood heory, he variace mari is he egaive iverse iformaio mari, which is he mari of secod parial derivaives of he log-likelihood fucio. The iformaio mari is I [ ] [ ] + 4 Noice ha sice he variace-covariace mari is assumed kow, he iformaio mari does o deped o he mea logsurvival parameers or he logsurvival observaios: i is a fucio of he variace-covariace mari ad he umber of Before ad Afer years as well as he umber of reame ad corol populaios. For coveiece, he mari is epressed as a pariioed mari coaiig four submarices as follows: 5 where is is a k k mari represeig he upper lef-had corer eries of, is he k k mari represeig he upper righ-had corer eries, is a k k mari represeig he lower lef-had corer eries, ad is a k k mari represeig he lower righ-had eries of. The variace mari is he calculaed as he egaive iverse of he iformaio mari, which yields varˆ The goal of he aalysis is o esimae he variace of he reame effec, o he variaces of he reame ad corol meas. The reame effec, however, is a simple fucio of he corol ad reame meas, amely, ˆ δ [ ]ˆ. Thus he variace of he reame effec esimae is

5 Page 5 var ˆ δ [ ] varˆ This formula shows how he variace of he reame effec esimae depeds o he values of k, k,, ad he eries of he variace-covariace mari. Usig his formula, i is he simple o calculae he sadard error of he reame effec esimae, which is equal o he square roo of he variace: se ˆ δ var ˆ δ 8 The coefficie of variaio is CV ˆ δ se ˆ δ / δ 9 Power is he probabiliy of rejecig he ull hypohesis of o reame effec whe he acual reame effec isδ ad is a fucio of he rue reame effec, he probabiliy of a ype I error usually called he alpha value, ad he sadard error of he esimaor. By maimum likelihood heory he esimaor of he reame effec is asympoically ormally disribued. I his case i may be show ha he esimaor is ormally disribued regardless of he sample size. This occurs because he variacecovariace mari is assumed kow ad he esimaor is a liear combiaio of radom variables ha are kow o follow a mulivariae ormal disribuio. Ay liear combiaio of ormally disribued radom variables is also ormally disribued. Thus, he power may be wrie as Π δ Φ z δ / se ˆ δ + Φ z δ / se ˆ α / α / δ where Φ z is he cumulaive disribuio fucio of a radom variable ha follows a sadard ormal disribuio a ormal disribuio wih mea zero ad sadard deviaio, ad z α / is he criical value such ha α / probabiliy lies o he righ of he value z α / i a sadard ormal disribuio. For eample, whe α. 5, he criical value is equal o.96. perimeers ofe choose a desig such ha power of.8 is achieved.

6 Page 6 Ackowledgeme This work was suppored by Boeville Power Admiisraio corac # Thaks o Charlie Paulse, Rishi Sharma, ad Tracy Hillma for heir valuable reviews. Thaks o Bria Maschhoff for implemeig his aalysis as a web ool a

7 Page 7 Refereces Gree, R.H Samplig desig ad saisical mehods for eviromeal biologiss. Wiley ad Sos, New York, New York. Mood, A.M, Graybill, F.A., ad D.C. Boes Iroducio o he heory of saisics, Third diio. McGraw-Hill, New York, New York. Oseberg, C.W. ad R.J. Schmid Deecig ecological impacs caused by huma aciviies. I Deecig cological Impacs: Coceps ad Applicaios i Coasal Habias, R.J Schmi ad C.W. Oseberg, diors. Academic Press, New York, New York.

8 Page 8 Appedi A. R code used o calculae saisical power for he BACI-ype eperime described i his repor. #Program o esimae sadard errors ad power i a BACI-ype eperime #s is year-o-year variace assumed equal for all populaios #rho is he correlaio of survivals bewee each pair of populaios # umber of before years # umber of afer years #k umber of corol populaios #k umber of reame populaios #me measureme error #alpha -- prob. ype I error Probabiliy of rejecig ull hypohesis whe rue. #dela -- rue reame effec represeig differece i aural log survival lsreame/scorol baci<-fucios,rho.9,5,5,k,k,melog.,alpha.5,delalog.5{ k<-k+k SIG<-maris*rho,colk,rowk diagsig<-s+me*me INVSIG<-solveSIG e<-rep,k se<-+*e%*%invsig%*%e e<-crep,k,rep,k e<-crep,k,rep,k de<-*e%*%invsig%*%e+*e%*%invsig%*%e de<-de**e%*%invsig%*%e-**e%*%invsig%*%e^ se<-sqrse/de #rule--rejec whe esimae eceeds.96 ses i absolue value wo-sided q<-qorm-alpha/ power<--pormq*se,meadela,sdse+porm-q*se,meadela,sdse reurlissese,cvse/dela,powerpower,ss,rhorho,,,kk,kk,meme, alphaalpha,deladela } #oupus #se -- sadard error #cv -- coefficie of variaio #power -- probabiliy of rejecig he ull hypohesis of o effec This R-code uses he added assumpios of a commo variace ad commo correlaio erms. This assumpio may be relaed, however, simply by specifyig SIG as a ipu o he fucio baci i place of he ipu variables s ad rho.