McGarvey: Estimating emigration rates from marine sanctuaries using tag-recovery data 



469 



In this study the data from two interacting tag-recov- 

 ery experiments were used to generate an estimate of 

 reserve emigration rate, namely of lobsters tagged and 

 released into the sanctuary and into the fished zone. 

 Thus, the product of a pair of linked hypergeometric 

 probability mass functions, each corresponding to a 2- 

 way contingency table, is the natural form of the likeli- 

 hood function for Pfj. 



The derivation of Equation 10 was made with two as- 

 sumptions, namely Equations 4 and 8. Incorporated in 

 the likelihood, the two assumptions constrain the eight 

 recapture numbers in the contingency tables. In the 

 likelihood formulation, a third constraint was needed 

 which is analogous to assumption 1 but which applies 

 to sanctuary releases. 



The derivation for constructing this likelihood from 

 a pair of linked hypergeometric probability functions 

 will proceed by 1) writing out the "raw" contingency 

 tables in terms of the eight recapture numbers (TV), 

 as denoted in the "Tag-recovery data" and "Notation" 

 sections, 2) algebraically re-expressing the elements 

 of the tables so that the parameter to be estimated is 

 explicit. 3) imposing the three constraints, and 4) writ- 

 ing out the likelihood, using the hypergeometric form 

 for contingency tables. 



For the lobsters tagged and released in the sanctuary, 

 the raw contingency table is 



The two hypergeometric probability mass functions 

 (pmfsi giving the model-predicted proportion of lobsters 

 that moved and were recovered, based on the two con- 

 tingency tables, are written as 



* Km)- 



N* +N 



ly M.R T ly M.NR 



N F 



N* 



-(N +N ) 



yly M.R TJV M.NR' 



N F 



\\ 



N F 



(12) 



N +N 



ly MR * NM.R ) 



Because the goal is to estimate the movement propor- 

 tion, Pf t (rather than any specific value of N), this pro- 

 portion will need to be made explicit in the likelihood 

 function as the sole freely varying parameter. Substitut- 

 ing from the definition of Py (Eq. 1), we have 



m" = ps . AT* _ MS 



ly M.NR 1 M iy T ly M.R- 



(13) 



Substituting for all occurrences of Nfj NR , Equation 11 

 becomes 



p«o= 



PmK)(N^(1-P^^ 



Ni 



AT? 



N* 



N s + N s 



ly M.R T ly NM.R ) 



(14) 



Writing the full joint-likelihood expression formed by 

 the product of the two hypergeometric pmfs gives 



L = 



N F +N F V N F -(N F +N r ) 



ly M.R T ly M.NR ly T yly M.ff T ly M.NR ' 



N F 



N F 



As formulated, the value of Nf !M R is still undeter- 

 mined by data or constraint. A third constraint is 

 therefore required. As with assumption 1 for the fished 

 zone (Eq. 4), we apply the assumed equivalence of tag- 

 and recapture-conditioned proportions to the sanctuary 

 releases: 



ps.rc »>-& /(/V s +/V i ) = P ' 



r M ly M.R ' yly NM.R T ly M.R' *M 



-(MS +MS )/ fjs 



~ Xly MR + IV M.NR" ly T- 



P<N m .r) = 



MS . AfS 

 ly M.R T ly M.NR 



N s 



Ms _ i \r> + ms , 



" T W, M.R ^ ly M.NR' 



N s 



M s +M S 

 ly M.R Tiv NM.R I 



In this application, Nf, M R is understood as the number 

 of lobsters that would have been taken if fishing had 

 1 1 1 1 not been excluded from the sanctuary. Solving for N^ M R 



yields the third constraint. 



MS -IMS .MS)/iMS ,JU* )_MS 



ly NM.R ~ yly M.R ly T " WV M.R^ ly M.NR 1 ly M,R' 



