468 



Fishery Bulletin 102(3) 



conditioned" and "recapture-conditioned" proportions. 

 A tag-conditioned movement proportion (Eq. li is the 

 total number of lobsters that moved (>3 km) divided by 

 the number originally tagged and released. It includes, 

 in the numerator, all tagged animals that moved, both 

 those that were recovered, as well as those that were 

 not recovered. With a recapture-conditioned movement- 

 rate estimate (e.g., Eq. 2), only counts of recaptured 

 lobsters are used. The estimate expresses the movement 

 proportion as the number of tagged animals that were 

 recaptured and that also moved l>3 km) divided by the 

 total number recaptured. These two definitions for the 

 movement proportion will be used to derive an estima- 

 tion formula in terms of the five data inputs. 



Step 1 The derivation begins by writing the estimate 

 for proportion of lobsters that moved (P|f) in tag-condi- 

 tioned form: 



N s + N s 



pS _ JV MR T .U..YW 



N* 



(1) 



This estimate of movement rate from the sanctuary is 

 based on a tag-conditioned proportion because we have 

 no observations of recaptured lobsters from the sanctu- 

 ary that did not move (no unbiased measure of Nfj MM ) 

 which a recapture-conditioned movement proportion 

 would have required. However we did have information 

 about N§f NR , the nonrecovery of tagged animals that 

 emigrate from the sanctuary into the fished zone. It can 

 be estimated (steps 2 and 3) with the second assumption 

 that recovery rate for lobsters moving from the sanctu- 

 ary equals that of lobsters moving (>3 km) within the 

 fished zone. 



Substituting Equations 2 and 3 into Equation 4 and solv- 

 ing for N F , XR , the number of lobsters that moved >3 km 

 within the fished zone but were not recovered, yields 



N F = N F 



N F 



N F +N 



1 



(5) 



Step 3 Assumption 2 permits the derivation of a for- 

 mula forWS yR . We first define the recovery proportions 

 of animals that moved within the fished zone (F) as 



f F = 



I m 



N F 



'^ MM 



M r +M F 

 1 v U .XR A * M M 



(6) 



and from the sanctuary (S) as 



f s 



I M 



Nt 



iV .U XR T .V.fl 



(7i 



Assumption 2, that the recovery rate (necessarily in the 

 fished zone) for animals that were tagged and released in 

 the sanctuary and that moved into the fished zone is the 

 same as for animals that were both released and recap- 

 tured after moving within the fished zone becomes 



f F = f s 



I M I M  



(8) 



Substituting Equations 6 and 7 into Equation 8 and 

 rearranging terms, we have 



iy M.XR 



N s (N F + N f ) 



N 



n; 



(9) 



Step 2 Under assumption 1, the two ways in which 

 movement proportion in the fished zone can be defined 

 (as tag- and recapture-conditioned proportions) are 

 equated. For fished zone releases, the recapture-condi- 

 tioned Crc') movement proportion is written 



N* 



N F +N F 



(2) 



For the recapture-conditioned estimate formula (Eq. 2), 

 all three quantities on the right-hand side are given as 

 data inputs. With only numbers of lobsters recovered, the 

 formula is, in this sense, conditional on recapture. 



The tag-conditioned I7c'» proportion of lobsters moving 

 >3 km of those released in the fished zone is written 



Af^ +N F 



,/• ,, _ ly M.R TJV M,.Vfl 



N F 



The first assumption is 



Din _ pf.l 



r M ~ r M 



(3) 



i4i 



Step 4 Substituting Equation 5 into Equation 9 and 

 substituting the result into Equation 1 yields a closed- 

 form estimation formula for the quantity we seek, the 

 proportion moving from the sanctuary in one year: 



P s =- 



N f 



■A rS 



/V s ' ■iN F +N h ) 



(10) 



Numerical estimator: double-hypergeometric likelihood 

 method 



A likelihood formulation of this estimator was also 

 constructed. The likelihood function describing a single 

 tag-recapture experiment is hypergeometric (Seber, 

 1982; Rice, 1995) because sampling is without replace- 

 ment. The set of possible outcomes from each of the 

 two tagging experiments can be formulated as a 2x2 

 contingency table for the experimental populations of all 

 lobsters originally tagged and released. The two pairs 

 of outcomes represented in each contingency table are 

 "moved" or "not moved" and "recovered" or "not recov- 

 ered," yielding the four possible outcomes from both sets 

 of tag releases (see "Notation" section). 



