176 



Fishery Bulletin 97(1), 1999 



This tag shedding model follows essentially the same line of thought as Xiao's (1996a) and can be readily 

 phrased in the standard terminology of competing risks in survival analysis (David and Moeschberger, 1978). 

 Also, notice that the left-hand side of Equation 1 sums to zero; the left-hand side of Equation 2 sums to n(i). 

 When a single fish is double tagged and released at time t^d), one of 16 mutually exclusive events can 

 happen at time t(i) (Equation 1 or 2). However, only three events are actually observable: the fish has, upon 

 recapture, retained both tags, retained tag A and lost tag B, or lost tag A and retained tag B, with respective 

 probabilities of C(i,A,B,t(i)), C(i,A,0,t(i)) and C(i,0,B,t(i)). The event that it has shed both tags upon recap- 

 ture, with a probability of C(i,0,0,t(i)), cannot be observed, for when both tags are shed, a fish cannot be 

 reliably distinguished from one that was never tagged. A likelihood function can be constructed to estimate 

 parameters in Equation 1 or 2 by following arguments in standard competing risk analysis, but these esti- 

 mates are substantially biased. To overcome this problem, we estimated model parameters by conditioning on 

 observations of three events only, i.e. by maximizing the conditional likelihood function for all reported recap- 

 tures with at least one tag retained 



i/— aj • ' J-^ ,-) ' -Lj oj 



with 



n 



C(h,A,Bj(h}) 



\C{h,A,B,t{h}) + C{h,A,0,t,{h)) + C{h,0,B,t(h)} 



n 



R{h,A,B,t{h))e(h,A,Bj{h) 



*-R(h,A,Bj(h))9{h,A,B,t{h)) + R{h,A,0,t(h))e(h,A,0j(h)) + R{h,0,Bjih))9(h,0,B,t{h)) 



/i = i 



^^=n 



C(j,A,0,tiJ)) 



J^C(j.A,BMj)) + C{j,A,Oj(j)) + C{j,0,Bj(j) 



n 



R{j,A,0,tij))e{j,A,Oj(j)) 



(3) 



R{j,A,BjAj))0{j,A.Bjij)) + RU,A,OMjmj^A,O,t(j)) + R{j.O,Bjij))9{j,O,Bj(j)) 



^3 



C(k,0,B.t(k)) 



\C{k,A.Bjik)) + C{k,A.Oj(k)) + C(k,0,Bj{k)) 



It 



n 



R(k,0,Bj{k})e{k,0,B,t{k) 



\R{k,A,B,t{k)}9{k,A,B,t(k}) + R{k,A,0,t(k))0{k,A,0,t{k}) + R{k,0,B,t(k))e{k,0,B,t{k)) 



- \[Hi.A.s\+lii,B.s)]ds 



d(i,A,B,t{i)) = p{i,A}p(i,B}e '»'" 



d(i,A,0,t{i)) = p(i,A)e 



Ja((,A,.s-Ic/s 



l-p(i,B)e 



P' 



ii.B.sUls 



