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Fishery Bulletin 88(2), 1990 



It then follows that 



Q{t) 



Note that as 6-*°°, 



- f L{u)du 



b + kt 



(3) 



>e-*'andL^A. Thatis, 

 b + Xt 



the variable-rate model reverts to the constant-rate 

 model. If a proportion 1-p of tags are shed immediate- 

 ly, then 



Q(0 = p 



b + U 



(4) 



Parameter estimation 



Because only tag returns with accurate recapture dates 

 are considered, an extension of the maximum likelihood 

 estimation procedure used by Kirkwood and Walker 

 (1984) can be used. Suppose fish are double tagged and 

 all tags not immediately shed have identical shedding 

 probabilities that are independent of their companion 

 tags' status (already shed or still retained). Then, if 

 Pzii ). Pi (t ). ^nd poit ) are the probabilities of a fish re- 

 taining two, one, and zero tags, respectively, at time 

 t(Q<t<°°) after release, 



P2(0 = Qit)^ 



and 



p,{t) = 2Q(t)[l-Q{t)] 

 P,it) = [i-Qit)V- 



A fish that has shed both tags will not normally be 

 distinguishable from a fish that was never tagged. Con- 

 sequently, the tag-recapture data consist of recoveries 

 of fish retaining one or more tags. Conditional on reten- 

 tion of at least one tag, the probability of a recaptured 

 fish retaining two tags at time t is 



P2(0 



1-Po(0 

 and that of a recaptured fish retaining one tag is 



Pijt) 

 l-Po(i) 



Following an initial release of double-tagged fish at 

 time t = 0, suppose m fish bearing two tags were recap- 

 tured at times t2,(i = 1, . . • ,to), and n bearing one tag 

 were recaptured at times tyij = 1,. . .,n). Then, the 

 log-likelihood L of the data, conditional on the recap- 

 ture times {<2i} and {^i^}, is 



L= 1 



P2(<2») 



1-Pofe) 



-t-5; In 



P ljtlj) 



1 - Poitij) 



Estimates of the model parameters incorporated in 

 P'zit ), Piit ), and pi){t ), and their asymptotic variances, 

 can then be obtained by maximizing L using standard 

 methods (e.g., Bard 1974). 



Model seiection 



For each of the experiments, we attempted to select 

 the most parsimonious model. Accordingly, we first 

 fitted model (4), which incorporates immediate tag 

 shedding and (with b<°°) a time-varying shedding rate. 

 Then likelihood-ratio tests (Kendall and Stuart 1961) 

 were carried out to test the null hypotheses that (1) 

 P = 1 (no immediate tag shedding), (2) 6 = <» (constant 

 shedding rate), and (3) both p = 1 and 6 = <». Unfor- 

 tunately, since these hypotheses are not fully nested, 

 some ambiguity can still remain. The choice is clear- 

 cut if none of the three hypotheses is rejected, or if all 

 but one are rejected. However, it is possible that while 

 hypothesis (3) is rejected, neither (1) nor (2) is rejected. 

 In that case, there is no reliable basis to distinguish 

 between models (2) and (3). 



Particularly when performing non-linear estimations 

 such as these, it is essential to examine the fit of the 

 estimated model to the observed data. Unfortunately, 

 since we are maximizing a likelihood conditional on the 

 observed recapture times, there is no exact way of do- 

 ing this. As an approximation, however, we can ex- 

 amine the fit of the estimated models to recovery data 

 grouped by time intervals. From the data in Table 2, 

 it is a simple matter (e.g., based on equation 9 in 

 Wetherall 1982) to calculate, for each k, estimates of 

 the proportions of tags shed (Pj.) by time t/f, the mid- 

 point of the A;-th time interval since release, as well as 

 exact 95% confidence intervals for these proportions 

 based on the binomial distribution. If it is assumed that 

 all tags shed during the A--th time interval were shed 

 at time 1^, then the fitted model and "observed" pro- 

 portions can be displayed on the same graph. For the 

 grouped data in Table 2, time intervals of one year were 

 used for the first five periods after release. The rela- 

 tively few longer-term recoveries were grouped into 

 periods at liberty of 5-10 years and 10-20 years. 



