FISHERY BULLETIN: VOL. 80, NO. 4 



r = r k 



-i 



(8) 



where m, estimated from double-tagging, is the 

 probability that a tag will still be attached at 

 time r r If returns from the double-tagging ex- 

 periment are too few to provide an estimate of 

 k, for each recapture interval, interpolation is 

 necessary. In any event, in this approach only 

 minimal assumptions need be made about the 

 manner in which shedding occurs. 



In most treatments of tag shedding, however, a 

 specific model is postulated for the shedding 

 process. Once the parameters of such a model are 

 estimated, appropriate adjustments for tag 

 shedding are made either to the single-tag recov- 

 ery data, as above, or directly to estimated popu- 

 lation parameters. A third strategy is to conduct 

 the experiment entirely with double-tagged fish, 

 and to estimate mortality rates and other popula- 

 tion parameters directly, in such a way that no 

 corrections are necessary. 



These various approaches are discussed now in 

 greater detail, assuming a continuous recapture 

 process. For situations in which tagged fish are 

 recaptured once at most, but only in point sam- 

 ples, some estimation procedures are given by 

 Seber (1973) and Seber and Felton (1981). For 

 multiple-recapture models of the Jolly-Seber 

 type, again with point sampling, the reader 

 should consult Arnason and Mills (1981). 



Estimating Adjustment Factors for 

 Single-Tag Recoveries 



Here we estimate x t , the probability of tag re- 

 tention at time t,, the midpoint of the ith recap- 

 ture period. We assume the shedding probabili- 

 ties for each tag are identical and independent of 

 the status of the other tag, and that recovery and 

 reporting rates are the same for recaptured fish 

 bearing either one tag or two. Under these condi- 

 tions the number of double-tag recoveries, r di , is 

 proportional to k, 2 and the number recovered 

 with only a single tag remaining, r s .,, is propor- 

 tional by the same factor to 2k, (1 — k { ). Of the total 

 number of recoveries from the double-tagging 

 experiment in the ?th period, the proportion 

 bearing two tags is therefore 



"it, — 



2 - K 



Maximum likelihood (ML) estimates of the k, are 

 692 



now easily derived. We assume the conditional 

 distribution of r dl , given (r d , + r si ), is binomial 

 with parameter P di . The likelihood of the /th re- 

 capture sample is thus 



Zi = 



n\ 



«i 



Ki 



ir x i 



r d ,\ r sl \J\2 - Ki/ \ 



The ML estimator of k, is easily found: 



2r di 



Ki 



r, + 2r, 



si di 



(9) 



This result is also given by Seber (1973) under 

 somewhat different assumptions. The asymptotic 

 variance of k 1 is 



Ou . - 



k,-(1 - *,-) (2 - K,f 

 2r, 



(10) 



As usual, numerical estimates are computed by 

 inserting k 4 in place of *,. 



Note that k, has a small negative statistical 

 bias. In fact, using a Taylor series expansion it 

 may be shown that 



E(k,) - 



«i [l - 



(1 - Ki) (2 



2rn 



«L~\ 



Bias increases with time out, i.e., as k, decreases, 

 and is inversely related to the total number of re- 

 captures. When k,  = 0.5 and r.% — 10, the negative 

 bias in k, is <4%. 



Note further that since the likelihood function 

 is conditioned on r,, inferences based on Equa- 

 tions (9) and (10) apply strictly only to the par- 

 ticular experimental outcome being studied, and 

 not to the broader class of results which might be 

 obtained in replications of the experiment. Pro- 

 viding that the approximation in Equation (7) is 

 valid, a more complicated unconditional model 

 would yield the same estimate of «,, but the vari- 

 ance of Ki would be greater, reflecting the sto- 

 chastic nature of the mortality and recapture 

 processes which lead to the r ( . Since our interest 

 is in estimating shedding rates and not mortality 

 rates, as a rule we consider only the simpler con- 

 ditional likelihoods. 



Above we have assumed the two tags are iden- 

 tical insofar as shedding rates are concerned. 

 When they are subject to different shedding 

 rates another set of estimators is required. Where 

 A and B tags are identified, the number of A- 



