FISHERY BULLETIN: VOL. 80, NO. 4 



experimental cohort, tagged fish are recaptured 

 at most once in a fishery which is essentially con- 

 tinuous, and the time at liberty is known exactly 

 for only a fraction of the recapture sample. 



The regression models studied by Chapman et 

 al. (1965), Bayliff and Mobrand (1972), and Kirk- 

 wood (1981) were extended to permit the Type II 

 shedding rate for each tagged fish to be a func- 

 tion of time. Both deterministic and stochastic 

 versions were presented and previously pub- 

 lished models were shown to be special cases. 



If all tags are subject to the risk of shedding, 

 i.e., if 6 = 1, and if data are available from several 

 recapture periods, a simple plot of In*, against 

 t, will reveal whether the average Type II shed- 

 ding rate, V(t), is constant; if it is, the relation- 

 ship will be linear. In this event the most parsi- 

 monious model consistent with the data will be 

 the deterministic model based on a constant 

 Type II shedding rate, L. In addition, if the 

 points suggest a negative intercept on the ordi- 

 nate the Type I retention rate, p, may be added to 

 the parameter set. One may then carry out the 

 parameter estimation using either the Bayliff- 

 Mobrand linear regression model, or the non- 

 linear regression of ln(r,,/2r rf( ) on r ( , depending 

 on which error structure is assumed. However, 

 since the plot of In*, versus r, is approximately 

 linear even with multiplicative error in the re- 

 capture process, it probably makes little differ- 

 ence which estimation method is used as long as 

 proper statistical weights are incorporated. 



If 8 = 1 and the plot of In*, versus r, is nonlin- 

 ear, one of the more complicated tag-shedding 

 models is called for. A trend which is concave 

 downward suggests that y(t) is increasing with 

 time and points to the stochastic model of Equa- 

 tion (5) or its deterministic counterpart. On the 

 other hand, upward concavity could be explained 

 either by a model in which the Type II shedding 

 rate decreased with time or by Kirkwood's(1981) 

 hypothesis, or by a combination of the two as in 

 Equation (5). 



Another useful diagnostic plot is 1 — k, against 

 r-. These are the variables considered in Kirk- 

 wood's nonlinear model. When L is constant the 

 plotted points will be traced by a line analogous 

 to a von Bertalanffy growth curve with asymp- 

 tote 8 and location parameter p, and they should 

 indicate which of these two parameters to in- 

 clude in the model and how much precision to ex- 

 pect in the resulting estimates. (In passing, it is 

 worth mentioning that if 8 is to be estimated 

 jointly with L, a longer experiment is required to 



ensure high precision in the parameter estimates 

 than if L alone is being estimated.) 



The treatment of recaptures from double-tag- 

 ging experiments with multiple cohorts was dis- 

 cussed in the context of the Bayliff-Mobrand 

 model. Alternative methods of combining infor- 

 mation from several cohorts to estimate common 

 shedding parameters were proposed, and a gen- 

 eral linear model approach was suggested for 

 situations where more elaborate structural as- 

 sumptions are made. A full numerical evaluation 

 of these procedures remains to be done. 



As an alternative to the least squares regres- 

 sion methods usually employed, some new ML 

 procedures were presented. These are more diffi- 

 cult to use than the regression techniques, but 

 offer advantages in some situations. For exam- 

 ple, when only two recapture periods are possible 

 one cannot compute the precision of regression 

 estimates in the Bayliff-Mobrand model, but 

 standard errors in the equivalent ML model are 

 still estimable. The most promising method for 

 deriving ML estimates in the general case may 

 be the iteratively reweighted Gauss-Newton 

 algorithm. Indeed, if one has access to the right 

 computer software (such as the BMDPAR and 

 BMDP3R programs supplied by BMDP) this 

 approach is nearly as easy to use as the simple 

 Bayliff-Mobrand linear regression method. A 

 sensible procedure would be to first study the 

 diagnostic plots suggested above for the regres- 

 sion analysis, and then fit the selected model 

 using an iteratively reweighted least squares 

 algorithm. 



The estimation procedures discussed above 

 are applicable when data are grouped by recap- 

 ture interval. For situations in which the exact 

 time at liberty is known for each recapture an 

 unconditional ML model was developed. This 

 may be applied not only to estimate shedding 

 rates but also to estimate mortality rates un- 

 affected by shedding. However, in its general 

 form the likelihood function is rather compli- 

 cated and only numerical solutions would be pos- 

 sible in most situations. Analytical estimators 

 for L and Z were derived for a simplified condi- 

 tional likelihood. Besides the more stringent 

 data requirements this model requires the extra 

 assumption of constant mortality rates during 

 the experiment. 



In the final section it was shown that through 

 double-tagging it is possible to estimate mortal- 

 ity rates free of tag-shedding biases even when 

 the recapture data are available only to interval- 



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