WETHERALL: ANALYSIS OE DOUBLE-TAGGING EXPERIMENTS 



r. 

 Z =T 



1 



+ 



1 



1 + a 1 + 2a 1 + 3a 



whence L — a Z. 



Estimating Mortality Rates by 

 Double-Tagging All Fish 



In most of the preceding sections we assumed 

 the basic purpose in double-tagging was to pro- 

 vide auxilliary information on shedding rates 

 which could then be applied to correct recapture 

 statistics or mortality rate estimates obtained in 

 a primary single-tagging experiment. An attrac- 

 tive alternative is to use double-tagged fish en- 

 tirely and to estimate the mortality rates and 

 other vital parameters in such a way that no bias 

 corrections are necessary. If exact recapture 

 times are recorded, the ML model just discussed 

 is appropriate. When recapture data are grouped 

 into n time intervals of length A, centered at 

 times r ( , a convenient context for developing this 

 approach is the single-tagging regression model 

 suggested by Chapman (1961) and discussed fur- 

 ther by Cormack (1968) and Seber (1973). This 

 takes the form 



In 



r> 



h A, 



= \n[qpNA0)] -Qfl ~ Xt, + e, (19) 



where f, = the nominal fishing effort during 

 period i 



f,' = 2 fj Aj H — — the estimated 



J=1 nominal effort 



up to time n 

 q = the catchability coefficient 

 c, = a random error term. 



In this particular model one obtains estimates 

 of q and X, and, since N s (0) is known, an estimate 

 of p as well. However, in the presence of Type II 

 shedding the exploitation rate for any period will 

 be underestimated, i.e., hidden in X will be the 

 term L. The usual Drocedure would be to correct 

 X by subtracting L, where L is obtained in an in- 

 dependent double-tagging experiment. Instead, 

 if we apply the model directly to recapture statis- 

 tics from a double-tagging experiment (AUO) 

 fish initially double-tagged) we will obtain an 

 estimate of X unaffected by Type II losses and in 



need of no corrections. This is accomplished by 

 substituting the dependent variable 



+ 2r di 

 2A, f, 



2r rfl - 



n, + 2r, 



(20) 



Further, it now transpires that the estimate of 

 the regression intercept term is free of Type I 

 shedding effects, i.e., one will estimate \n[qN,i(0)] 

 rather than \n[qpN d (0)]. 



If we assume r, ( and r tl , are complementary bi- 

 nomial variables given r. it and that r, is Poisson, 

 then approximately correct weights for the re- 

 gression employing Equation (20) are 



Wi 



(n, +2r rf ,-) 2 r di , 



When effort statistics are not available so that 

 a constant fishing mortality rate must be as- 

 sumed, or when there is a linear dependence be- 

 tween the two independent variables f ,' and r, , 

 then separate estimates of q and X are not pos- 

 sible using the single-tag model of Equation (19) 

 unless Type I errors are absent. Nor is p esti- 

 mable. Instead, one may only regress ln(rj/A,) 

 on t, to yield estimates of ln[pFM(0)] and (F + X). 

 But when the model is applied to a double-tag- 

 ging experiment under the same restrictions, it 

 is still possible to estimate both q and X un- 

 affected by shedding. 



Note that the dependent variable of Equation 

 (20) from the double-tagging experiment is anal- 

 ogous to the one of Equation (19) appropriately 

 corrected for tag shedding, as in Equation (8). In 

 both cases the recaptures r di and r,., are assumed 

 to be point samples taken exactly at t,. Thus 

 while in the example above k, = p exp(— Lr,), the 

 correction procedure of Equation (8) and the 

 method outlined here are independent of assump- 

 tions on the manner of tag shedding (cf. Seber 

 1973:281), provided the recapture intervals are 

 reasonably small (say 1 yr or less). 



SUMMARY AND CONCLUSIONS 



The aim of this paper has been to extend the 

 theory and methodology of estimating tag-shed- 

 ding rates through double-tagging. Attention 

 was focused on the situation most commonly en- 

 countered in fishery applications, wherein two 

 identical tags are placed on each member of an 



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