WETHERALL: ANALYSIS OF DOUBLE-TAGGING EXPERIMENTS 



E A (r di ) = (r B , + r dt ) (1 - J a (t,)). Thus an itera- 

 tively reweighted Gauss-Newton algorithm mini- 

 mizing 



S' B = 2 w Bi (r di - E B (r d ,)Y' 



1=1 



with w Bi = w Bi = [(r Ai + r di ) J B (r,)(l - Jb(t,))]' 1 is 

 used to compute ML estimates of shedding pa- 

 rameters for the B class of tags. A parallel pro- 

 cedure gives ML estimates of parameters for the 

 A class. Note that the two sets of parameter esti- 

 mates are not independent. 



Unknown Recapture Times 



A tacit assumption in the foregoing procedures 

 is that the time between release and recapture 

 for each returned fish is known to "interval accu- 

 racy," and that exact recapture time information 

 is available for only a fraction of the recoveries, 

 so that all recoveries are grouped into the n time 

 intervals to permit estimation. This will often be 

 the case. However, in some fisheries it is conceiv- 

 able that only the crudest sort of information is 

 available on recapture times. For estimation 

 purposes, all that is known is r d and r s , the total 

 number of recaptures in each class over the ex- 

 perimental period (0, T). When T is relatively 

 small, say 1 yr or less, then estimation of a single 

 Type I or Type II shedding parameter is possible, 

 as in Equation (15) or (16). In an experiment of 

 longer duration this is not feasible. However, it is 

 possible under certain circumstances to estimate 

 the ratio of the Type II shedding rate to other 

 Type II losses. Let fishing be constant, contin- 

 uous, and uniform at an instantaneous rate F. 

 Assume further that the total instantaneous mor- 

 tality rate is a constant, Z, and that shedding of 

 tags occurs at an instantaneous rate L. If there 

 are no Type I losses, the ratio of E(r g ) to E(r d ) in 

 a double-tagging experiment approaches 



2L 



x 



Z + L 



as T — °°. Thus if L = aZ, a moment estimator 

 of a is provided by 



x 



a = 



2 - x 



and if one has an estimate of Z which has a syste- 

 matic bias due to Type II shedding, say Z\ then 

 a corrected estimate may be obtained, i.e., Z' 

 = Z\\ +a)- 1 . 



This method may also be used where single- 

 tagging and double-tagging experiments are 

 run concurrently. Then if A^(0) fish are released 

 double-tagged and M(0) with single tags, let r' d 

 and r'g be recaptures from each group still bear- 

 ing the initial complement of tags. Under the 

 same assumptions as above this leads to 



a 



r/AUO) ~ rJAE(O) 

 2riN s (0) - r,'N,(0) 



1 < d < 2. 

 riNM 



—^ , 0<-<2 (17) 



2r d - r s r d 



Exact Recapture Times 



Turning now to the other end of the spectrum, 

 under ideal conditions it is possible that the exact 

 time out will be known for each fish returned. 

 When exact recapture times are available for all 

 fish the returns from a single-tagging experi- 

 ment may be analyzed using ML procedures first 

 developed by Gulland (1955) and later elabor- 

 ated by Chapman (1961) and Paulik (1963). 

 These rest on the assumption of binomial recap- 

 ture probabilities based on constant Type II loss 

 rates and on a resulting conditional recapture 

 time distribution which is truncated negative 

 exponential. Chapman et al. (1965) extended the 

 same concepts to returns of fish initially double- 

 tagged and still retaining both tags upon recap- 

 ture, and showed that the difference between the 

 estimated total Type II loss rate in a double-tag- 

 ging experiment and the corresponding total 

 Type II loss rate in a single-tagging study yielded 

 an estimate of L. They noted that this is the best 

 estimate of L possible using only the recapture 

 information from a single-tagging experiment 

 and from fish put out and returned with two tags. 

 Left open was the possibility of combining this 

 information with recapture times for fish initial- 

 ly double-tagged but returned with only one tag 

 still attached. For this class of fish the distribu- 

 tion of recapture times is more complicated. 



We now consider an exact recapture time mod- 

 el for an experiment based exclusively on fish 

 initially double-tagged. Suppose JV rf (0) double- 

 tagged fish are released at time 0. Over the course 

 of the experiment, terminating at time T, a total 

 of r d fish are recaptured and returned with both 

 tags intact, and r s with only a single tag remain- 

 ing. In addition, for each tagged fish returned 



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