WETHERALL: ANALYSIS OF DOUBLE-TAGGING EXPERIMENTS 



type recaptures, rAi, is proportional to ka,(1 - k Hi ) 

 while rsi is proportional to k b ,(1 — k a ,). The num- 

 ber of double-tagged recaptures is proportional 

 to KAi kb, . We assume the number of recaptures 

 in the three classes are trinomial given Ka% + >'n, 

 + r di , with conditional probabilities 



Pm = 



B, 



di 1 '"(I ~ KA,)(1 "- KB,) ' 



These assumptions lead to the ML estimates 



and 



'di 



«A, 



r B , + r 



and kb, = 



r d , 



di 



r A, + ^di 



Estimating Parameters of 

 Specific Models 



Regression Methods 



Despite the directness and simplicity of the 

 general adjustment procedure outlined above, 

 most double-tag analyses have aimed at unravel- 

 ing specific underlying mechanisms of the tag- 

 shedding process. The probability of tag reten- 

 tion, k,, is then seen as a continuous function of 

 time and a vector of model parameters, 6, to be 

 estimated from the recapture data. In the termi- 

 nology established above, k ( =1 —J(t,). Thus if k, 

 or some transformation of k, is plotted against r, 

 the form of an appropriate shedding model may 

 be revealed. In fact, this is the approach adopted 

 in much of the recent tag-shedding literature, 

 and various weighted regression procedures 

 have been developed to handle the parameter 

 estimation. The general formulation of these is: 

 find 6 such that 



S(6) = S w.Uj,-/^)) 2 



,=i 



is minimum. In the two-parameter Bayliff-Mo- 

 brand model y ; = ln/c,-, where k, is given in Equa- 

 tion (9), and /, (0) = lnp — Lr . In the four-param- 

 eter Kirkwood model 



Vi = 1 ~ "i 



Me) = 6 



,6 + At, 



In both cases the authors suggest setting 

 h; = r.i. This is not optimal in a statistical sense, 

 but is clearly preferable to equal weighting. It 

 should be noted further that neither the Bayliff- 

 Mobrand model nor the Kirkwood model is based 

 on an explicit consideration of error structure for 

 the observations. For example, there is consider- 

 able support in the literature for a multiplicative 

 error in the recapture process, i.e., r„ = E(r s ,) 

 exp(csi) and r di = E(r di ) expi^,) and in this case 

 the algebra leads one to the nonlinear model 



2r 



Jin) 



1 " J(r) 



+ e, 



where e, has mean and variance al r Appropri- 

 ate weights for this model are 



Wi 



_2 'si 'di 



r-i 



The regression models discussed here have as- 

 sumed that recaptures are obtained from a single 

 cohort of tagged fish. However, it is often the case 

 that several lots of tagged fish are released at dif- 

 ferent times, so the recaptures in a particular in- 

 terval may come from different cohorts. In this 

 event the analysis may be applied to each of the 

 m cohorts separately, provided these are fairly 

 large. When multiple releases are made but the 

 individual cohorts are small, so that relatively 

 few recaptures are expected from each cohort, 

 the usual procedure is to assume mortality rates 

 and shedding rates are constant and identical for 

 each group and to simply aggregate the recap- 

 ture statistics from the several releases. Let the 

 recapture intervals be of equal length, A, and let 

 r\,j and r dl] denote the number of single-tagged 

 and double-tagged fish from the jih cohort re- 

 captured during the ith interval following that 

 cohort's liberation. Employing the Bayliff-Mo- 

 brand linear regression model, one can estimate 

 lnp and L in the usual manner as 



£ = {x 1 w xr x 1 w y 



(11) 



where /? = [lnp L] T 



X = {x,j} is the augmented data matrix 



693 



