WETHERALL: ANALYSIS OF DOUBLE-TAGGING EXPERIMENTS 



where L, is the regression estimate correspond- 

 ing to the jth cohort. 



A shortcoming of many double-tag analyses is 

 that attention has been focused on estimating 

 constant p and L despite the existence of multiple- 

 release statistics. In fact the multiple-release ex- 

 periment permits a more elaborate assessment 

 of shedding processes, with the level of detail 

 determined by specific characteristics of the 

 experimental design. To illustrate this, consider 

 an experiment with six recapture periods of 

 equal length, A. Two cohorts of double-tagged 

 fish are released, at the beginning of the first and 

 fourth periods. We assume there is a unique Type 

 I shedding rate associated with each cohort, and 

 that the Type II shedding rate is the same for 

 each group but is a function of time following 

 release. Specifically, we assume the latter rate is 

 constant for two recapture intervals following 

 the release of any cohort, may then change to 

 another constant level for two more periods, and 

 so on. 



The recapture statistics from the experiment 

 may be arrayed as follows: 



Release 

 group 



1 



2 



Recapture interval: 



2 3 4 5 



r su 



T sl2 

 V d\2 



r 



sl3 

 ^13 



V sU 

 r dU 

 r s21 

 r d2l 



r sl5 

 r dl5 

 V s22 

 V d22 



6 



V sl6 



r d\e 



V s23 

 V d23 



Note that m\ = m 2 = m 3 = 2 and ra 4 = wis = me 

 = 1. 



The parameter vector p = [lnpi lnp 2 Li L 2 L 3 ] T 

 may now be estimated from Equation (11) with 

 Y as given in Equation (13) and the data matrix 

 defined as 



X = 



As usual, the covariance matrix of (5 is estimated 

 by V = (X T WX)-\ 



With a little imagination this general linear 

 model can easily be adapted to accommodate a 

 wide variety of multiple-release experimental 

 designs. Standard analysis of covariance tech- 



niques may be applied to test the associated 

 hypotheses concerning /3. 



Maximum Likelihood Methods 



As an alternative to the least squares methods 

 we now describe some ML procedures for esti- 

 mating the model parameters in the single-re- 

 lease case. Given the total number of recaptures 

 in the tth period we again assume the numbers 

 falling in the various classes are multinomially 

 distributed. Thus when the A and B tags are 

 identical there are just two classes, and the num- 

 bers in each are binomial variables with condi- 

 tional expectations 



E(r dl ) = r, P n 



di 



= r.A 



— r,- 



and E(r si ) = r. ( (1 - P di ). Assuming further that 

 the statistics for successive periods are mutually 

 independent, the joint likelihood function for the 

 double-tag recovery data {r dl r d2 , ..., r dn ] given 

 {r.\, r.2, ..., r n } is 



>=i\r d il r si \/ 



p d rw - p di ) 



rgi 



(14) 



where P di is a function of r, and the vector of 

 parameters to be estimated, 6. 



When the A and B tags are not identical, the 

 recaptures are partitioned into three disjoint 

 classes, and the numbers in each are trinomial 

 with expectations 



Jb(tv) (1 - Ja(t,)) 



and 



695 



