Hampton: Estimates of tag-reporting and tag-shedding rates for tuna in the tropical Pacific Ocean 
77 
these, 29 returns were of the primary tag and 39 were 
of the companion tag. The cumulative binomial prob- 
ability of 29 or less of either the primary or compan- 
ion tag being found in a sample size of 68 is 0.275, 
indicating that there is a reasonable chance of the 
assumption being satisfied. 
Fourth, in the analysis carried out here, it was 
assumed that tag pairs (from fish recovered with two 
tags) are reported (or not) as a pair — i.e. either both, 
or none are reported. Furthermore, it was assumed 
that the probability of reporting a tag pair was the 
same as that of reporting a single tag. I will refer to 
this as the dependent hypothesis. An alternative 
hypothesis is that the reporting of individual tags 
forming a pair is completely independent; whether 
or not one tag of a pair is reported has no effect on 
the probability that the other will be reported. I will 
refer to this as the independent hypothesis. Under ei- 
ther hypothesis, we can define the probability, U(t), that 
a tag is retained at recapture time t and is reported as 
U(t) = pQ(t ! p,L) = p(l- p)exp(-Lt). (2a) 
However, the probabilities of two, one, and no tags 
being retained at recapture time t, and, in the case 
of at least one tag being retained, also reported, are 
different under the two hypotheses, as follows: 
Pd’^d) = pQ(t I Pd;A d ) 2 
Pdfo I Pd ’^d'; = 2 pQ(t I p d ,L d )[l-Q(t i p d ,L d )\ 
Pdoit ! Pd > L d ) = PQtt * Pd'L d ) 2 - 2 pQ(t \ p d ,L d )+ 1 
(3a) 
and 
P l2 (t\p„L i ) = p 2 Q(t\p l ,Lf 
P il (t\p i ,L i ) = 2pQ(t\p i ,L i )[l-pQ(t\ pi , L )] 
Pioit I Pi, Li) = [1- pQ(t I Pi,Li)f, (3b) 
where the d and i subscripts indicate the dependent 
and independent hypotheses, respectively. 
It can be shown that substitution of the right-hand 
sides of Equations 3a into the log-likelihood Equa- 
tion 4 produces an identical result to substitution of 
Equation 3; the p’s cancel out and reporting rate has 
no influence on the estimates p d and L d when the 
dependent hypothesis is true. This is therefore 
equivalent to using Equation 2 as the tag-shedding 
and reporting model, as I have done in this study. 
Under the independent hypothesis, substitution of 
the right-hand sides of Equations 3b into the log- 
likelihood Equation 4 does not result in a canceling 
out of p terms, and therefore p must be included in 
the tag-shedding and reporting model as shown in 
Equation 2a. However, p is totally confounded with 
1 -p., and cannot be estimated from the double-tag- 
ging data. If an independent estimate of p is avail- 
able (for example, from a tag-seeding experiment), 
Equation 2a can be applied and p ; estimated free of 
the effects of p. 
For most double-tagging experiments, it will not 
be known with any certainty whether the dependent 
or independent hypothesis is more appropriate. The 
following procedure may provide some insight in this 
regard: 
1 Obtain tag-shedding parameter estimates p d and 
L d , assuming that the dependent hypothesis is 
true (using Equations 2 and 3). 
2 If an independent estimate of the reporting rate, 
p , is available, obtain tag-shedding parameter 
estimates p, and L, , assuming that the indepen- 
dent hypothesis is true (using Equations 2a and 
3b). Small values of p (less than 1 - p d ) will usu- 
ally result in p, entering an unreasonable (nega- 
tive) domain. Alternatively, if p, is constrained to 
be nonnegative, L t will differ from L d and the fit 
to the data will degrade (i.e. >Q d ). In either 
case, this indicates inconsistency between the re- 
porting rate estimate p and the independent hy- 
pothesis. In the present study, the estimated re- 
porting rate (0.586) was much smaller than 1 - p d 
(0.941, see Table 1). If p is applicable to the 
double-tagged tuna, this implies that the indepen- 
dent hypothesis is inappropriate for these data. 
In reality, it is likely that the actual situation with 
respect to the reporting of tag pairs will lie some- 
where between completely dependent and completely 
independent reporting. It is possible to generalize 
the tag-shedding model with respect to these hypoth- 
eses by defining a coefficient of independence, c, such 
that 
U[t) _ p( 1- p)exp(-Lt) 
c(l— p) + p 
Setting c=0 is equivalent to the dependent hypothesis, 
c=l is equivalent to the independent hypothesis, while 
0<c<l implies partial independence. For the RTTP 
double-tagging data and p = 0.586, c<0. 088 allows an 
unconstrained p to remain nonnegative. This range of 
possible values of c implies that the dependent hypoth- 
esis is likely to be appropriate for these data. 
The tag-shedding model fitted to the double-tag- 
ging data assumes that the rate of tag shedding is 
constant over time. Kirkwood (1981) and Hampton 
and Kirkwood (1990) found that, in some cases, a 
model that allowed the probability of shedding to 
