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Fishery Bulletin 91 12), 1993 



having larger standard errors. Nevertheless, James' 

 method provides us with more robust parameter esti- 

 mates. 



This paper illustrates how critical is statistical meth- 

 odology to the estimation of growth parameters from 

 tag-recapture data. We show, by simulation and 

 through analysis of actual data, that different estima- 

 tion methods can easily lead to very different param- 

 eter estimates. With the help of simulation, we feel 

 that realistic choices can be made among these esti- 

 mators. For the actual datasets chosen for this paper, 

 we are able to compare growth-curve parameter esti- 

 mates from tag-recapture data with parameter esti- 

 mates calculated from length-at-age data, using ages 

 estimated from directly counting annuli. This provides 

 another criterion for choosing among estimators. 



Methods 



We apply estimators considered by James (1991). The 

 "unweighted Fabens" estimates (Fabens 1965) is the 

 widely used historical method. What we call the 

 "weighted Fabens" estimates, and James' estimates, 

 were derived by James (1991). The weighted Fabens' 

 method weights the residuals by an inverse variance 

 estimate and appears to work well for the observa- 

 tional error model. James' estimators appear to be less 

 prone to bias than the unweighted Fabens' estimators, 

 and more robust to the presence of variability in L„ 

 than the weighted Fabens' estimators. A more theo- 

 retical description of these estimators can be found in 

 James (1991). 



First we present a simulation study that corrobo- 

 rates the main findings of James ( 1991). Our study is 

 more systematic than that of James ( 1991 ), using three 

 estimation methods and error structures and calculat- 

 ing mean square error. Our simulation is based on 

 parameters estimated for a marine teleost, Pacific whit- 

 ing Merluccius productus, while James' simulations 

 dealt with two shellfish species. Nevertheless, these 

 simulations should be considered an addendum to 

 James' original work. 



Using notation similar to James (1991), consider a 

 population having von Bertalanffy parameters (L , K,,) 

 which we wish to estimate. L refers to the average 

 value of L. in the population, where L. might vary 

 among individual fish. We assume K=K,> does not vary 

 among fish. Also, recall that t„ cannot be estimated 

 from tag-recapture data alone. To determine t„ we need 

 the average length of at least one age-group which can 

 be supplied by direct age data (i.e., ages obtained from 

 directly counting annuli), or the modal length-frequency 

 of a dominant year-class of known age. 



Suppose we observe the size at release (y,,) and re- 

 capture (y 2l ) of i=l n fish, and the ages at release 



and recapture are (t„ t,+d,). Suppose also that inde- 

 pendent normally-distributed errors (e,„ e 2i , e 3l ) enter 

 into these observations in three possible ways: 



y u = (L„ + £.,,)[ 1.0 - exp(-K,,t,)] + e,, 

 y 2 , = (L„ + e 3l )[1.0 - exp(-Ko(t, + d,))] + e 2i . 



We refer to e,, and e 2i as "observational errors" and e :! , 

 as "variability in L_." 



The three estimators of (L , Kq) considered in this 

 paper are all based on the "residuals": 



n, = y 2 , - y., - (L - y„)[1.0 - exp(-Kodi)]. 

 These estimators are: 



1 Unweighted Fabens, estimated by minimizing In,, 2 



2 Weighted Fabens, estimated by minimizing 

 Ilri.VU.O+exp^Kod,))]. 



3 James, estimated by solving the simultaneous 

 equations £n =0 and ld,n,=0. 



Unweighted and weighted Fabens' estimates, and 

 their standard errors, were calculated using nonlinear 

 least-squares methods. Define g^Iq, and g 2 =Id,q,. 

 James' estimates were calculated as suggested by 

 James ( 1991) by solving for "L (l in terms of K,, from g l 

 followed by substitution into g 2 " and then applying the 

 bisection method to estimate K,, (see Press et al. 1986). 

 This method appeared computationally robust and suit- 

 able for simulation studies. The variance estimates of 

 these parameter estimates were calculated according 

 to James (1991): 



cov(L,K) = G'DfG')- 1 , 



where 



G 



dg- 2 dg 2 

 , dK n dL (l , 



£ = Ir| 2 



1.0 d, s 

 \ d, d 2 ij 



and the right-hand expressions are evaluated at the 

 estimated parameters. 



Our simulation considered three different error as- 

 sumptions. 



Simulation 1 The observational error model where 

 all fish are assumed to have common growth param- 

 eters, with variation due wholly to observational er- 

 rors e,„ e 2l ~ N(0,o 2 ). Here, all e 3l are assumed to be 

 zero. 



Simulation 2 The variation in L. model, where 

 all variation in the observations are due to e 3l ~ N(0,o 2 ), 



