580 



Fishery Bulletin 89(4), 1991 



Model 4: Kirkwood and Somers model with model 

 error and release-length-measurement error Be- 

 cause the tagging operation involves the handling of 

 powerful, often violently struggling fish, it is quite 

 reasonable to expect that release length will be mea- 

 sured with error. If possible, this error should be in- 

 dependently estimated and included in the growth 

 model. On the other hand, measurement of a dead fish 

 at recapture should not involve a significant error if 

 competently carried out. In this study, recapture 

 lengths are assumed to be measured without error. 



If growth is assumed to be negligible, a comparison 

 of lengths-at-release and lengths-at- recapture (1 2 ) for 

 animals at liberty for a very short time should provide 

 a good estimate of release-length-measurement error. 

 Such a comparison was made for 251 tag returns in 

 which the release length was 50 cm or more (see 

 Residuals Analysis section) and the period at liberty 

 was 10 days or less. For this data set, the mean mea- 

 surement error /i m = (Z{l 2 -li}/251) was 0.4861cm 

 with a variance, o m 2 , of 5.2428cm 2 . At least some of 

 this Mm might be attributed to growth, since the 

 average growth increment over a 10-day period is ap- 

 proximately 0.5 cm. In this paper, Mm is assumed to be 

 zero. 



Because c5 1 ; now depends on another random vari- 

 able (1 ; ), it is convenient to assume that the length-at- 

 recapture is a normally-distributed random variable. 

 Equation (1) can be modified to describe the ith recap- 

 ture length as 



2i = [ml -0i + Ei)] [l-e- Kt i] + 1, + £l + e 



(5) 



where e t is the normally-distributed release-length- 

 measurement error. The expected value of U; is given 



by 



E(l 2i ) = K,-E(li)] [l-e-Kti] + E(l,), 



where E(l;) is equal to lj + Mm- Collecting terms in 

 Ei, Equation (5) is rewritten as 



!•>, = [ML ro -l,][l-e- Kt .] + l 1 + e- Kt 1£ , + e, 

 The variance of l 2i is then given by 



var (l 2i ) = o L 2 (l-e- Kt .) 2 + o e 2 + o m 2 e" 



Model 5: Sainsbury model Sainsbury (1977, 1980) 

 described a model that recognised individual variation 

 in K, as well as in L^. He assumed that both were 

 independent random variables, with K following a 

 gamma distribution and L^ being normally distributed. 

 He-also assumed that, as an approximation, 6\ t is 

 normally distributed for given lj and t ; , pointing out 

 that this should involve little error if L^ is normally 

 distributed (Sainsbury 1977). With mk and o K 2 

 denoting the mean and variance, respectively, of K, the 

 relevant equations are 



E(dli) = [ml -1,] 



and 



var(dli) = d o L 2 + C 2 ( ML -li) 2 , (6) 



where 



Ci = 1 - 2 



Mk 



1 + 



ok 2 tj 



MK 



1 + 



2o K 2 tj 



Mk 



°K 2 



and 



mk 



■w 



1 + 



2o K 2 t; 



MK 



1 + 



o K 2 tj 



MK 



Maximum-likelihood estimates of ml„. °l„ 

 o K 2 can be obtained as shown earlier. 



Mk, and 



2Kt; 



Model 6: Sainsbury model with model error As 



shown in model 3, a model error can be included sim- 

 ply by adding the model error variance term to Equa- 

 tion (6) giving 



var(dli) = C, a L J + C 2 (ml„- 1j) 2 + ° e 2 . 



Solution by maximum likelihood now requires that a 

 fifth parameter, o e 2 , be estimated from the data. 



Given estimates of M m and o m 2 , maximum-likelihood 

 estimates of ml„> °lJ, K, and o e 2 can be obtained as 

 before. 



Model 7: Sainsbury model with model error and 

 release-length-measurement error Using logic sim- 

 ilar to that developed in model 4, length-at-recapture 

 is now considered as the random variable and assumed 



