588 



Fishery Bulletin 89(4). 1991 



model error to both the Kirkwood and Somers model 

 and the Sainsbury model resulted in significantly be- 

 tter fits to the data. This suggests that growth models 

 that include individual variability should, if possible, in- 

 clude model error in the estimation procedure. 



Release-length-measurement error did not prove to 

 be a significant source of error in this case. However, 

 its effect should, if possible, be tested in tag-recapture 

 growth studies and, if significant, incorporated into the 

 model as shown in this paper. This will be essential if 

 negative length increments are included in the data set 

 being analysed. 



Sainsbury (1980) showed that an underestimate of 

 mean K can result if substantial individual variability 

 in K is present but ignored. There are a number of in- 

 dications from southern bluefin data that the level of 

 individual variability in K is not substantial. First, there 

 is virtually no difference in estimates of j4 L „ an d Mk 

 between models 3 and 6 and between models 4 and 7. 

 (The only difference in the models in each case is that 

 both the latter models incorporate individual variabil- 

 ity in K.) If the real level of individual variability in K 

 was substantial, we might expect, on the basis of 

 Sainsbury's (1980) observation, that models 3 and 4 

 would have given overestimates of /^ Loo and underesti- 

 mates of ^ K compared with models 6 and 7. Second, 

 analyses of length-frequency modes suggest that the 

 variance of length-at-age increases with increasing age 



(W.S. Hearn, CSIRO Div. Fish., GPO Box 1538, Hobart 

 7001, Aust., pers. commun.). This, as Sainsbury (1980) 

 notes, is more the rule than the exception in fish popula- 

 tions, and is indicative of variation in L^ having the 

 dominant effect on overall variation in length-at-age. 

 Third, modes are clearly visible in the length-frequency 

 data for at least the first four age-classes (Kirkwood 

 1983). Majkowski et al. (1987) give a general condition 

 for the visual or statistical separability of length-fre- 

 quency modes as \\i{- \i\_i\<2mm(p\, o 1 + 1 ), where 

 Hi and o\ are the mean length and its standard devia- 

 tion, respectively, of age-class i. Applying this condi- 

 tion to the mean-length and standard-deviation-at-age 

 data given in Table 2, we find that only for models 3 

 and 4 (no variation in K) and models 6 and 7 (o k //j k 

 = 0.07) is the condition satisfied for separating age- 

 classes 3 and 4. On this basis, we could conclude that 

 the visibility of at least four modes in the length- 

 frequency data would preclude levels of K variability 

 much greater than those derived for models 6 and 7. 

 The problem of selecting the most appropriate model 

 for use in stock assessment was addressed, using 

 likelihood ratio tests. These indicated that model 1 was 

 inadequate, and that a significantly better description 

 of the data was provided by incorporating individual 

 variation in L^ (model 3). However, the incorporation 

 of individual variation in K or release-length-measure- 

 ment error could not be justified. 



