870 



Fishery Bulletin 92(4). 1994 



In addition to these methods which allow specific 

 constraints, some iterative methods require starting 

 or initial values for some parameters, such as num- 

 ber of components, corresponding mean lengths at 

 age, proportion in each age class and SD's of the com- 

 ponent distributions (e.g. Akamine, 1982, 1984, 1985). 



Our results can be used to select appropriate con- 

 straints and starting values for measures of disper- 

 sion for LFA methods. We have shown that the mag- 

 nitudes of SD and V are dependent to a large extent 

 on life history parameters. Therefore, if the LFA user 

 has estimates of growth parameters, the multiple 

 linear regression models in Table 1 can be used to 

 estimate the SD for the species and size in question. 



However, in most cases the objective of LFA is to 

 estimate growth parameters, which are therefore not 

 available for input into the predictive models. In this 

 case, the CV may be more useful as a constraint. 

 While the magnitudes of SD and V of mean length at 

 age are related to characteristics of each species, rela- 

 tive variability in length at age (CV) is similar in 

 species that differ greatly in life history parameters. 

 Furthermore, while there are no consistent age- and 

 size-dependent trends in absolute measures of vari- 

 ability, relative variability decreases in a predictable 

 manner in almost all cases. 



This was confirmed in a previous investigation of 

 the shapes, magnitude, and age and size dependence 

 of length-at-age distributions of marine fishes 

 (Erzini, 1994). Analysis of 415 individual data sets 

 showed that in 97% of the data sets the CV was nega- 

 tively related to relative length at age, and the slope 

 was significant (P<0.05) in 53% of the sets. CV val- 

 ues were similar for all species. A negative relation- 

 ship between CV and size and decreasing variation 

 with size are to be expected because changes in vari- 

 ability with growth are typically of smaller magni- 

 tude than changes in size with growth. 



In contrast, although there was no dominant size- 

 dependent or age-dependent trend for the SD, the 

 most common pattern was that of increasing vari- 

 ability to a maximum at an intermediate age or size. 

 This trend for increasing variability to a maximum 

 at an intermediate size is illustrated in Figure 1, 

 where the SD is plotted against relative length and 

 asymptotic maximum length (LJ. It is particularly 

 evident for species with large L n values. 



In conclusion, we believe that the practical impli- 

 cations for LFA are that these empirically derived 

 relationships between measures of dispersion, size, 

 age, and life history parameters can be used to se- 

 lect starting values and to impose constraints on 

 measures of dispersion corresponding to particular 

 lengths at age. This is useful as there are no well 

 established rules or guidelines for this process, which 



consequently has been highly subjective and depen- 

 dent on each LFA user. 



The choice of model depends on the availability of 

 the data for the independent variables of the mod- 

 els. In the absence of any such data, the simplest 

 model of the CV as a function of relative length can 

 be used. As a preliminary step, length-frequency dis- 

 tributions should be examined and the number of 

 possible component distributions and modes that 

 may represent mean lengths at age identified visu- 

 ally. An estimate of L n obtained from the literature 

 or on the basis of the maximum observed size can be 

 used to convert lengths to relative lengths. The esti- 

 mated CV values and their corresponding confidence 

 intervals for these modes can then be estimated with 

 the models presented in this study. One possible ap- 

 proach is to use the estimated CVs as starting values 

 and the confidence intervals as lower and upper con- 

 straints. Such a strategy would provide realistic start- 

 ing values, reasonably narrow constraints, and would 

 improve the often arbitrary choices which are made. 



Acknowledgments 



We would like to thank the Scientific Editor and an 

 anonymous reviewer for their suggestions which 

 greatly improved the manuscript. 



Literature cited 



Akamine, T. 



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1984. The BASIC program to analyze polymodal 

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1985. Consideration of the BASIC programs to ana- 

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Basson, M, A. A. Rosenberg, and J. R. Beddington. 

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1990. Sample size and grouping of data for length 

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