NOTE Erzini and Castro: Measures of dispersion for length-frequency analysis 



869 



data grouped by the growth parameter K 

 (Fig. 5, A-J). With the exception of group- 

 ings for K<0.15 (Fig. 5, A and B), which have 

 smaller intercept and slope values, the re- 

 gressions are similar (Fig. 5, C-J). The re- 

 gression line and the 95% confidence inter- 

 vals are also shown and the associated sta- 

 tistics are given in Table 4. The slopes of the 

 regressions are all significantly different 

 from0(P<0.001). 



Discussion 



A number of LFA methods, especially those 

 that estimate parameters by maximum like- 

 lihood methods, allow constraints on mea- 

 sures of dispersion. For example, the simplex 

 method of Kumar and Adams (1977) incor- 

 porates linear constraints on the standard 

 deviations (SD's) of normal components. The 

 SD's can be equal or fixed and the coefficient 

 of variation (CV) can be fixed or constant in 

 the program MIX (Macdonald and Pitcher, 

 1979; Macdonald and Green, 1985). The SD's 

 can be linear functions of mean length or of 

 age in the Schnute and Fournier (1980) 

 method. MULTIFAN (Otter Software, 1988) 

 allows age-dependent or length-dependent 

 trends in SD's. A common CV between 0.01 

 and 0.5 for all lengths at age or SD's that 

 increase linearly with mean length was pro- 

 posed for LFA constraints by Liu etal. (1989) 



Figure 5 



The coefficient of relative variation as a function 

 of relative length for data grouped by K. RL is 

 relative length (LJL x , length-at-age divided by 

 Lj. The interval classes and the regression sta- 

 tistics are given in table 4. Parallel lines are 95% 

 confidence intervals. 



c 

 o 



CO 

 > 



C 



o 

 it 



(D 



o 

 O 



2 4 06 01 10 12 : 4 06 OS 10 12 



00 02 04 06 Oi 10 12 00 02 04 06 0B 10 12 



) 2 



2 2 i 



RL 



RL 



