Chen et al : Developing a growth-transition matrix for stock assessments of Strongylocentrotus droebachiensis 



739 



information, justification for dividing the Maine coast, and 

 selection of the habitats can be found in Vadas et al. ( 1997) 

 and Vadas et al. (2002). 



Vadas et al. (1997) modeled the size-at-age data using 

 the von Bertalanffy growth function (VBGF) described as 



L, = L„{\-e 



-All-I,,) 



(1) 



where L, = size at age t; 



L^ = defined as the mean asymptotic length that 



the sea urchin may attain; 

 K = the Brody growth parameter; and 

 tg = the hypothetical age of size (Ricker, 1975). 



For each area and habitat, a VBGF was used to fit the size- 

 at-age data. Three parameters in the VBGF (i.e. L„, K, and 

 tf,) and their standard errors were estimated by using the 

 nonlinear least squares method. These estimates were pre- 

 sented in Vadas and BeaP and Vadas et al. (2002), and were 

 made available to the authors of the present study (Table 

 1). Clearly there were large differences in the estimates of 

 L^ and K and their associated variations among different 

 areas and habitats (Table 1). 



The LJs estimated for different areas and habitats 

 ranged from 63.1 (northeast region with barren habitat) 

 to 95.2 mm (southeast region with kelp habitat) (Table 1) 

 and tended to be smaller than some large individuals ob- 

 served in the fishery (about 100 mm, Vadas, 1977; Hunter, 

 unpubl. data). This likely resulted from relatively small 

 sample sizes that covered relatively small areas, in a 

 relatively short period, compared with the fishery catch, 

 which targeted larger-size individuals. The exclusion of 

 individuals in the fishery catch that were larger than the 

 L.,'s estimated in Vadas and Beal'^ and Vadas et al. (2002) 

 from the calculation of the growth-transition matrix may 

 underestimate the variability in sea urchin growth, thus 

 introducing errors in stock assessment. Based on the data 

 collected in the Maine sea urchin fishery (Hunter, unpubl. 

 data) and previous studies (Vadas, 1977), 100 mm was 

 considered a reasonable value for the average asymptotic 

 size (LJ for sea urchins on the coast of Maine. However, 

 more extensive sampling needs to be done in the future to 

 verify this estimate. 



We might be able to derive an estimate of L^ for the 

 Maine sea urchin stock based on the examination of the 

 data collected from the fishery and other studies (Ricker, 

 1975; Moreau, 1987; Chen et al., 1992). An estimate of K 

 for the whole Maine urchin stock is, however, more difficult 

 because K is an abstract rate describing how fast organ- 

 isms approach the L_.^ and there are no observations or 

 background information with which to compare estimates 

 (Ricker, 1975; Moreau, 1987). We thus need to develop an 

 approach to estimate K for the Maine sea urchin stock 

 which corresponds to the value we assumed for the L^. 

 Many studies have indicated that estimates of K and L„ 

 tend to be highly and negatively correlated (e.g. Moreau, 

 1987; Chen and Harvey, 1994). Thus, a fish population or 

 species with a large L^^ tends to have a low ii" value, and vice 

 versa (Gallucci and Quinn, 1979; Chen et al., 1992). This 

 suggests a strong relationship between L^ and /f estimates 



(Pauly, 1980; Stergiou, 1993). Such a relationship may be 

 used to estimate K for a given L„ or to estimate L„ for a 

 given K. In this study we developed and used the follow- 

 ing empirical approach to derive K for a given value of L^ 

 and its associated uncertainties in the development of a 

 growth-transition matrix: 1) conduct a regression analysis 

 for K and L„ estimated for different areas and habitats 

 along the coast of Maine (Table 1); 2) calculate coefficients 

 of variation (CV) for each K and L„ (Table 1) as 



„ standard error for K 



CV(K) = and 



CV(L,) = 



K 

 standard error for L^ 



(2) 



and conduct a regression analysis of CV(/0 and CV(L„) 

 estimates of different areas and habitats (data in Table 1); 

 3) use 100 mm to approximate L^„ and use this L„ to esti- 

 mate K from the regression analysis between K and Lj, 

 and 4) calculate the average CV for LJs of different areas 

 and habitats and then use the average CV(L„) to estimate 

 CW(K) from the CW{K)-CV(LJ regression equation. 



Because K and L„ were estimated for different areas 

 and habitats and had different precisions, outliers might 

 arise in the regression analyses. To avoid possible bias in- 

 troduced by outliers, we used a reweighted least squares 

 (RLS) method for the regression analyses (Chen et al., 

 1994). This method involves conducting a robust least 

 median of squares (LMS) analysis to identify outliers 

 (Rousseeuw and Leroy, 1987) and justifying the identified 

 outliers by using background information, followed by a 

 weighted LS analysis where justified outliers are weighted 

 by and other data have a weight of 1 (Chen et al., 1994). 

 In the two regression analyses (i.e. steps 1 and 2), L^ and 

 CViLJ were used as the independent variables and K and 

 CW(K) were used as the dependent variables. The reason 

 for this choice (instead of the other way around) is thatL„ 

 is often estimated more reliably and with much smaller 



