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



741 



Ln(/.. 



0.0 - 



4 



-0.5 



-1.0 

 -1.5 

 -2.0 

 -2.5 J 



4.2 



4.3 



4.4 



4.5 



4.6 



 Observed 



 Outlier 

 — Predicted 



Figure 2 



The regression analysis of logarithmic K and /-„ for different 

 locations and habitats of Maine sea urchins. 



larger than the estimates for other locations and habitats 

 (Table 1). This was the only site where the K estimate was 

 not significantly different from (thus the VBGF was not 

 significant). We thus concluded that this data point was 

 an outlier because of the poor fit of the VBGF, and subse- 

 quently it was given a zero weight in the RLS analysis. The 

 RLS regression equation for K and L^ was estimated by 



LniK) = 8.653 - 2.3777 LniLJ, 



P=0"0038, adj. r2=0.94. 



(10) 



The standard deviations for the intercept and slope were 

 1.2605 and 0.28923, respectively. The P value for Equation 

 10 indicates that the regression model is significant. The 

 adj. r^ is the coefficient of determination adjusted for the 

 sample size, suggesting 94% of the variance in \n{K) could 

 be explain by the model. 



The LMS analysis of the CVs of parameters K and L_ 

 also suggested that the barren habitat in the southwest 

 area was an outlier because it had an exceptionally large 

 CV for K (Fig. 3). We thus concluded that this data point 

 was an outlier and should be given a weight of zero in the 

 RLS analysis. The RLS regression equation for the CVs of 

 parameters K and L„ was estimated by 



50 100 



Midpoint of size class (mm) 



150 



Figure 4 



The expected annual growth increment for Maine sea 

 urchins of different size classes. 



CVm = 0.189 + 1.5602 CV (LJ, 



P=0.034, adj. 



r2 = 0.76. (11) 



The standard deviations for the intercept and slope were 

 0.0561 and 0.42319, respectively. The P value suggested the 

 regression model was significant (P<0.05). The value of r^ 

 suggests 76% of the variance in CV(K) could be explained 

 by the model. 



The average CV for LJs of different areas and habitats 

 was 15%. The L^ was assumed to have a value of 100 mm 

 in this study as discussed previously. This gave the L^ a 

 standard error estimate of 15.0 mm, making its 95% con- 

 fidence intervals 70 mm to 130 mm. The iiT value was esti- 

 mated to be 0.1006 using Equation 10 and L^ of 100 mm. 

 Using Equation 11 and the CV for L,^ the CV for K was 

 estimated to be 42.3%, which yielded the value of 0.0426 

 for the standard error for K. 



The annual expected growth increment decreased quick- 

 ly with sea urchin size (Fig. 4). The largest expected annual 

 increment was 6 mm for the smallest size class (39.5-40.5 

 mm) included in the study. The variance for annual growth 

 increments calculated by using Equation 8 was large for 

 small sea urchins. It decreased initially with size, reaching 

 the smallest value at the 59 mm size class (58.5-59.5 mm), 

 followed by a progressive increase with size (Fig. 5). The 

 expected annual growth increment for the largest size class 

 included in this study had the highest variance, which was 

 over eight times as high as the smallest variance (Fig. 5). 



The probability distribution of annual growth increment 

 varied among size classes (Fig. 6), reflecting the differences 

 in variances associated with different size classes. The last 

 size class was a plus class, with the probability of staying 

 m the same size class being 1. Figure 6 clearly indicated 

 that no negative growth was allowed. 



Discussion 



Great variation in growth was observed in the Maine sea 

 urchin stock (Vadas et al., 2002). Such a pattern of variation 

 was reflected in estimating the VBGF parameters for dif- 



