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Fishery Bulletin 101(4) 



variations (Chen at al., 1992; also see Table 1), whereas 

 K is often less reliably estimated (Moreau, 1987). One of 

 the basic assumptions for a regression analysis is that the 

 independent variable is error free. In practice, this assump- 

 tion is often relaxed when the independent variable has a 

 much smaller error than the dependent variable (McArdle, 

 1988). The violation of the normal distribution assump- 

 tion for the errors in the regression analyses may bias the 

 test for the significance of the regression model and its 

 parameters using common parametric tests {F- or /-tests), 

 but does not necessarily result in biases in the regression 

 analysis (Sen and Srivastava, 1990)." 



Given K and L_, the growth increment during a unit of 

 time (i.e. year) can be calculated as 



Ai. =(L„-L„)(l-e"'). 



(3) 



where K and L^ are the true values without errors; n 

 indexes size class; and L,, is the middle point of the n"' size 

 class. With Equation 3, we can develop two approaches to 

 estimate the growth-transition matrix. One approach is a 

 Monte Carlo simulation. We can randomly sample H sets 

 ofK and L^ values from their joint distributions ( thus con- 

 sider their covariance) and then use them in Equation 3 

 to calculate H sets of AL for each size group. We can then 

 derive the probability distribution for AL from these H sets 

 of AL values for each size group. The Monte Carlo simula- 

 tion approach is straightforward but requires extensive 

 calculations, in particular when there are a large number 

 of size groups. It is also inconvenient to update the growth- 

 transition matrix when there are new growth data or large 

 changes in growth due to changes in the environment. The 

 second approach is analytic and not so straightforward, 

 but it is easy to update with new information and is less 

 computationally intensive. It is likely that the growth- 

 transition matrix for the Maine sea urchin fishery will 

 need to be updated because of possible changes in growth 

 caused by changes in the sea urchin population size and 

 its ecosystem. Thus we used the second approach, which is 

 described as follows. 



Assuming the uncertainties associated with the VBGF 

 parameters L^ and K are AL„ and AK respectively, where, 

 AZ._ e N{0.ai_ ) and AK e N{0.al ), we have 



L = L+AL ami K = K + AK. 



(4) 



where Z,„ and K^ are the estimated parameters. Replacing 

 the true values of L„ and K in Equation 3 with Equation 

 4 and using the approximation e-^ = 1 + AX for small AX, 

 we have 



AL„=(L„-L„)(\ -€-'') + 

 [aL_(1 -e *)-(!„- Z.JAA:^"*' -AL„AA:e''l = aZ„ -I- f„, 



(5) 



where 



AL„=(L„-L„){\-e-'') (6) 



e„=ALJ\-e-^)-{L^-L„)AKe-^ -AL„AKe~^- (7) 



Thus, the expected (mean) value of AL,, is AL^ and vari- 

 ance of AL^ can be estimated from Equation 7 as 



Var(AZ,„) = CT^d -e '^ ) +(L„ - L„)-(7j.e" 

 2Cov( L„. Ar)( 1 - e"'' )(ZL - i,„ )?'*". 



(8) 



Items with the order of three and above for AL„ and AK 

 are omitted in deriving Equation 8 from Equation 7. From 

 Equation 8, it is clear that the variance of the growth incre- 

 ment varies among different size classes. 



From AL,, estimated in Equation 6, an expected average 

 yearly growth increment was calculated for each size class. 

 The variability for the average yearly growth increment 

 was assumed to follow a normal distribution with a mean 

 of AL„and variance ofVar (AL,,) estimated from Equation 8. 

 This distribution was used to determine the vector of prob- 

 abilities of growing from size class k to other size classes. 

 If d,^„, and d„ are the lower and upper ends of size class d, 

 the probability of a sea urchin growing from size class n to 

 size class d can be computed as 



P..^., = I /( 



.v|AL„,Var(AL„)(/.v, 



(9) 



where x is a random variable having a density probability 

 distribution defined by /(.vj AL„,Var(AL„ )) with its expected 

 value of AL,;, and variance of Var(AL,^) (Quinn and Deriso, 

 1999). In the present study we assumed that the .v variable 

 was a normal density distribution function with a mean of 

 AL^, defined by Equation 6 and with a variance of Var ( AL,^ ) 

 defined by Equation 8. The probability of a sea urchin grow- 

 ing from one size to another was estimated for all size classes 

 to form the matrix. Negative growth increments were not 

 permitted. The largest size class acts as a plus group; there- 

 fore sea urchins in this group have a probability of 1 of 

 remaining in the group. The model contains 61 size classes, 

 each with 1-mm interval width, ranging from 40 mm in 

 size (midpoint value for size class from 39.5-40.5 mm) 

 to 100 mm. 



Because no negative growth was allowed, the summation 

 of the probabilities of a sea urchin of size class k growing 

 into all other size classes was smaller than 1 (because the 

 normal distribution is symmetric). This problem was avoided 

 by standardization which involved dividing the probability 

 of an urchin in a given size class n growing into each size 

 class by the summation of the probabilities of growing from a 

 given size n to all the size classes. All calculations were done 

 in MS-Excel©( Microsoft Office 2000, Microsoft Corporation, 

 Redmond, WA). A worksheet for estimating a growth-transi- 

 tion matrix as described above is available upon request. 



Results 



The LMS analysis suggested that the logarithmic K and 

 L^ data for the barren habitat in the Southwest area was 

 an outlier in the K and L,, regression analysis (Fig. 2). The 

 estimated K and L^ values for the barren habitat in the 

 Southwest had CVs over 120% and 24%, respectively, much 



