NOTE Lenarz: Estimation of weight-length relations from group measurements 



199 



I then obtained an estimate of a (a') corresponding 

 to p" by using the standard least squares linear re- 

 gression with zero intercept method. I used a non- 

 linear least squares procedure to obtain the estimate 

 of (3 ( B )■ This procedure was analogous to finding 

 the transformation, Lf, that minimized the sum of 

 squares about the linear regression (Eq. 4). Using 

 this procedure, I estimated brackets for ensuring 

 that the searching range included P with the proce- 

 dure MNBRAK (Press et al., 1989). Then I used the 

 iterative procedure BRENT (Press et al., 1989) to 

 obtain the final estimate. BRENT uses parabolic 

 interpolation to minimize the sum of squares as a 

 function of (3'. Convergence is assumed when the 

 procedure does not change the value of P' more than 

 a tolerance specified by the user. As previously 

 stated, observations were weighted by n to stabilize 

 the variance. I implemented the WLRAW method in 

 double precision using Sun FORTRAN for a Sun 

 SPARC2 work station. 



Bootstrap approximations of confidence intervals 

 about the line were calculated for the new method. 

 The literature contains a variety of bootstrap meth- 

 ods proposed to approximate confidence intervals 

 (e.g., DiCiccio et al., 1992). I used the nonparamet- 

 eric BC method of Efron (1987) because it often 



a 



produces good results and is relatively easy to use. 

 BC a stands for accelerated bias corrected boot- 

 strap confidence intervals. Efron (1987) showed that, 

 in the parametric case, the method is approximately 

 correct if a transformation to a normally distributed 

 variable exists. The transformation does not need to 

 be known and the variance does not need to be con- 

 stant. While the correctness of the BC has not been 



a 



mathematically proven for nonparametric cases, such 

 as the WLRAW, Efron (1987) stated, "...empirical re- 

 sults look promising." The BC a confidence limits of an 

 estimate of parameter 8, 0, are 



IBS(N(z[a]))<6<IBS(Nlz[l-a])) 



(5) 



IBS(P) is the value of 6 that corresponds to the per- 

 centile P of the cumulative bootstrap frequency dis- 

 tribution. N(Z) is the percentile of the cumulative 

 normal probability distribution that corresponds to 

 the standard deviate Z. z[a] is given by Efron 

 (1987) as 



z[a] = z n + 



Zn+Z 



l-a(z 0+ z"») 



(6) 



z ,al is the standard deviate that corresponds to the 

 a percentile of the normal cumulative distribution. 



2 is the standard deviate of the normal cumulative 

 distribution that corresponds to the percentile that 

 corresponds to in the cumulative bootstrap fre- 

 quency distribution. Efron (1987) called z Q the bias 

 constant. Efron called a the acceleration constant. It 

 is related to the skewness of the bootstrap frequency 

 distribution. Efron gave the following approximation 

 for a: 



a ~ 



<2>J 



;=i 



(7) 



where U , 



3(A) 





ef ] -e 



A 

 estimate of 6 whenyth sample has 

 a very small amount of extra 

 weighting (A). 



If a and z are zero, then Equation 7 becomes the 

 percentile method that is the most frequently used 

 bootstrap method in the fisheries literature (e.g., 

 Sigler and Fujioka, 1988). 



I chose to approximate 90% confidence bands 

 rather than 95% or 99% bands because 90% non- 

 parametric bootstrap intervals tend to perform bet- 

 ter than intervals that cover a wider portion of the 

 distribution (Efron, 1988). Following the advice of 

 Efron, I used 1,000 bootstrap replicates. 



Cammen (1980) used the general nonlinear re- 

 gression program of BMDP to estimate the param- 

 eters of Equation 2, except that he assumed that the 

 error term is multiplicative and used total sample 

 weight instead of average weight as the dependent 

 variable. The BMDP program uses the Gauss-New- 

 ton algorithm. I used the same algorithm in the 

 nonlinear regression program of the SAS package 

 (SAS Institute Inc., 1989) on a Sun SPARC2 to com- 

 pare parameter estimates and execution times with 

 the new method. Since the correct error model is not 

 known, I also estimated the parameters using 

 no _weighting__and weight set to 1/W ,1/W, 2 , 

 n l IW r and n ] I W", and compared asymptotic stan- 

 dard errors of the parameter estimates. The new es- 

 timation procedure is simpler than the Gauss-New- 

 ton approach because it searches for the least 

 squares by iteratively changing the value of one 

 parameter instead of two. 



I used data collected on chilipepper rockfish 

 {Sebastes goodei) by a cooperative landing sampling 

 program of the California Department of Fish and 



