200 



Fishery Bulletin 92|1). 1994 



Game and National Marine Fisheries Service to 

 examine utility of the WLRAW method. Samplers 

 collected two groups of fish from each sampled land- 

 ing. For each group a container that holds 22.7 kg 

 of fish was filled with fish regardless of species. 

 Then the sampler obtained total group weights to 

 the nearest lb (0.45 kg) for each species and the total 

 length of each fish was measured to the nearest mm. 

 I converted weights to kg. I changed lengths to deci- 

 meters to minimize potential scaling problems in the 

 computations. Before using the WLRAW method, I 

 combined groups within a landing because they may 

 not be independent. 



I first used data for all months during 1991 from 

 all ports between Morro Bay and Crescent City, 

 California, to develop, test, and time the software. 

 Results of the test runs are described briefly in the 

 Results and Discussion section. More detailed re- 

 sults are presented for a more typical application of 

 the method. Investigators are more likely interested 

 in results from a smaller number of samples taken 

 from more restrictive scales of time and area than 

 from data sets like the one used in the preceding 

 example. I used data for chilipepper rockfish taken 

 during July and August 1991 from Morro Bay to 

 illustrate use of the method. 



Results and discussion 



The data from all ports consisted of measurements 

 from 7,687 fish taken in 186 samples. The procedure 

 required 1.6 seconds, compared with 18.8 seconds for 

 the Gauss-Newton method. The Gauss-Newton and 

 new methods produced parameter estimates that 

 were identical to six decimal places. Predicted 

 weights were very close to the results of Phillips 

 (1964), who used data from individually measured 

 fish. Sums of squares plotted against P' indicated 

 that there were no local minima. Residuals were not 

 related to weight, indicating that the additive error 

 assumption is correct. Sometimes transformation of 

 (3' to ln(P') when estimating parameters of power 

 equations avoids problems due to curvature (Rat- 

 kowsky, 1983). Transformation was tried and pa- 

 rameter estimates were identical to the results when 

 P' was not transformed. When P' was transformed, 

 the procedure required more time to complete, so the 

 transformation was not used. 



Data were available for 583 fish taken from 13 

 samples taken in Morro Bay, during July and Au- 

 gust 1991. There were no strong trends between the 

 residual and expected weight (Fig. 1). There was a 

 tendency for absolute values of residuals to be nega- 

 tively correlated with the number offish in a sample 

 (Fig. 2A). The tendency was reduced when residu- 



04 0.5 06 07 0.8 0.9 1 



Expected Weight (kg) 



Figure 1 



Residual of average weight (kg) as a function of 

 expected weight (kg) for chilipepper rockfish 

 (Sebastes goodei) collected in samples taken from 

 Morro Bay during July and August 1991. 



als were multiplied by sn , as expected under the 

 assumption that variance is proportional to the in- 

 verse of sample size (Fig. 2B). Also, n produced the 

 lowest asymptotic standard errors of the parameter 

 estimates of the six weighting factors explored 

 (Table 1). The results shown in Table 1 and Figures 

 1 and 2 indicated that the additive error model with 

 weighting by n was appropriate for these data. 

 Bootstrap estimates of standard error using the new 

 method were higher than asymptotic estimates us- 

 ing the Gauss-Newton method. The bootstrap and 

 asymptotic normal confidence intervals were narrow 

 and similar within the range of most observed av- 

 erage weights but diverged when expected weight 

 was greater than 0.75 kg even though individual 

 fish of larger size occurred in many of the samples 

 (Table 2). The bootstrap confidence intervals were 

 skewed at the larger sizes. However, the bootstrap 

 estimates of absolute bias were less than 0.01 kg 

 except they were -0.01 kg for 450-mm fish and -0.02 

 kg for 500-mm fish. All estimates of the absolute 

 value of a were about 0.015, which indicated that a 

 could have been ignored for this set of data. 



The new WLRAW method performed well. Good 

 fits to the data were obtained and the residuals 

 agreed with the assumptions. Approximate confi- 

 dence limits indicated that precise estimates of ex- 

 pected weight are obtained with a small number of 

 samples under field conditions for sizes of fish 

 within the range of most observed average weights. 

 The method is fast when used on a work station or 

 on a modern personal computer. The new method is 

 10 times faster than using the Gauss-Newton ap- 



