1 1( 



Fishery Bulletin 91(1). 1993 



that resulting estimates be normally distributed 

 according to the Central Limit Theorem. However, 

 Hansen et al. (1983) state, "When surveys use rela- 

 tively small samples, the samples may be too small for 

 the application of the theory [for large samples] to be 

 essentially assumption-free." Our approach is model- 

 based. As long as one is restricted to samples that may 

 not be considered "acceptably large" (and further ham- 

 pered by considerable skewness caused by extreme val- 

 ues), use of a model-based approach is not unwarranted 

 (Little 1983). 



With regard to robustness, Myers & Pepin (1990) 

 argue that exclusive use of a lognormally-based esti- 

 mate can be sensitive to model assumptions, leading 

 to possible bias and reduction in efficiency. Because 

 contamination of a lognormal distribution with data 

 from similarly-shaped distributions (e.g., Weibull or 

 gamma) is difficult to detect for sample sizes less than 

 40, they suggest using lognormally-based estimators 

 of abundance only when there is evidence that the 

 underlying population is lognormal. Obviously, use of 

 transformations and model-based estimators is a 

 "double-edged sword," and these procedures should not 

 be applied indiscriminately. When appropriate (e.g., 

 Table 1 ), however, significant improvement in the rela- 

 tive efficiency of the sample average and, in particu- 

 lar, the estimated variance, can be realized (e.g., Finney 

 1941, Koch & Link 1970, Myers & Pepin 1990). 



A comparative approach 



If nothing is known about the spatial distribution of 

 an organism, the sampling plan must be designed to 

 determine distribution patterns as well as population 

 size. Knowledge of the distribution pattern aids in se- 

 lection of the proper estimation procedure. Based on 

 the arguments presented above, combined with the 

 rather ubiquitous nature of overdispersion in the ma- 

 rine environment, we prefer an approach based on three 

 estimators, namely the arithmetic mean, the geomet- 

 ric mean, and the Finney-Sichel estimator of u. By 

 taking a comparative approach, one may be reason- 

 ably certain of apparent trends in the data if the trend 

 is consistent for the three estimators. For the estua- 

 rine crab population illustrated in Figure 7a, the par- 

 allel behavior of the estimates corresponded to chang- 

 ing values of Y coupled with nominal changes in s 2 Y 

 (Table 1). In this case, there is no evidence to suggest 

 that conventional analysis of catch data (i.e., using the 

 AM method) was less than adequate. However, trends 

 in the estimates may, on occasion, be opposed to one 

 another, as was demonstrated for the coastal crab popu- 

 lation (Fig. 7b). The AMs suggest a precipitous drop in 

 abundance occurred during the interval between 



Cruises 3 (with two extreme values) and 4, whereas 

 both the FM and GM procedures indicated a moderate 

 increasing trend during the same period. From the 

 behavior of the three estimators, we conclude that be- 

 tween Cruises 2 and 4, u lN (and u) may have increased 

 slightly, but cr 2 LN (and thus the skewness of the distri- 

 bution) probably increased and then decreased, affect- 

 ing the FM and AM estimates (the latter more strongly) 

 but not the GM estimate. This is verified by checking 

 the Y and s 2 Y values in Table 1. Changes in skewness 

 relate directly to the size of the larger catches and, 

 hence, the degree of spatial aggregation in the popula- 

 tion. Plotting the three estimators and relating the 

 trends back to changes in Y and changing s 2 Y has 

 yielded some insight into the behavior of the estima- 

 tors. It has also allowed us to extract more informa- 

 tion about the crab population than if we had used 

 only one estimator, the AM. In cases such as this, where 

 there is significant disagreement among the estima- 

 tors, the data set should be carefully evaluated as to 

 its underlying probability distribution and the most 

 appropriate index selected. If the lognormal distribu- 

 tion is reasonable, the GM may well be the preferred 

 estimator (Aitchison & Brown 1969); use of the GM 

 may be advantageous because it is relatively insensi- 

 tive to extreme values ( particularly so for highly-skewed 

 data) in terms of accuracy and precision. Since catch 

 coefficients are not routinely considered with trawl- 

 survey data of this type, the resulting stock-size 

 estimates are, strictly speaking, indices of abundance 

 (Caddy 1986). Under these circumstances, it may 

 be advantageous to use an alternative estimator of 

 central tendency, such as the GM, to generate the 

 index. 



Acknowledgments 



We thank Drs. David Armstrong and Donald Gun- 

 derson of the University of Washington School of 

 Fisheries for access to unpublished data and for their 

 comments and suggestions. Their research program 

 was supported by an institutional grant from Wash- 

 ington Sea Grant (#NA86AA-D-SG044 Project R/F-68) 

 and the U.S. Army Corps of Engineers (#DACW67-85- 

 C-0033). In addition, we thank the following for help- 

 ful comments and suggestions on previous drafts 

 of this paper: Dr. Robert Crittenden (University of 

 Washington, School of Fisheries), Charles Douglas 

 Knechtel (FishStat Statistical Helps Service, Seattle), 

 Stephen J. Smith (Marine Fish Division, Bedford 

 Institute of Oceanography), and two anonymous 

 reviewers. 



