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Fishery Bulletin 94(3). 1996 



1993). For current surveys with sampling intensity 

 proportional to stratum area, it would likely be better 

 to combine the small strata into ones with larger sample 

 sizes for calculating abundance estimates. Another way 

 to increase sample sizes for future surveys and to im- 

 prove survey efficiency in general would be to reduce 

 tow duration and use the time saved to sample at more 

 stations (Pennington and Volstad, 1991, 1994). 



It has been suggested that since the lognormal 

 model may be incorrect or not robust, the sample 

 average and variance are the preferred estimators 

 (Jolly and Hampton, 1990; Myers and Pepin, 1990; 

 Smith, 1990). Using finite population techniques, 

 Smith ( 1990) examined the performance of the esti- 

 mators based on the A-distribution and concluded 

 that for small populations the estimators are biased 

 and not robust to deviations from the model. But the 

 sort of model-based bias that Smith considered is not 

 a concern for marine surveys. Because for most, if 

 not all, marine surveys, the population size, i.e. the 

 total number of tows that could be made, is effec- 

 tively infinite, whereas Smith's simulations were 

 samples from populations of size 30. There is no rea- 

 son that the A-estimators should be unbiased if ap- 

 plied to samples from small populations. For Smith's 

 simulations, the usual properties of the lognormal- 

 based estimators are apparent if the small samples 

 (n-3) are assumed to be from a larger population. 

 That is, if the samples of size 3 are assumed to come 

 from a survey for which the possible number of tows 

 (the population size) is large, then the estimators are 

 unbiased (see Table 1 in Smith, 1990). The model- 

 based bias that Smith observed is a function of popu- 

 lation size, not a property as such of the A-estima- 

 tors or the size of the sample. 



What would cause concern is the possiblity that 

 the underlying distribution may have appeared to 

 be approximately lognormal but was not and that 

 the departure from lognormality caused the lognor- 

 mal-based estimates and inferences to be mislead- 

 ing. Myers and Pepin ( 1990) have claimed, motivated 

 by some simulations, that lognormal-based estima- 

 tors are very sensitive to undetectable deviations 

 from lognormality. But to test a model fairly, the al- 

 ternative models should be realistic. The nonro- 

 bustness that they observed was simply due to the 

 contamination of lognormal distributions with very 

 small values, the opposite of what causes the impre- 

 cision of abundance estimates from marine surveys, 

 i.e. the large catches (Pennington, 1991). It was not 

 only that the contaminating values were small, but 

 there was a relatively high probability that small 

 values would occur. Since In .r goes to minus infinity as 

 x approaches zero, these small values resulted in large 

 negative values on the log scale, which caused the ex- 



treme instability of the lognormal-based estimators in 

 Myers and Pepin's simulations. Aitchison ( 1986, p. 270) 

 made the same point when discussing a sensitivity 

 analysis of another log-based procedure. Analyzing ar- 

 tificial data is no different from analyzing real data; all 

 aspects of the simulated data should be examined care- 

 fully (see, e.g. McConnaughey and Conquest, 1992) to 

 ensure that the resulting conclusions are relevant. 



In practice, even if such small values were statis- 

 tically "undetectable" (Myers and Pepin, 1991), one 

 would know ( e.g. by looking at the data ) whether val- 

 ues could be arbitrarily close to zero and, if so, deal 

 with them appropriately as in Pennington ( 1991 ). The 

 small values that may occur after transforming abun- 

 dance data for a particular length class with an age- 

 length key (Myers and Pepin, 1991) will not cause 

 any problems if the original catch at length data are 

 distributed lognormally. This is because ln(o.r ) = In a 

 + In x, and, therefore, the log-based estimate of the 

 mean of ax is a multiplied by that for .r. 



The reason most often given for using the sample 

 estimates and not employing any modeling tech- 

 niques is that the sample average and variance are 

 always unbiased estimators (Myers and Pepin, 1990; 

 Smith, 1990). Lognormal-based estimators may be 

 slightly biased for some applications but they are not 

 overly influenced by the occasional huge catches and 

 therefore can have a considerably smaller mean 

 square error than the sample estimates for highly 

 skewed distributions (Conquest et al., 1996). 



There are problems if the sample estimates are 

 used for marine data (Lo et al., 1992). The estima- 

 tors are very sensitive to large catches and therefore 

 may be rather inefficient. Another difficulty is that 

 for the sample sizes common for marine surveys, the 

 distribution of the sample average may be far from 

 normal for these highly skewed distributions 

 (Sissenwine, 1978; McConnaughey and Conquest, 

 1992; Conquest et al., 1996). Thus the central limit 

 theorem cannot be invoked to assess the uncertainty 

 associated with the estimates or to make inferences. 

 Likewise, the distribution of the A-estimator may not 

 approximate a normal distribution for small samples, 

 but for skewed distributions it appears to converge 

 to normality more quickly than does the sample mean 

 (Conquest et al., 1996). For small to moderate sample 

 sizes, methods based on the lognormal model can be 

 used to make confidence statements. 



Acknowledgments 



The work presented in this paper was done while I was 

 visiting the Alaska Fisheries Science Center. The hos- 

 pitality and support shown me are greatly appreciated. 



