Fishery Bulletin 91 fl), 1993 



than that for the AM. Aitchison & Brown (1969) note 

 that "since the arithmetic mean involves both the lo- 

 cation and dispersion parameters, it is not a pure mea- 

 sure of the [response variable] under the lognormal 

 hypothesis: for this the geometric mean or median is 

 to be preferred." 



Monte Carlo simulations based on 

 crab trawl data 



Monte Carlo simulations consist of calculations made 

 on data sets whose elements are randomly selected 

 from specified probability distributions. This approach 

 permits an evaluation of various point-estimation pro- 

 cedures on the basis of expected outcomes. It also al- 

 lows a closer examination of individual cases than is 

 possible with a purely analytical approach and per- 

 mits an evaluation of the effects of sample size. For 

 this investigation, single values of mean density and 

 standard deviation were calculated for each cruise in 

 the two trawl locations along the Washington coast. 

 The means of these statistics were used to define two 

 representative lognormal distributions, which are iden- 

 tified as lognormal (4,2) for the coastal area and log- 

 normal (6,1) for the estuarine area. These distribu- 

 tions have means of 4.0 and 6.0, and standard 

 deviations of 2.0 and 1.0, respectively, for the log- 

 transformed variable; they will be referred to as LOGN 

 (4,2) and LOGN (6,1) (Fig. 1). From these two prob- 

 ability distributions, we created 1000 sets of simu- 



lated density data for each of 13 sample sizes 

 (2,4,6,8,10,15,20,25,30,35,40,45,50) using a pseudoran- 

 dom number generator (Minitab, Inc., University Park 

 PA). Sample sizes were selected to encompass the range 

 of values associated with ongoing trawl surveys. Table 

 2 presents descriptive statistics for each of these data 

 sets. (These data sets are archived on magnetic tape, 

 and access can be arranged through the authors. ) 



We investigated three methods of estimating cen- 

 tral tendency. The AM method consisted of computing 

 arithmetic means and traditional confidence intervals 

 (e.g., at 90% confidence) based on the Student's 

 /-distribution. The FM method used the Finney-Sichel 

 estimator for the mean of a lognormal distribution as 

 presented in Eq. (1) and (2). For confidence limits, the 

 method by Land (1971, 1975) as presented in Eq. (3) 

 and (4) was used. The GM method used e 1 as an esti- 

 mate of e^ LN , the geometric mean (or median) in the 

 untransformed scale. A 90% confidence interval was 

 derived as follows: 



exp 



Y-L 



1y_ 



exp 



y+ '-' t 



(5) 



This method estimates a different parameter (the me- 

 dian rather than the population mean) than the first 

 two methods. However, because the median is asymp- 

 totically a function of only a single parameter, u LN , the 

 GM method tends to give more stable estimates of its 

 parameter, and it is worthwhile to compare its perfor- 

 mance as another index of central tendency to the first 

 two methods. 



Comparison measures to evaluate 

 performance of the estimators 



We used the following measures of comparison to evalu- 

 ate the performance of the estimators: root mean 

 squared error (RMSE), deviation of the estimate from 

 the true parameter (BIAS), average length of the 90% 

 confidence interval (AVL), standard deviation of the 

 90% confidence interval length (SDCI), and percent 

 containment of the parameter by the confidence inter- 

 val estimate (PERCON). These were estimated as 

 follows: 



RMSE= 



V (estima ted pa ra meter 



^ from ; lh data set — true value)' 2 . 



1000 



Root mean squared error is a measure of the average 

 variation in the estimated mean relative to the true 



