The Usage and Estimation of DELTA Means 



statement of the Problem 



The average number of marine organisms 

 caught per tow in trawl surveys, or observed in 

 a sample in other monitoring work involving dif- 

 ferent sampling schemes, is often used as an index 

 of a species abundance. When large areas are 

 sampled and the target species occupies only a 

 part of the total area, the occurrence of samples 

 with no organisms (i.e., zero observations) is un- 

 avoidable. Even in situations where a species is 

 known to be present in the entire area, the oc- 

 currence of zero observations in varying propor- 

 tions is still common. The frequency of zero data 

 is particularly high for mobile organisms such as 

 fish, and for plankton, which generally exhibit a 

 high degree of spatial variability or "patchiness". 



The presence of zero observations in monitor- 

 ing data complicates the data analyses on two 

 accounts. First, zero data are diRicult to interpret 

 because they may arise from natural patchiness, 

 low population density, undetected sampling gear 

 problems, and other reasons singly or combined; 

 and second, the presence of zero observations in- 

 creases both the coefficient of variability (i.e., the 

 variance-to-mean ratio) and the skewness of the 

 data. Because both higli variability and high 

 skewness contribute to non-normality in data, es- 

 timation methods based on normal theory do not 

 apply. Since logaritlunic transformations to cor- 

 rect skewness are not effective when many zero 

 observations are present, the usual approach is to 

 use order statistics such as the sample median and 

 nonparametric variance estimators of unknown 

 power. In extreme cases where over 50% of the 

 data are zeros, the median cannot be used because 

 it would always be zero regardless of obvious 

 differences among samples. 



This study addresses the estimation problems 

 described above and suggests the use of the ?>-mean 



as a more desirable statistic for describing the 

 relative abundance of marine organisms when 

 large numbers of zero observations are present in 

 the data. The performance of the ?>-mean relative 

 to three other statistics commonly used to estimate 

 population abundance is investigated through nu- 

 merical simulation. The results of this simulation 

 also serve to illustrate how the four statistics are 

 affected by the presence of zeros in lognormally 

 distributed data. 



The Delta Distribution 



The delta distribution first introduced by 

 Aitchison (1955) and later described by Aitchison 

 and Brown (1969), is a generalized form of the 

 lognormal model in which some of the observa- 

 tions may be zeros and the nonzero values follow 

 the lognormal distribution. The latter has two 

 parameters (\i) and (a^) which are the mean and 

 variance of the log-transformed observations 

 (Hastings and Peacock 1975). The delta distri- 

 bution has the same two parameters ( \i and a ) 

 of the undcriying lognormal model, plus a third 

 parameter (5) which is the proportion of zeros in 

 the data. Thus, the lognormal distribution is a 

 particular case of the delta distribution in which 

 the parameter (5) is zero (i.e., when the data do 

 not contain zero observations). 



Because the abundance of living organisms is 

 the result of an inherently multiplicative process 

 (i.e., the number of female parents times a fecun- 

 dity rate, times a survival rate), random samples 

 of naturally occurring organisms tend to follow 

 the lognormal distribution (Demetrius 1971). In 

 the case of marine organisms, fish in particular, 

 it has been shown that recruitment variability is 

 generally well described by the lognormal distri- 

 bution (Ilennemuth et al. 1980; Peterman 1981; 

 Ililbom 1985). Therefore, the delta distribution 

 appears particularly well suited to describe the 



DELTA Means 



311 



