Finally, the lognormal data were converted into 

 "delta-distributed" data by adding an equal number 

 of zeros to each data set and increasing that num- 

 ber in each of the 15 simulations conducted. The 

 zeros added in these simulation were 5, 10, 15, 

 20, . . , up to 75, resulting in data sets with 105, 



110, 115, , 175 observations and 4.8 to 43 



% of zeros. 



The h-mean, the sample mean, the median, 

 and the geometric mean with their respective stan- 

 dard errors were computed separately for each of 

 the three lognormal data sets prior to adding any 

 zeros, and then recomputed after adding zeros in 

 each of the 15 simulations. The 95%C/'s for 

 each statistic were also computed and normalized 

 as the percentage of the statistic value represented 

 by the total width of its confidence interval. The 

 purpose of this normalization was to facilitate the 

 comparison of 95%C/'s among simulations and 

 across data sets with different variability. Direct 

 comparison of 95"/oC/'s between statistics, how- 

 ever, can be misleading because the standard er- 

 rors of the sample and geometric means underes- 

 timate the true variance. As a result, the 95%C/'s 

 for the sample mean and the geometric mean 

 reflect a coverage which is less than the nominal 

 95%, whereas the 95%C/ for the 8-mean reflects 

 an accurate coverage and the 95% C/ for the me- 

 dian is very conservative (i.e., its coverage could 

 be closer to 99% than to 95%). The 95%C/ of 

 the median was estimated using the two order 

 statistics given by Snedecor and Cochran's (1980) 

 nonparametric formula: 



n+ 1 



1.96 7« 



(V) 



where /; is the sample size ( 1 00 to 175 observations 

 in this simulation study). 



Simulation results: The simulation results are 

 presented in Figure la- lb for low variability data, 

 in Figure 2a-2b for moderate variability, and in 

 Figure 3a-3b for high variability. The magnitudes 

 of the four statistics for each simulation are shown 

 in Figures la, 2a, and 3a. Although the biased 



sample mean and the unbiased 5- mean did not 

 coincide, they tracked each other remarkably well 

 over the entire range of zeros and variability sim- 

 ulated. Also the median and geometric mean 

 tracked each other well for moderate and high 

 variance data, but the geometric mean was badly 

 affected by the increasing number of zeros when 

 variability was low. Except for highly variable 

 data, the geometric mean was always smaller than 

 the median (very much so at low variance). This 

 indicates that, in the presence of zeros, the geo- 

 metric mean is not a reliable estimator of the 

 middle point of the data which is always accurately 

 described by the median. The two estimators of 

 the population mean (i.e., the sample mean and 

 the 8- mean), were always located to the right of 

 the middle point of the data (as expected) and 

 the separation increased with the variability in the 

 data. Except for the geometric mean in the case 

 of low variability data, the magnitude of the four 

 statistics declined at similar rates in response to 

 the addition of zeros to the lognormal data. 



The relative widths of the 95 %C/ for each 

 statistic are shown in Figures lb, 2b, and 3b. The 

 95%C/ of the h-mean (the only CI known to be 

 accurate for "delta" data) increased with the data 

 variability, but its relative witdth was almost un- 

 affected by the number of zeros in the data. Al- 

 though the 95%C/ of the sample mean was also 

 quite insensitive to the number of zeros present, 

 its coverage was not accurate because the sample 

 mean standard error underestimates the true vari- 

 ance. Finally, the 95%C/'s of the geometric mean 

 and median were erratic and had very different 

 widths, especially with moderate and high vari- 

 ability. Except at low variability, both 95%C/'s 

 were greatly affected by the number of zeros in 

 the data. 



In summary, the 5-mean and its 95%C/ be- 

 haved predictably and consistently over the ranges 

 of data variability and proportion of zeros simu- 

 lated. The 8-mean increased in magnitude with 

 the data variability and decreased as the proportion 

 of zeros increased. The decrease caused by the 

 presence of zeros was more pronounced when the 

 data had high variability. The relative width of 



DELTA Means 



315 



