Edwards and Perrin: Annual dolphin mortality 



633 



BYAVEm, = YAVE Mltl - known kill 



J mjki 



SOijkl ' 



L ijk > 



where BYAVE- Mlkl is the bias of YAVE m , and V(YAVE Mjkl ) 

 is the variance of YAVE Mlkh calculated as 



V(YAVE 50:iH ) = (1/49) * l(YHAT ljklm - YAVE m )* 



(Cochran, 1977). 



The preceding equations produce 10 YAVE 50 's, 10 

 RBY M 's, and 10 CVY 50 's for each combination of fleet 

 size, percent coverage, and dolphin group type. 



The average of these 10 RBY^s is 



RBY m , k = (VIO) *liRBY 50l]kl ) 



and the average of the 10 CVY 50 's is 



CVY i0tjh = (1/10)* I CVY ; 



bOijkl  



The analytic confidence intervals for the 50 estimates 

 illustrate the influence of various combinations of fleet 

 size and observer coverage level, on the estimated 

 precision that may be associated with any individual 

 estimate. 



The analytic confidence intervals for individual rep- 

 licate estimates of mortality were calculated by the 

 International Mathematical and Statistical Library 

 (IMSL) routine SMPRR for ratio estimates (IMSL, 

 1987). This routine calculates confidence intervals for 

 ratio estimates using the analytic formula for approxi- 

 mate variance of a ratio. The procedure is based on 

 the assumption that a normal approximation to the 

 ratio variance is appropriate (Cochran, 1977; IMSL, 

 1987). Where data are sparse (fewer than 30 data 

 points in the data set; i.e., in most of the cases in 

 these simulations) this assumption is generally inap- 

 propriate (e.g., Cochran, 1977), but for single repli- 

 cates we had no computationally simple alternative. 

 Bootstrapping confidence intervals for these individual 

 replicates would have eliminated any need for a nor- 

 mal approximation but would have required signifi- 

 cantly more computer time to convey essentially the 

 same gross patterns and general message. 



Sampling distributions and 

 confidence intervals 



To facilitate interpretation of patterns seen in relative 

 bias and coefficients of variations, we plotted both fre- 

 quency distributions and analytic 95% confidence in- 

 tervals for an arbitrarily selected single set of 50 indi- 

 vidual replicate estimates of mortality derived under 

 various combinations of conditions. The frequency dis- 

 tributions and confidence intervals illustrate, in par- 

 ticular, variations due to differences between replicates 

 in the fleet selected, in contrast to the relative bias 

 and coefficient of variation, which pertain to sampling 

 properties of the estimator. Only one set of 50 repli- 

 cates, of the 10 sets generated under each combination 

 of fleet size and coverage level, is illustrated for each 

 combination because the general messages conveyed 

 by the figures were the same for all sets. 



Frequency distributions and confidence intervals are 

 plotted only for the cases of 5 and 20 boats at 25% and 

 75% coverage. Combinations of these values spanned 

 the range of fleet sizes investigated and enabled us to 

 examine whether problems might occur even with cov- 

 erage as high as 75% when fleet size is as small as 20 

 (or worse, 5) boats. 



The frequency distributions of the 50 estimates il- 

 lustrate graphically the influence of various combina- 

 tions of fleet size and observer coverage level on the 

 behavior (dispersion) of the estimator itself (kill/day). 



Results 



Relative bias (RB) 



RB was generally small overall, but exceeded the man- 

 agement objective of 5% for common dolphins and 

 whitebelly spinner dolphins when coverage was low 



(25%; Fig. 5). 



Coefficients of variation (CV) 



CV decreased with increasing percent coverage and 

 with increasing fleet size in both the "best" data group 

 (offshore spotted dolphin; Fig. 6) and in the "interme- 

 diate" data group (whitebelly spinner dolphin). CV de- 

 creased with increasing percent coverage but showed 

 no consistent effect of fleet size in the "worst" data 

 group (common dolphin). 



Sampling distributions 



Frequency distributions were affected somewhat by 

 fleet size (primarily by shifting the central tendency), 

 noticeably by percent coverage (primarily by decreas- 

 ing the spread of the distribution), and very strongly 

 by dolphin group type (primarily in terms of the num- 

 ber of modes in the distributions; Figs. 7, 8, and 9). 

 Bias and variability increased with small sample sizes 

 and non-smooth data distributions. 



