Anganuzzi: Detecting trends in dolphin abundance indices 



185 



6000 



5000 



" 4000 



! 3000 



• 2000 



1000 



1992 



Figure 1 



Smoothed trends in abundance of the northern offshore stock of spotted dolphin Stenella 

 attenuata in the eastern tropical Pacific Ocean. Broken lines indicate -85% confidence 

 limits. Horizontal lines correspond to 85 r t confidence limits for the 1990 estimate. If 

 both the 1990 confidence limits lie above the upper limit for an earlier year, abundance 

 has increased significantly between that year and 1990 lp<0.05); if both limits lie below 

 the lower limit for an earlier year, abundance has decreased significantly. 



is often not met, but it can be 

 shown that results are robust-to- 

 moderate departures from it. The 

 third condition is not met for es- 

 timates that are close in time, 

 due to the correlation introduced 

 by smoothing, and the relative 

 importance of this effect is dis- 

 cussed further below. 



Simulation comparison 

 between tests 



To illustrate the difference in per- 

 formance between methods, a 

 simulation study was carried out. 

 Series of estimates were simu- 

 lated by assuming different sce- 

 narios of underlying trends in the 

 population over a period of 25 yr. 

 The following scenarios were 

 chosen. 



To obtain confidence intervals of the smoothed esti- 

 mates, Buckland et al. (1992) combined smoothed esti- 

 mates and bootstrap replication using the following 

 procedure. First, they obtained 79 bootstrap estimates 

 of the abundance index for each year. Next, they built 

 bootstrap replicates of the series of estimates by tak- 

 ing, for example, the first bootstrap estimate for each 

 year to obtain the first replicated series. They smoothed 

 each replicated series, thereby obtaining 79 smoothed 

 estimates for each year. Finally, they sorted the 

 smoothed estimates within each year and obtained 85% 

 confidence intervals based on the percentile method 

 (Buckland 1984). The median of the smoothed boot- 

 strap replicas is considered to be the best smoothed 

 estimate. 



The confidence intervals thus calculated allow a di- 

 rect comparison between estimates. If the confidence 

 intervals for two estimates do not overlap, then they 

 are significantly different at a level of -5%. An ex- 

 ample based on estimates of relative abundance for 

 the northern stock of offshore spotted dolphin is shown 

 in Fig. 1. Since the last and first smoothed estimates 

 are too variable, Buckland et al. (1992) recommend 

 excluding them from the comparisons. The significance 

 level is approximate, since it depends on normality of 

 the estimates, homogeneity of the variances of esti- 

 mates, and their independence. The second condition 



Stable population Population 

 exhibits no trend over the simu- 

 lated period. This scenario pro- 

 vides us with an estimate of the 

 probability of a Type-I error de- 

 tecting a trend when, in fact, there is none, or obtain- 

 ing a "false positive." Under this scenario, a percent- 

 age of detected trends close to 5% would be expected 

 for a test based on a significance level of 5%. 



Rapid decline Population remains at a constant level 

 for a period of time, and then declines sharply over a 

 period of 3yr to 50% of its previous level. After the 

 decline, the population recovers at a rate of 5% per yr. 



Steady decline Population declines exponentially at 

 an annual rate of 10%. 



Steady cycle Population follows a sinusoidal change 

 over the simulated period, completing one cycle over 

 the 25 yr. Amplitude of the cycle is 30% of the original 

 population size. 



These scenarios are intended as a means of high- 

 lighting properties of the different tests and not as 

 an exhaustive list of possible situations. Each 

 scenario was replicated 100 times for different combi- 

 nations of sources of variation. Variability in the esti- 

 mates was assumed as coming from two different 

 sources: (1) Interannual variability, resulting in a 

 point estimate for each year t of 



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