Power to detect linear trends In 

 dolphin abundance: Estimates from 

 tuna-vessel observer data, 1975-89 



Elizabeth F. Edwards 



Peter C. Perkins 



Southwest Fisheries Science Center, National Marine Fisheries Service, NOAA 

 P.O. Box 271, La Jolla, California 92038-0271 



Trends in abundance of dolphin 

 stocks affected by the tuna purse- 

 seine fishery in the eastern tropical 

 Pacific Ocean (ETP) are of intense 

 interest to a number of organiza- 

 tions concerned about the stocks' 

 continued survival (Hammond and 

 Laake 1983, Gerrodette 1987, Holt 

 et al. 1987, Buckland and Anga- 

 nuzzi 1988, Anganuzzi and Buck- 

 land 1989, Anganuzzi et al. 1991). 

 The most straightforward method 

 for estimating such trends is linear- 

 regression analysis of relative abun- 

 dance indices across time (e.g., 

 Anganuzzi and Buckland 1989). 

 Such abundance indices can be 

 derived from data collected by ob- 

 servers aboard the tuna vessels 

 (Buckland and Anganuzzi 1988, 

 Anganuzzi and Buckland 1989, 

 Anganuzzi et al. 1991). Linear 

 trends in abundance over successive 

 5-year periods have been reported 

 by Buckland and Anganuzzi (1988), 

 Anganuzzi and Buckland (1989), 

 and Anganuzzi et al. (1991). 



Power analysis provides a method 

 to quantify the probability of not 

 detecting low rates of change in 

 abundance over a specified time- 

 period. It also provides a method, in 

 cases where no statistically-signifi- 

 cant trends are apparent, for deter- 

 mining the steepness of change 

 necessary for its statistical detec- 

 tion given observed variability in 

 the data, i.e., detectable trend (Ger- 

 rodette 1987, Peterman 1990). We 

 use power analysis here to assess 

 the efficacy of weighted linear- 

 regression analysis for estimating 



linear trends in abundance of eight 

 stocks of ETP dolphins. While it is 

 instructive to evaluate the power of 

 conclusions about observed trends, 

 it is perhaps even more important 

 to determine the magnitude of 

 change required for detection of a 

 trend, given observed variability in 

 the dolphin abundance estimates. 

 Therefore, we also calculate detect- 

 able trends, in addition to power of 

 observed trends. 



We present here estimates of 

 observed trend, power to detect 

 trends, and detectable trends for 

 eight stocks of ETP dolphins, over 

 time-series of 5, 8, and 10-years, 

 assuming a two-sided hypothesis 

 with a = 0.10, using the noncentral 

 ^distribution for the alternative 

 hypothesis. Detectable trends were 

 estimated assuming Type I (a) and 

 Type H (/?) error levels equal 0.10. 

 We estimated power and detectable 

 trends for all three sets of time- 

 series to determine how much im- 

 provement might be expected by 

 increasing the number of years in- 

 cluded in the trend estimate. We did 

 not include longer time-series, as it 

 is unlikely that even a population 

 with reproductive and individual 

 growth rates as relatively slow as 

 ETP dolphins would follow a linear 

 trend for more than a decade, if that 

 long. 



Methods 



Relative abundance indices and 

 their associated bootstrap standard 

 errors for eight stocks of ETP 



dolphins during the years 1975-89 

 (Table 1, Fig. 1) formed the data- 

 base for the regression analyses 

 presented here. Indices and stan- 

 dard errors for 1975-87 were taken 

 from Anganuzzi and Buckland 

 (1989), and for 1988 and 1989, from 

 Anganuzzi et al. (1991). The eight 

 dolphin stocks included northern 

 offshore and southern offshore 

 stocks of the pantropical spotted 

 dolphin SteneUa attenuata, the east- 

 ern spinner dolphin SteneUa longi- 

 rostris orientalis, northern and 

 southern stocks of whitebelly spin- 

 ner dolphin (hybrid/intergrades be- 

 tween SteneUa I. orientalis and Ste- 

 neUa I. longirostris [Perrin 1990]), 

 and northern, central and southern 

 stocks of the common dolphin 

 Delphinus delphis. 



Observed trends 



We estimated linear trends in rela- 

 tive abundance for each of the eight 

 stocks over sequential series of 5, 

 8, and 10 years, using standard 

 weighted least-squares regression 

 (Wilkinson 1989). The slope of the 

 regression (b) estimates the trend in 

 abundance. The estimated standard 

 error of the estimated trend (st,) 

 indicates the variability associated 

 with the trend estimate. Weights 

 were the reciprocal of the square of 

 the bootstrap standard errors 

 (Buckland and Anganuzzi 1988). We 

 eliminated the estimate for 1983 

 from all analyses, because the pres- 

 ence of a very strong El Niiio that 

 year caused biologically unreason- 

 able estimates of abundance for 

 many of the stocks, in particular for 

 northern offshore spotted dolphins, 

 the stock affected by the fishery in 

 greatest numbers (Buckland and 

 Anganuzzi 1988, Anganuzzi et al. 

 1991). In the absence of any objec- 

 tive criteria for choosing which 

 stocks were (or were not) affected 

 by the El Nino, we elected to treat 



Manuscript accepted 1 June 1992. 

 Fishery Bulletin, U.S. 90:625-631 (1992). 



625 



