NOTE Edwards and Perkins: Detecting linear trends in dolphin abundance 



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Detectable trends 



We estimated detectable trends (rj) for all data-series. 

 All estimates of r^ assume error levels a = p = 0.lO. 

 Detectable trends were estimated by determining the 

 value of Delta (do.g) that returns a power value of 0.90 

 from the net algorithm. As before, input value for IDF 

 was 71 - 2, and for si, was the value estimated from the 

 weighted regression. Then the value of b generating 

 the desired power level (bnct) is 



bnct = f^o.g/Sb 



and the detectable initial trend per year is 



Td = bnct/Ai. 



Results 



Observed trends 



The majority (151/192; 79%) of the series showed no 

 significant trend (specific data available from the 

 authors). Of those that did, most showed decreases 

 prior to the mid-1980s and no consistent trends since. 

 Where population-abundance indices changed relatively 

 regularly over time, successively longer time-series re- 

 tained the same general patterns as found in shorter 

 series. For example, observed trends were significantly 

 negative for northern offshore spotted dolphin during 

 the 5-year series 75/79 and 77/81, the 8-year series 

 75/83, 76/83, and 77/84, and the 10-year series 75/84 

 and 76/85. Similarly, 5-year negative trends were also 

 reflected in 8- and 10-year series for southern off- 

 shore spotted dolphin, eastern spinner dolphin, north- 

 ern whitebelly spinner dolphin, and central common 

 dolphin. 



Data were so sparse and variable for southern white- 

 belly spinner dolphin and southern common dolphin 

 that little can be said about trends in these stocks. 

 Northern common dolphin were the only species for 

 which trends may have switched during the period of 

 investigation (from negative during earlier years, to 

 positive more recently); but it is obvious that here, as 

 in the other series, the pattern in trend estimates is 

 simply a function of the length of the series selected 

 and its placement in time. 



Power 



Power to reject a false null hypothesis increases with 

 increases in either or both of series length (as degrees 

 of freedom increase) or 6 (offset) (Fig. 2), but for TVOD 

 the increases generally were not sufficient to be of 



practical use. Where no significant trends (slopes) were 

 found, power to detect a false null hypothesis was low, 

 averaging 20-30% in most cases and never exceeding 

 60%. Power for each test was small because the alter- 

 native hypothesis for these power calculations was 

 taken to be the observed slope, which was usually fair- 

 ly small, and also because scatter around the regres- 

 sion line tended to be large. Therefore the null and 

 alternative distributions overlapped considerably. The 

 low power of these tests simply means that if the true 

 slope equaled the observed slope, the power to distin- 

 guish the true slope from a slope of zero (i.e., no change 

 in abundance) would be quite small in most cases. 



Detectable trends 



The range of detectable trends decreased rapidly with 

 increasing series-length in all cases (Fig. 3), as this in- 

 creases the degrees of freedom (number of data points). 

 The decrease is misleading in most cases, however. 

 Although the improvement in ability to detect smaller 

 trends with longer time-series appears dramatic, in 

 most cases even the smallest detectable trends are still 

 much too large to be of use. 



Even with as many as 10 years of data in a series, 

 linear trends less than about 10% per year could 

 be detected consistently only for northern offshore 

 spotted dolphin. For all other stocks, trends of at least 

 15-20% per year would be required to produce a signifi- 

 cant result (Fig. 2). Series lengths would have to be 

 such that populations more than doubled or decreased 

 to zero in order for the change to be statistically detect- 

 able. This would require series lengths of at least 10 

 years. 



In many cases, where significant trends were found, 

 these trends were of lesser magnitude than the esti- 

 mated detectable trend. This occurs because the esti- 

 mated detectable trend is the expected value of the 

 alternative distribution. Any trend value which falls 

 below this expected value, but which also falls above 

 the Type-I error limit for the null distribution, will be 

 assumed significantly different from the null even 

 though the trend could actually belong to either 

 distribution. For example, if the Type-I error limit for 

 the null distribution occurs at a trend value of 0.75 (i.e., 

 if the cut-off point for values assumed to belong to the 

 null distribution is 0.75), and the expected value (i.e., 

 the mean) for the alternative distribution falls at 0.85, 

 any trend value within the range 0.75-0.85 will be 

 assigned to the alternative distribution even though it 

 is smaller than the expected value of the alternative 

 distribution. In practice for the ETP data, this effect 

 is unimportant compared with the overall problem of 

 high variability obscuring the possibility of detecting 

 managerially-relevant trends in abundance (i.e., the 



