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Fishery Bulletin 9 1(2), 1993 



variability) may still affect estimates from year to year. 

 This problem may not be exclusive to the TVOD esti- 

 mates; interannual variability also seems to affect 

 estimates of relative abundance derived from research- 

 vessel data (Wade & Gerrodette 1992). These biases 

 and imprecise estimates will affect the performance of 

 statistical tests designed to detect trends and, ulti- 

 mately, our ability to draw conclusions about the sta- 

 tus of populations. 



For the analysis of trends in the EPO dolphin stocks, 

 Buckland & Anganuzzi (1988) applied a linear test for 

 trends over a moving period of 5 yr, although they ex- 

 pressed concern about the low power of such a test. 

 Edwards & Perkins (1992) extended the moving time- 

 frame to 10 yr to increase the power. However, such a 

 test still shows some undesirable properties. Given the 

 inadequacy of the tests based on linear regressions, 

 Buckland et al. (1992) proposed a different procedure, 

 based on a nonparametric regression, which addresses 

 some of the problems exhibited by the linear test. 



In this paper, the characteristics of these tests are 

 discussed and compared by analyzing their performance 

 in a number of simulated scenarios. 



Current tests for trends 



Linear tests 



Buckland & Anganuzzi ( 1988) tested for linear trends 

 over successive 5 yr periods by carrying out a weighted 

 linear regression of abundance index vs. time. Each 

 individual estimate was weighted by the inverse of its 

 variance, calculated by applying a bootstrap procedure. 

 The null hypothesis for the test is that no change has 

 occurred in the population, i.e., that the slope of the 

 regression is equal to zero. As the authors noted, the 

 test has low power since it estimates precision from 

 the deviations of only five estimates from a straight 

 line. Power can be increased by extending the moving 

 time-period to incorporate more years in the test. Un- 

 fortunately, this also increases the probability of vio- 

 lating the assumption of a constant rate of change 

 implicit in the linear model being fitted to the esti- 

 mates (Edwards & Perkins 1992). 



The linear test also fails to consider the precision of 

 estimates adequately. Variances of the estimates are 

 not taken into account except as weights in the regres- 

 sion. As a consequence, only the ratios of the variances 

 between estimates are relevant, and not their absolute 

 values. For example, if for any given series of esti- 

 mates we double the variance of each individual esti- 

 mate, the results of the test will remain unchanged. 



Weighting by the inverse of the variance can also 

 present other problems. Suppose, for example, that 



the variance of the estimate is not independent of 

 the estimate itself, but that the variance is correlated 

 with the estimate, i.e., the coefficient of variation 

 (CV=ratio of standard error to point estimate) is con- 

 stant. In this case, a very low estimate (and especially 

 in the case of an outlier) with a correspondingly 

 small variance will become an influential observation. 

 A linear test for trends will then indicate that there 

 was a decline in the population if that estimate is 

 at the end of the moving period, or a significant in- 

 crease if it is at the beginning. An example from the 

 EPO dolphin abundance estimation is the case of the 

 1983 index of relative abundance for the northern 

 stock of offshore spotted dolphin Stenella attenuate!, 

 which was an anomalous index as a result of a very 

 strong El Nino event (Fig. 1). In such cases, where 

 the error distribution of the estimates seems to be 

 better approximated by a lognormal distribution, 

 it would be more appropriate to apply weights (wt) 

 defined as 



wt = ln(l+CV 2 ) 1 . 



(1) 



For the comparisons in this paper, two versions of the 

 linear test are applied: the original 5yr linear test 

 with inverse variance weighting applied by Buckland 

 & Anganuzzi (1988), and a 10 yr linear test with 

 weights as described by Eq. 1. 



Smoothed trends 



The approach taken by Buckland et al. (1992) differs 

 considerably from the method just described. First, they 

 replaced the assumption of a parametric model for the 

 underlying change in population with a nonparamet- 

 ric model. Among the many possible choices for such a 

 model, they selected the smoothing algorithm known 

 as '4253H, twice' (Velleman & Hoaglin 1981) on the 

 basis of a comparison described by K. L. Cattanach 

 and S. T Buckland (SASS Environ. Model. Unit, 

 MLURI, Craigiebuckler, Aberdeen, Scotland, unpubl. 

 manuscr. ). The adoption of a nonparametric model in- 

 creases the robustness of the test to model mis- 

 specification, a problem that affects the linear test. 

 Furthermore, the procedure, which involves the use of 

 a compound running median, incorporates the infor- 

 mation from nearby years into calculation of the 

 smoothed estimate for a particular year, therefore re- 

 ducing the influence of possible outliers and increas- 

 ing the precision of each smoothed estimate. The 

 smoothed test also provides a different way of looking 

 at the trend. Instead of the trend being described by a 

 single parameter (the slope of a linear regression), the 

 sequence of smoothed estimates constitutes the best 

 estimate of the underlying trend. 



