Fishery Bulletin 90(1), 1992 



values of the series. This is known as smoothing 

 "twice." For example, if we define the residuals of the 

 time-series smoothed by 4253H to be {E(t )} = {X(t ) - 

 •54253 (i )}. then the values of the times-series smoothed 

 by "4253H, twice" can be defined by 



{•S4253H, twice (0} = {54253h(0 + 'S4253h(-E'(0)}- 



Thus the "4253H, twice" running median method 

 uses a 4253 running median to smooth the time-series, 

 estimates the endpoints of the smoothed series, and 

 then smooths the resultant series by Manning. The 

 residuals of the series are calculated and are also 

 smoothed, using the same method as above. The 

 smoothed values of the residuals are then added to the 

 smoothed values of the time-series to produce a time- 

 series smoothed by "4253H, twice." The advantage of 

 using running medians is that the magnitude of an 

 extreme estimate does not affect the resultant 

 smoothed time-series. The above method is sufficient- 

 ly complex that its behavior cannot be readily under- 

 stood. However, simpler methods were found to suf- 

 fer from one or more of the following shortcomings: 

 Estimated trends were not always smooth; implausible 

 rates of change were sometimes indicated; trends near 

 the start or end of the sequence of estimates were often 

 poorly estimated. 



Nonparametric bootstrap replicates are generated as 

 described by Anganuzzi and Buckland (1989). We select 

 here the bootstrap estimates that correspond to an 85% 

 confidence interval for relative abundance in each year. 

 The rationale for the choice of confidence level is that 

 if two 85% confidence intervals do not overlap, the 

 difference between the corresponding relative abun- 

 dance estimates is significant at roughly the 5% level 

 (P<0.05); whereas if they do, the difference is not 

 significant (P>0.05). If the abundance estimates are 

 assumed to be lognormally distributed, each with the 

 same coefficient of variation, then the exact confidence 

 level that gives this property is 83.4%. If one estimate 

 has twice the coefficient of variation of the other, the 

 confidence level increases slightly to 85.6%. Thus a 

 choice of 85% makes some allowance for variability in 

 the coefficient of variation. 



. For each abundance estimate, 79 bootstrap replicates 

 are run, so that the 6th smallest and 6th largest boot- 

 strap estimates provide an approximate 85% confi- 

 dence interval (Buckland 1984). If this procedure is 

 carried out independently for each year, confidence 

 intervals are wide. Provided the assumed stock area 

 spans the whole range of the stock, numbers of dolphins 

 within it are unlikely to vary greatly in successive 

 years, and a procedure that calculates confidence in- 

 tervals for a given year incorporating information from 

 years immediately preceding and following that year 



is more informative. For a given stock, we achieve this 

 by carrying out one bootstrap replication for each year 

 that a relative abundance estimate is available. These 

 estimates are smoothed using the routine described 

 above, and the process is repeated 79 times. For each 

 year, the 6th smallest and 6th largest smoothed 

 estimates provide approximate 85% confidence limits. 

 We use the sequence of medians of the smoothed boot- 

 strap estimates (i.e., the 40th estimate of each ordered 

 set of 79) as the "best" indicator of trend, so that it 

 is calculated in a comparable manner to the confidence 

 limits. Larger numbers of bootstrap replicates are 

 preferable, but available computer power was limited. 

 Repeat runs for the northern offshore stock of spotted 

 dolphins were carried out, to assess the Monte Carlo 

 variability. 



By using overlapping confidence intervals to test for 

 a difference between years, independence between 

 smoothed estimates for different years is assumed. 

 Given the strong positive correlation in the smoothed 

 estimates between successive years, the test is unlike- 

 ly to detect a large change between one year and the 

 next, but should be reliable for detecting trends over 

 a period of perhaps five or more years, for which cor- 

 relations between smoothed estimates are small. 



Results 



Figures 1-10 show the estimates of underlying trend 

 for each of the main stocks associated with tuna in the 

 eastern tropical Pacific Ocean. Since stock boundaries 

 and stock identity are both uncertain, we also show 

 trend estimates after pooling data from stocks that are 

 not differentiable in the field. The broken horizontal 

 lines in these plots correspond to the upper and lower 

 85% confidence limits for the 1988 relative abundance 

 estimate. Years for which the entire confidence inter- 

 val lies outside the region between the broken horizon- 

 tal lines show a relative abundance significantly 

 different from that for 1988. Because the smoothed 

 estimate for the first or final year of a sequence can 

 be poor, we show the unsmoothed estimate and cor- 

 responding 85% confidence limits for the first and last 

 year on each plot. 



Figures 1 and 2 show estimated trends for northern 

 offshore spotted dolphins, with and without the abnor- 

 mally low 1983 estimate, which corresponded with a 

 very strong El Nino event. It is clear that the 1983 

 estimate affects the smoothed estimate of trend, but 

 its effect is no greater than if it had been just smaller 

 than the 1984 estimate. Thus abnormal estimates may 

 be more safely retained when using this procedure, and 

 subjective decisions of whether to treat an estimate as 

 an outlier are avoided. 



