Helser and Hayes: Quantitative management advice based on stock abundance indices 



297 



where the value of the fitted index is less than the 

 value of the lower quartile (Fig. 6). The probability 

 that the fitted 1990 index lies below the estimated 

 value of the lower quartile (-1.9) is approximately 

 60% (Fig. 6). Further, because of the shape of the 

 parent distribution of the lower quartile, the prob- 

 ability that the 1990 index lies below the reference 

 point increases rapidly for higher and higher values 

 of the reference point. For example, the probability 

 that the 1990 index lies below a value of-1.85, which 

 is nearly as likely as the bootstrapped mean (Fig. 5), 

 increases to almost 75% and reaches nearly 85% at 

 a value of -1.8 (Fig. 6). As an alternative to making 

 inferences about the level of the population at only 

 one point in time, fishery managers may wish to con- 

 sider the likelihood of two (or more) consecutive years 

 jointly falling below the value of the reference point. 

 To address this question, we computed the joint prob- 

 ability of the 1990-91 and the 1990-92 fitted survey 

 indices falling below our chosen reference point. The 

 fitted survey indices over the 1990-92 period were 

 as likely as those over the 1990-91 period to be be- 

 low the reference point: approximately a 57% prob- 

 ability that the indices jointly fell below the refer- 

 ence point. This indicates that current population 

 levels (as indexed by the fitted abundance indices) 

 are considerably below the prescribed threshold level 

 if the lower quartile of the fitted time series were 



Lower quartile 

 Figure 6 



Probability that the value of the 1990 spring survey index 

 predicted by an integrated moving average (IMA) model 

 fitted to 1,000 bootstrapped generated realizations of the 

 survey time series 1968—92 lies below the value of the lower 

 quartile (25th percentile) derived from the predicted 

 bootstrapped indices 1968-92. 



adopted as the acceptable reference point. It should 

 be emphasized that an "acceptable" reference point 

 in this example refers more to choice of the range of 

 years used for computation of the reference than to 

 the choice of the interquartile, specified in this case 

 to be the 25th percentile. We advise using a range of 

 years for the abundance index which represents rea- 

 sonably high population sizes and then using that 

 fixed set of years for computation of the reference 

 point even as the time series lengthens. This would 

 prevent a ratcheting effect where the reference point 

 declines as the abundance index declines, while al- 

 lowing a characterization of the uncertainty in the 

 reference point. 



In conclusion, this technique represents an ad- 

 vancement for index-based assessments in the pro- 

 vision of quantitative advice for the management of 

 fish populations surveyed by research vessels that 

 are otherwise lacking in data sources. This approach 

 provides an examination of the joint cumulative prob- 

 ability for the condition Priindex in year t, t+l, ... , 

 t+m < lower quartile) and is important because it 

 allows the likelihood of correctly deciding whether 

 or not a stock is below a prescribed threshold abun- 

 dance or reference point to be ascertained quantita- 

 tively. We emphasize that the computation of a ref- 

 erence point from the time-series data is arbitrary 

 and should be based on a series of observations rep- 

 resenting reasonably high as well as low stock abun- 

 dances. Finally, we illustrated these procedures with 

 trawl survey data for wolffish in the natural log scale, 

 which when the data are differenced, produces ho- 

 mogeneity of variance and stationarity of the time 

 series (Nelson, 1973). It should be noted that 

 retransformation back to the linear scale is likely to 

 result in some bias. We did not examine the effects 

 of transformation bias in this study but suggest that 

 these effects be investigated in the future. 



Literature cited 



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