374 



Fishery Bulletin 89(3), 1991 



smaller, but still substantial, change of 5% per year 

 (total 155% increase or 37% decrease) would have a 

 very low chance of being detected at this level of a. If 

 small changes are to be detected, then a may have to 

 be set higher. 



The most extreme form of raising a-levels is ac- 

 complished by considering only the sign of the estimate 

 for the covariate coefficient in the ANCOVA, thus set- 

 ting a = 1. In this case, the direction, rather than the 

 presence, of a trend is tested for. This approach max- 

 imizes power, and may be an alternative for situations 

 where power cannot be improved through other means 

 (i.e., increasing the number of surveys conducted). For 

 harbor porpoise trend estimation, the roughly equal 

 fractions of positive and negative covariate coefficient 

 estimates, 6 (Table 5) indicate that such an analysis is 

 not biased towards detecting either trend direction. 



With a = 1.0, power to detect the correct trend in 

 harbor porpoise abundance ranges from 0.67 to 1.00 

 for 5-10 survey years and ± 5% and ± 10% annual 

 population changes. Power of 0.80 or higher is achieved 

 with a = 1.0 after 5-6 survey years for a 10% annual 

 change, or after 8 survey years for a 5% annual change. 

 However, since the cost of low power in this case is 

 a y-error, power should be higher than when a is set 

 at the traditional level of 0.05. In this case, eight survey 

 years may provide high enough power to detect an 

 annual 10% change, whereas even 10 years may not 

 yield sufficient power to detect the smaller 5% annual 

 change. 



The magnitude of the y-error when a = 1.0 can be 

 demonstrated with Figures 6 and 7. The three curves 

 in these figures represent the distribution of covariate 

 coefficients, 6, for 500 simulated data sets with annual 

 changes of (A) -10%, (B) 0%, and (C) +10%. The 

 y-error is represented by the area under curves A and 

 C which lies on the incorrect side of zero. If this area 

 is small or equal to zero, as when 10 annual surveys 

 are conducted (Fig. 6), then the analysis will have a high 

 probability of detecting the direction of a trend correct- 

 ly. However, if the area is large, as when only five 

 annual surveys are conducted (Fig. 7), then the pro- 

 cedure will not be able to detect the direction of trends 

 accurately. The large degree of overlap between the 

 three curves in Figure 7 also reflects the low power 

 to detect trends. The dotted line marks the location of 

 the covariate coefficient estimate (6) for the 1986-90 

 survey data. It is apparent that the estimate could 

 reasonably come from any of the three distributions. 



Setting a = 1.0 is valid only if the costs of interpreting 

 a nonexistent trend in a stable population are small in 

 relation to the costs of failing to detect an existing 

 trend. This may be the case if one needs to determine 

 whether an existing level of take from a commercially 

 exploited population is sustainable. The cost of not 



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COVARIATE 



Figure 6 



Distribution of covariate estimates (6) representing yearly 

 change in abundance (from ANCOVA) for 500 simulations 

 each of (A) 10% annual decrease, (B) no change, and (C) 10% 

 annual increase in abundance over 10 survey years. Shaded 

 area under curves A and C which lies on the incorrect side 

 of zero represents the y-error when a = 1.0; here, it is essen- 

 tially zero. 



detecting a decreasing trend could be extinction of the 

 population and the permanent loss of a resource. On 

 the other hand, eliminating or reducing exploitation on 

 a stable population which is incorrectly thought to be 

 decreasing would cause smaller, short-term costs. In 

 the case of marine mammals in the United States, 

 existing laws mandate that all species be maintained 

 at sustainable levels, so extinction represents an un- 

 acceptably high cost. 



Several assumptions of the above procedures must 

 be discussed. The most critical assumption is that the 

 five years of data collected during 1986-90 characterize 

 the level of variability expected in a longer time series. 

 In addition, the results of the simulations are only 

 accurate if the ANCOVA model is appropriate. The 

 results indicate that the chosen model fits the data well 

 (P<0.0001). 



