Lindley et al : Application of statistical power analysis to recovery of an endangered species 



763 



0.9 - 



5 



o 



CL 



Number of observations 



Figure 3 



Type-I error rate (a) of the proposed Mest as a function of the power n and the number of 

 observations n. Power-level is with respect to detecting a mean gi'owth rate of p = (which 

 is projected to lead to quasi-extinction over 50 years with probability 0.05). The value of the 

 noncentrality parameter in this case is 5 = -0.57/(0.552/ \A7j. 



provide an adequate level of protection for winter chinook 

 salmon through timely identification of low gi-owth rates, 

 without incurring too many false positive results. 



The suggested protocol, therefore, would be to apply the 

 i-test annually, with a fixed power-level of SOC-J for detect- 

 ing a mean population growth rate of p = (which is 

 projected to lead to quasi-extinction over 50 years with 

 probability 0.05). The choice of k and p is somewhat arbi- 

 trary, reflecting the perceived costs of type-I and type-II 

 errors, and is discussed in a later section of this paper. 



The obser\'ed growth rates are defined as /■, = log(A^y 

 N^_jK and the protocol would commence with the / = 1997 

 and 1998 spawning runs — both runs having benefited 

 from the 1996 shift in ocean harvest regulations designed 

 to reduce fishing mortality on winter chinook salmon.' 

 Each year after 1998, the additional observed growth 

 rates would be added to the test data set, until five 

 growth rates are obtained. Beyond the year 2001, the test 

 would be limited to the most recent five growth rates. 



at which point the a-level will stabilize at 0.11. The pro- 

 tocol's moving five-year data frame will facilitate identi- 

 fication of shifts in winter chinook salmon survival and 

 strengthen the basis for the assumption that the Ir^j are 

 identically distributed (discussed below). Survival shifts 

 might be expected in response to naturally arising or 

 management-related changes in the freshwater or marine 

 environment. 



To illustrate in concrete terms the proposed monitoring 

 and analysis protocol, we applied it to the historical time 

 series of adult returns i = 1970,..., 1996. as if the protocol 

 had commenced with the / = 1970 and 1971 spawning runs. 

 The calculations and results of this application are pre- 

 sented in Table 2. Throughout the historical time series, 

 r failed to reach the target level of 0.57, and this failure 

 would have been declared significant in all years except 

 for 1983-89 and 1996. If the test used a higher k, the 

 number of failures declared significant would be higher. 

 For instance, using k = 0.85 results in 21 null hypothesis 



