762 



Fishery Bulletin 98(4) 



Equation 4 for t^. and substituting this into Equation 3 

 gives 



a = T[T-Hn)]. 



(5) 



Equation 5 indicates the type-I error rate, a, associated 

 with the test when conducted at specified power-level n. 

 Note that, n having been specified, a also depends on 

 the magnitude of the underlying difference p - p^,„o/, the 

 variability a in gi'owth rates, and the sample size n, all 

 through the noncentrality parameter 8. 



Relation between a, n, n, and quasi-extinction 



Although Equation 5 gives the value of a corresponding 

 to a specified power-level n, the formula itself does not 

 reveal the nature of the relation between a and n, and 

 how this relation is affected by p and n. We illustrate 

 these relationships below and consider their consequences 

 in the context of a proposed test for detecting low growth 

 rates in the Sacramento River winter chinook salmon 

 population. 



To apply Equation 5, we must specify o, p. and n. We 

 want to know (with probability n) that winter chinook 

 salmon growth is not less than p^^^^i by a critical amount. 

 Because the ESA is invoked to prevent extinction, we 

 want to guard against growth rates that could lead to 

 extinction. We used the winter chinook salmon popu- 

 lation viability model developed by Botsford and Britt- 

 nacher (1998) to identify growth rates corresponding to 

 quasi-extinction probabilities of 0.05. 0.50, and 0.99 over 

 50 years. Quasi-extinction occurs when a population falls 

 below some threshold level, in this case 200 adults in 

 three consecutive cohorts. We initialized the viability 

 model simulation with winter chinook salmon spawning 

 escapements from the 1989-93 base period, and set a = 

 0.552, the observed standard deviation of growth rates 

 during the base period, assuming that this value will 

 continue to hold in the future. The mean growth rates 

 corresponding to the specified quasi-extinction proba- 

 bilities were found to be about 0.0, -0.14. and -0.40, 

 respectively. We note that if indeed p = p , = 0.57, quasi- 

 extinction is an extremely unlikely event according to 

 this model. 



For each of the three growth rates, 5 was computed 

 according to Equation 2, and Equation 5 was then used 

 to determine the type-I error rate a over a range of spec- 

 ified power-levels jz and sample sizes n. We calculated 

 Equation 5 by using the Matlab Statistics Toolbox func- 

 tion NCTI>A^ (Jones, 1996). (Fortran and S-PLUS subrou- 

 tines are also available for the noncentral /-distribution 

 from the Carnegie Mellon University Department of Sta- 

 tistics' StatLib and Oxford Univei'sity Department of Sta- 

 tistics' FTP archive, respectively. Alternatively, because /^^ 

 is distrib uted as a r atio of independent random variables, 

 iZ + S)/ ^x!, 1 /<« - l),whereZis a standard normal variate 

 and xj^_y is a chi-square variate with ;? - 1 degrees of free- 

 dom, a large number of draws of /_.; could be simulated, and 

 the 100 X ;r'th percentile could be taken as an approxima- 

 tion to T, '(;rl in Equation 5.) 



The results for p = 0.0 (Fig. 3) display the general behav- 

 ior expected: 1) for fixed k, a decreases with n;2) for fixed 

 n, the higher n is set, the gi-eater a becomes; and 3) for 

 fixed a, power increases with increased sample size. For 

 example, if the power-level were fixed at it = 0.8 and a set 

 accordingly, a type-I error would be expected about 229c of 

 the time with 3 years of data, 11% of the time with 5 years 

 of data, or 3% of the time with 10 years of data. On the 

 other hand, if the type-I error rate were fixed at a = 0.05. 

 there would be roughly an 809^ chance of detecting this 

 value of p with 7 years of data, but the power of detection 

 would drop to n: < 35% with only 3 years of data. 



Table 1 lists results for mean growth rates of 0.0, -0.14 

 and -0.40. Notice that while the type-I error rate required 

 diminishes for a given power-level and sample size as the 

 underlying growth rate declines, use of a fixed « = 0.05 

 even in the most dire case of p = -0.40 would provide very 

 low power for n < 3. 



Monitoring protocol 



A monitoring protocol cannot be designed solely on the basis 

 of statistical consideration.s — it must be guided by manage- 

 ment policy. In this instance, the management policy is to 



