760 



Fishery Bulletin 98(4) 



to r simply as the "growth rate" of the cohort returning 

 in year i. Denoting the underlying mean growth rate by p. 

 the goal of at least a 31% increase in the average cohort 

 replacement rate over the observed mean of 1.35 for the 

 1989-1993 period is equivalent to a goal, on the log scale, 

 ofPgoai ^ log(l-35 X 1.31) = 0.57. 



A natural recovery benchmark then is to compare the 

 observed sample mean gi-owth rate r in future years to 

 p ,. If recovery efforts have the desired effect, and if no 

 increased mortality occurs from other causes, recovery will 

 proceed as desired and r will likely exceed Pgoai- However, if 

 r < p ^1 , recovery may not be proceeding as desired and fur- 

 ther conservation measures may need to be implemented. 

 This possibility raises the question of how one should evalu- 

 ate whether an observed r <Pg„,i warrants concern. 



In our paper, we propose using a one-sample, one-sided 

 /-test to evaluate the statistical significance of a differ- 

 ence between the observed mean gi'owth rate and the 

 target rate. We depart from the usual procedure (Lehm- 

 ann, 1986), however, by conditioning the test on a specific 

 level of statistical power, rather than on a fixed type-I 

 error rate, in order to provide an adequate detection prob- 

 ability for dangerously low population gi'owth rates. Appli- 

 cation of the test requires choosing a particular power 

 level and specifying what constitutes a "dangerously" low 

 population growth rate. Together, these quantities deter- 

 mine the sensitivity of the test for detecting population 

 growth rates below the target, and the likelihood of false 

 positives, i.e. concluding that population growth rate is 

 below the target when it in fact is not. The level of danger 

 posed by a certain growth rate is evaluated by using a 

 population viability model. 



Figure 1 



Map of Northern California depicting the Sacramento 

 River, its upper tributaries, and mainstem dams. 



Hypothesis testing and statistical power 



Evaluating whether or not winter chinook salmon are 

 meeting the recovery goal requires a statistical test 

 because of the variability inherent in r. We propose that 

 a one-sided /-test be used to decide whether an observed 

 mean population growth rate falls significantly short of 

 the goal — in which case further regulatory action may 

 be necessary. The null hypothesis of the test is that the 

 underlying mean growth rate p (estimated by r ) is greater 

 than or equal to p,„„,,, and the alternative hypothesis is 

 that p is less than p^„„,. That is, 



^O: P ^ P^oal 

 ff.V P < Pf^oal 



with, in this case, p^,,,,,, = 0.57. Given a set of;; > 1 obsei-ved 

 r, values {r,, r,,,... ,/■„), with mean r and standard devia- 

 tion s, the test statistic is 



P - P,.^ 

 a I yjn 



(1) 



Assuming that the |r,) are independent and identically 

 distributed normal random variables, / has a central /-dis- 



tribution with n - 1 degrees of freedom if p = p,,,^,, , and a 

 noncentral /-distribution if p ^^ p^,,,^,, (Lehmann, 1986). The 

 /■test rejects //„ in favor of // ^ when / is less than some 

 critical value /^. specified a priori by the investigator. 



A /-test has four possible outcomes, two of which result 

 in correct inference: the test can accept H„ when it is true, 

 and it can reject //,, when it is false. The test could also 

 reject H„ when it is in fact true — a "type-I" error, or it could 

 fail to reject //q when it is false — a "type-II" error. The 

 expected rates of type-I and type-II errors are convention- 

 ally denoted as a and /3, respectively. The probability of 

 correctly rejecting //,, is known as the power n of the test, 

 and K = I - p. 



The type-I and type-II error rates are determined by the 

 value of/, selected for the test. A smaller value of/, results 

 in a lower type-I error rate a and a higher type-II error 

 rate p. A larger value of /, has the opposite effects. In all 

 cases, the two error rates change in opposite directions 

 when the value of/, is changed. Thus, for a given data set 

 of size n, a and /J cannot be simultaneously minimized by 

 adjusting /, . 



Given the endangered status of winter chinook salmon, 

 it is clearly necessary to ensure that the statistical test 

 has enough power to detect dangerously low population 

 growth rates. To achieve this goal we propose that the 

 power ;rof the test be held at a fixed level, rather than the 



