LANDER: PROBLEM OF BIAS IN MODELS 



Applying Equations 10 and 11 and the actual 

 values of Zj from Equations 1 and 5 when 

 C is unknown: 



5(z;)= [T-t^]-'[-\naEi/NJ 

 + \n(E^/NQ)-\n(\-m)]. 



(12) 



+ [t, -t^]-'[\n{EjN^ -in m] . (13) 



^(_-2) = [T-t^]-'[-\n(E^/N^)] 



+ [T-t,][\niEjNj-\mEJNJ 

 - In m +ln(l - m)] . (14) 



Fredin's (1964) Model 



Ricker's model accounts for the magnitude 

 but not for the actual timing of preultimate 

 returns (again, for the magnitude of Ei in 

 Equations 1-5 here). Fredin (1964) accounts 

 both for magnitude and actual timing. His 

 Model 1 is based on Equation 11 of Parker 

 (1962), who separates relatively high natural 

 mortality on small juveniles plus returning 

 adults — both in inshore waters — from relatively 

 low natural mortality in offshore waters. 

 Parker's results actually derive, however, from 

 his Equations 13 and 14 and utilize data from 

 paired groups of marked smolts. I consider 

 only the information available from a single 

 group of smolts (marked or unmarked). 



Thus Model 1 of Fredin (1964) assumes with 

 Ricker (1962) that Z* = Z, ^Zg; more strictly, 

 the assumption is again that Mj = Mi = .1/2 

 l)ecause Fredin deals also with data not in- 

 fluenced by offshore fi.shing. Substituting the 

 assumption in Equation 5 gives an estimator 

 in the relation: 



EJN^ = e-^4(T -'o)-(£JiVo)e-^T(^-'i>.(15) 



Equation 15 is solved by trial and error 

 unless one applies more advanced mathematical 

 properties. Equation 15 differs from Equation 

 10, so values of Z* from the same data obviously 

 will differ. We anticii)ate less bias in Z* from 

 Equation 15 because it accounts for the actual 

 timing of the Ei jacks. With no e.xplicit defini- 



tion for Z^ in Equation 15, however, relations 

 for bias (e.g., Equations 12-14) cannot be 

 written and Equation 15 is later evaluated 

 numerically. 



Finally, Fredin's Models 2-4 employ various 

 assumptions to accord with the reality, Z\ > Z2 

 (or M] > M2). Although no "estimators" in the 

 sense of Equation 15 are available from his 

 Models 2-4 in the indirect api)roach, it is 

 interesting that calculated maturity schedules 

 were relatively insensitive to the different time 

 distributions of mortality Fredin assumed 

 between his Models 1-4. 



Cleaver's (1969) Model 



This model was developed specifically to 

 utilize inshore and offshore catch data as known 

 from a landmark study which evaluated the 

 bioeconomic contribution of 1961-64 brood Chi- 

 nook salmon from Columbia River hatcheries 

 (Worlund et al., 1969). Its basic assumption in 

 the context of Equations 1-5 is that M-z = 

 during T - tx. The result is one-sided limits 

 for certain parameters: 



E^I{C + E^) ^ s^^ ^ e-^2L^'-h\ (16) 



'^iL = E^l^2v = C + E^. (17) 



m^, = EJiE^+N^,). (18) 



As actually applied (Cleaver, 1969; Henry, 

 1971), the model used data on four escapements 

 to the river (Ei) at ages 2-5 from a given smolt 

 class and data on offshore catches (Cj) of 

 immature plus currently maturing fish at ages 

 3-5 (unfortunately, marked recoveries in the 

 offshore catch were not sampled for maturity — 

 the catch being taken by small vessels and 

 landed dressed with heads on). Values of E^ 

 and Ci led to rejection of the hypothesis that 

 Zj was constant for ages 3-5 in offshore waters 

 (certain resulting values of m^ > 1.0 implying 

 more mature fi.sh than were present in the total 

 marked po))ulations). Rejection of constant 

 Zj was deduced (by values of Cj and E^) to be 

 mainly from variation in F;. Recalling the 

 basic assumption (M^ = during the la.st year 

 or 3/2 = here), the authors then examined 



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