LANDER: PROBLEM OF BIAS IN MODELS 



not fishing mortality operating in offshore 

 waters. 



4. Mortality estimates from the Ricker (1962) 

 and Fredin (1964) models (Equations 10 and 15, 

 respectively) both utilize for biomass computa- 

 tions the independent evidence that natural 

 mortality coefficients are highest in small, 

 "prerecruit" stages of ocean life and lowest 

 in later stages when fish are large enough to 

 be ex|)loited in offshore as distinct from inshore 

 waters; both models give an average rate of 

 natural mortality which is an upper limit for 

 "prerecruits" and a lower limit for "post- 

 recruits" (quotation marks above indicate these 

 models were develoi)ed and api)lied to hypo- 

 thetical as distinct from actual offshore fish- 

 eries). Ricker's model treats preultimate re- 

 turns or spawning escapements as if none of 

 those fish mature and all survive until the 

 end of ocean life as observed with the last 

 actual return. Equations 12-14 express abso- 

 lute bias in mortality estimates from the Ricker 

 model in terms of the unknown maturity 

 parameter and observable data without off- 

 shore fishing or with an offshore catch un- 

 known. Fredin's model accounts for actual 

 timing of all returns and therefore approxi- 

 mates more closely than Ricker's model the 

 actual mortality coefficients for any set of 

 data taken when no fishery operates offshore. 

 Fredin's model is solved by trial and error so 

 explicit bias equations cannot be written. When 

 offshore fishing occurs but the origin-specific 

 offshore catch is unknown, it is reasonable to 

 expect from both models less (unknown) bias 

 in mortality coefficients than without offshore 

 fishing; i.e., to use calculated values along 

 with the fixed relations in Equations 1 and 5 

 as estimates of actual mortality rather than 

 as limits for natural mortality. 



5. Biomass computations of Ricker (1962). 

 Parker (1963), and Fredin (1964) overestimate 

 minimum losses in yield. Although the evi- 

 dence indicates growth typically exceeds mor- 

 tality in potential and actual postrecruits, the 

 implicit assumption of full availability is in 

 error: management restrictions, fleet size, and 

 bad weather always prevent continuous, com- 

 plete availability of a stock in offshore waters. 



6. Cleaver's (1969) model is the first actually 



to utilize known offshore catches (but see 

 Equation 7 of Ricker, 1964). Its basic assump- 

 tion is that natural mortality is absent during 

 the last year of ocean life; the assumption 

 leads to certain one-sided limits, e.g., to the 

 minimum population about a year before the 

 last return (Equations 16-18). The model does 

 not give unique estimators or bias equations 

 but is the basis for the limit-mean model. 



7. The limit-mean model with offshore fish- 

 ing absent or offshore catch unknown assumes 

 a) the survival between times of outmigration 

 and of the first return does not exceed survival 

 between times of the first and second returns 

 (Equations 19-20) and b) the latter survival 

 is less than 1.0 (Equation 21). The result is 

 lower and upper limits on the product of these 

 two survival fractions. The geometric mean 

 of limits is taken arbitrarily as an estimator 

 of the product (Equation 22), then the fixed 

 relations among parameters and observable 

 data (Equations 1-5) give estimates of the 

 survival, maturity, and abundance schedules 

 (Equations 23-27). With an offshore catch 

 known by origin, the first assumption above 

 (Equation 20) gives an upper limit for the 

 number of immatures at sea near the time of 

 the first return (Equations 2 and 29). The 

 additional assumptions of no fishing or natural 

 mortality after recruitment offshore gives lower 

 and ui)per limits for the number recruited 

 (Equations 28 and 31). The geometric mean 

 is taken arbitrarily as an estimator (Equation 

 32) to give estimates of the exploitation rate, 

 fishing, and natural mortality coefficients (Equa- 

 tions 33-37). Other estimates for survival, 

 mortality, maturity, and abundance are then 

 available. The limit-mean model involves many 

 substitutions and the cumbersome bias equa- 

 tions are not given. 



8. Numerical examples of relative bias for 

 a hypothetical smolt class of coho salmon are 

 ex])lained and i^resented in Tables 1 and 2. 

 Values are chosen for close agreement with 

 data and estimates for coho salmon actually 

 reared at Columbia River hatcheries. The four 

 examples in Table 2 show a wide range of 

 bias in estimates of different parameters from 

 the same model, and also for the same jiara- 

 meter as estimated from different models in 



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