FISHERY BULLETIN: VOL. 71, NO. 3 



into the observed categories with certain 

 probabilities 0,- (/ = 1 •• 4) as follows: ^ 



-6 Ml 



Probability of i^i = ^i = me' 



(1) 



Probability of C =02 

 (1 - m)e 



-6Mig 



n .g-4.5(F + M2;1 



7.5M9| ^ 



[F + M2J 



(2) 



Probability of £2 = ^3 = 



Probability of D =04 = 1-01-02-^3' 



where D = Dq + Di + D2 • (4) 



The maximum likelihood estimators of 

 the 01 are: 



01 = E^INq. (5) 



02 = CjNo. (6) 



03 = E2IN0. iV 



04 = 1-01-02-^3- (8) 



A maximum likelihood estimator of a function 

 of the parameters 0, is obtained by replacing 

 the parameter values by the corresponding 

 maximum likelihood estimates, 0,. Beyond 

 that, however, there exists no unique trans- 

 formation or function to obtain maximum likeli- 

 hood estimates of Mi, m, M2, and F: any given set 

 of observed data can generate a variety of 

 combinations of parameter estimates. 



METHODS OF ESTIMATING 

 PARAMETERS 



Estimations Based on Selection of;// or AI2 



Since no unique solution exists, the only prac- 



3 We are indebted to Jerome Pella who, as editorial 

 referee, suggested that the relationships affecting the coho 

 salmon be depicted in this manner and pointed out a 

 relation between the proportion maturing as jacks (m) and 

 the rates of natural mortality that we had not considered 

 earlier. Pella's suggestions greatly improve the understand- 

 ing and description of the actual situation. 



tical solution is to assume some values for one 

 of the unknown parameters and solve the equa- 

 tions for the remaining parameters. This in 

 effect is what Cleaver (1969) and Henry (1971) 

 did for hatchery chinook salmon, 0. tshawyt- 

 scha, (with 3-4 spawning escapements from a 

 given release group). In their calculations, how- 

 ever, they assumed various values for M2 (nat- 

 ural mortality during the last year of life) and 

 then calculated values for the remaining param- 

 eters. In applying this method to the 1965-66 

 brood coho salmon data, the appropriate equa- 

 tions, based on the time periods and notations 

 shown in Figure 2, would be: 



CIE2 = 



N. 



E2 



E, 



- (g4.5F+4.5M2.j) 



F+M2 



solve for F 



^^g4.5M2+4.5F 



solve for N^ 



g-12M2-4.5F^.^) 



(9) 



(10) 



"TTT 



solve for m. 



(11) 



The resulting values for these computations 

 at six different levels of natural mortality for 

 all hatcheries combined for each brood year are 

 listed in Table 3. It is apparent from these data 

 that the changes in natural mortality have a 

 relatively greater effect on the proportion matur- 

 ing as 2-yr-old fish (m), and on the number of 

 recruits to the third year of life (Ni) and to the 

 fishery (Nr), than on the fishing mortality (F). 



As mentioned previously, it is difficult to 

 analyze the data separately from the four sec- 

 tions of the river. For the 1966 brood two 

 additional groups of marked fish were released 

 —approximately 92,000 D-Ad-LM in the Middle 

 River areas and about 73,000 D-Ad-RM in the 

 Uppermost River area (see footnote 2). Ob- 

 viously, any fin regeneration or fins missed in 

 sampling from this group would confound data 

 from the other marked groups, making a com- 

 parison of individual markings for the 1966 

 brood much more questionable. Nevertheless, 

 in Table 4 are listed the parameter values for 



682 



