LAI: ESTIMATING AGE COMPOSITION 



For a random age subsample, N* is obtained by 

 substituting Equation (11) into Equation (5) 



N* = ib,/r* + ho)ID- 



n* = r*N* 



min. C = CiN* + c^n* 



(18) 

 (19) 

 (20) 



Because < r < 1, Equations (7) and (11) indicate 

 that 



C1/C2 < a2/ai for fixed age subsample and 



C1/C2 < 62/61 for random age subsample 



must be held for the optimum allocation. When 

 equality holds, /' = 1 and then ALK degenerates to 

 a simple random sampling for age samples. 



EXAMPLES 



Three sets of ALK data are used for the exam- 

 ple: Pacific cod, aged by the scale method, from the 

 Washington coast (Kimura 1977); sablefish, aged by 

 the otolith method, from the Gulf of Alaska; and 

 walleye pollock, aged by the otolith method, from 

 the eastern Bering Sea (Lai 1985). The parameters 

 of Vartot, a's and 6's, are calculated and summarized 

 in Table 1. Accurate cost estimates are difficult to 

 determine; therefore, time measurements required 

 for observing a length and determining an age of 



fish are used for c■^ and Co. The total cost C is thus 

 the total time required to build an ALK. The mea- 

 surements of C] and Cg for the three species (Table 

 1) are primarily based on the author's experience. 



Given a total of 120 working days and 6 working 

 hours per day to a fisheries scientist, the optimum 

 allocation is summarized in Table 2. Under this 

 budget, a random age subsample can provide higher 

 precision than a fixed age subsample (improved 

 10%, 15%, and 18% respectively for Pacific cod, 

 sablefish, and walleye pollock). Also, for this budget, 

 the error of estimated cumulated age proportion is 

 less than 2.5% for the three species using either 

 fixed or random age subsamples. 



Using Equations (7), (15), (16), and (17) for a fixed 

 age subsample and Equations (11), (18), (19), and (20) 

 for a random age subsample, costs under various 

 desired D are minimized for the three species (Table 

 3). At the same level of D, a fixed age subsample 

 requires much larger sample sizes of length and age 

 than a random age subsample does for the three 

 species. The greatest benefit of using random age 

 subsample is that it drastically reduces the total cost 

 required to obtain the same level of Z). The total cost 

 is reduced by 35%, 45%, and 55% respectively for 

 Pacific cod, sablefish, and walleye pollock for any 

 given D when a random instead of a fixed age sub- 

 sample is used. 



Figure 1 shows the relationships of D and total 

 cost. Whether a fixed or a random age subsample 

 is used for the three species, it is obvious that D 

 decreases rapidly until C = 10,000 minutes, which 

 is nearly 70 working days. A point of diminishing 



Table 1 . — Parameters of Vartot and costs for opti- 

 mum allocation. 



'In minutes, assuming 120 working days for each species 

 and 6 working hours per day. 



Table 2.— Optimum allocation of minimizing Vartot for Pacific cod, 

 sablefish and walleye pollock. (D = \A/artot.) 



Table 3.— Optimum allocation minimizing total cost for various 

 desired precision level (D = \/Vartot). Parameters and c, and 



Cj are listed in Table 1. 



Pacific cod 



Sablefish 



Walleye pollock 



181 



