OPTIMUM ALLOCATION FOR ESTIMATING AGE COMPOSITION 



USING AGE-LENGTH KEY 



Han-Lin Lai^ 



ABSTRACT 



A new optimum allocation method for age-length keys (ALK) was developed by applying Kimura's Vartot, 

 an error index of estimated age composition. The method is applied to Pacific cod, sablefish, and walleye 

 pollock. At the present working capacity, the total of 10,000 minutes (about 70 working days) will 

 approximate the most effective cost to estimate age composition for the three species. Increasing costs 

 beyond this level will show no more gain. 



Age-length keys (ALK) are widely used for esti- 

 mating age compositions in fisheries. The theory of 

 ALK is based on a double sampling technique with 

 stratification (Tanaka 1953). The first stage involves 

 a simple random sampling for a relatively large size, 

 less costly length sample. The second stage involves 

 a stratified random sampling for a smaller size, more 

 costly age subsample from each length stratum. 

 Following the approximation of Kutkuhn (1963) and 

 Southward (1963), the proportion of fish at the ith 

 age class (p,) and variance of p, are estimated 

 as 



Var(p,) = I 





^j gu (1 - gy) h (Qy - Vi) 



+ 



n: 



N 



(1) 



(2) 



where l^ is the proportion of fish that fall into the 



jth length stratum, 

 A^ is total length sample size, 

 n^ is the size of age subsample in the jth 



length stratum, 

 q,j is the proportion of Uj fish classified into 



the ith age class, 

 A is the number of age classes, and 

 L is the number of length strata. 



Kimura (1977) defined Vartot as the sum of all 

 variances of the p, : 



'School of Fisheries, WH-10, University of Washington, Seattle, 

 WA; present address: Center for Quantitative Science, HR-20, 

 University of Washington, Seattle, WA 98195. 



Vartot = Z Var(p,) = £; 



t=i 



^ iPi - Vt)^ 



i=l 



(3) 



Manuscript accepted February 1987 

 FISHERY BULLETIN: VOL. 85, NO. 2, 1987. 



which is an error index for assessing precision of 

 the ALK. Furthermore, Vartot is the expectation 

 of the squared distance between the estimated age 

 composition P' = (p^, pg, . . . , p^) and the true age 

 composition of the population P' = (pj, pg, . . . , p^). 



Then, D = \/Vartot can be interpreted as a kind of 



average Euclidean distance between P and P in an 

 A-dimensional space. Kimura (1983) indicated that 

 D can be viewed as the percent error of the esti- 

 mated accumulated age proportion (i.e., the percent 

 error of Zp, = 1). 



This paper derives a new method for the optimum 

 allocation of ALK, applying the properties of Vartot 

 and Cauchy-Schwarz inequality (Kendall and Stuart 

 1977). The optimum sizes of length sample and age 

 subsample are determined so that either Vartot is 

 minimized subject to a fixed total cost or the total 

 cost is minimized subject to a desired level of Vartot. 

 Although this method is basically derived for the 

 problem that all age classes are of equal interest, 

 it can be modified by adding weighting factors to 

 the ages which are important to population dynam- 

 ics. This method was applied to Pacific cod, Gadus 

 macrocephalus, from the Washington coast; sable- 

 fish, Anoplopoma fimbria, from the Gulf of Alaska; 

 and walleye pollock, Theragra chalcogramma, from 

 the eastern Bering Sea. 



METHODS 



Two subsampling schemes related to ALK are fre- 

 quently used by fisheries biologists: 1) fixed age sub- 



179 



