Optimal sampling design for using 

 the age-length key to estimate age 

 imposition of a fish population 



CO 



Han-Lin Lai 



Washington State Department of Fisheries 

 II I I Washington Street SE, PO. Box 43 1 44 

 Olympia. Washington 98504-3 1 44 



An age-length key (ALK) is the tra- 

 ditional method for estimating age 

 composition of an application of fish 

 population. Tanaka (1953) showed 

 that the ALK method is a double- 

 sampling technique (Cochran 1977). 

 The first stage uses simple random 

 sampling to collect a very large, but 

 relatively inexpensive, length sample. 

 The second stage subsamples a small 

 number of fish from the first-stage 

 length samples for relatively costly 

 age determination (age subsample). 

 Age subsampling in the second stage 

 can be taken in two fundamental 

 ways, one in which age subsamples 

 are randomly taken from the entire 

 first-stage length samples (random- 

 age subsampling; Kimura 1977 ) and 

 another in which age subsamples are 

 taken from each stratified length- 

 stratum (stratified-age subsampling; 

 Tanaka 1953). In stratified-age 

 subsampling, fixed-age subsampling 

 (a constant number of age speci- 

 mens is taken at each length-stra- 

 tum) and proportional age sub- 

 sampling (age specimens is propor- 

 tional to the random length-fre- 

 quency) are the most popular 

 (Ketchen 1949). However, a general 

 stratified-age subsampling (number 

 of age specimens varied at each 

 length-stratum) can also be used. 

 Stratified-age subsampling is the 

 major focus of this paper. The simi- 

 larities in results obtained from 

 proportional and random-age 

 subsamplings are given in the 

 Discussion. 



This paper was motivated by two 

 articles, Lai (1987) and Jinn et al. 



(1987), using optimal sampling de- 

 signs to estimate age composition 

 of a fish population using ALK. The 

 two articles differ significantly. Lai 

 (1987) was based on the classic 

 double-sampling technique and de- 

 rived the optimal allocation of 

 length samples and age subsamples 

 using Kimura's Vartot (Kimura 

 1977) and the Cauchy-Schwartz in- 

 equality (Kendall & Stuart 1977) for 

 fixed-age subsampling and propor- 

 tional-age subsampling. Lai (1987) 

 incorrectly used random-age 

 subsampling for proportional-age 

 subsampling. In contrast, Jinn et 

 al. ( 1987) used a Bayesian approach 

 to estimate age composition, vari- 

 ance, and covariance for a general 

 stratified-age subsampling. They 

 used the iterative method of Roa & 

 Ghangurde (1972) to obtain the op- 

 timal allocation of length samples 

 and age subsamples for each length- 

 stratum, with a set of known per- 

 unit costs for ageing a fish in each 

 stratum. 



The length-based optimal sam- 

 pling design of Jinn et al. (1987) 

 has advantages. Because the age of 

 a fish can be expressed as a func- 

 tion of its length (e.g., a von 

 Bertalanffy growth relationship) 

 and because older fish are more dif- 

 ficult to age, the per-unit cost of age- 

 ing a fish can be used to estimate 

 the difficulty of ageing older (larger) 

 fish. In addition, the covariance 

 components are important statistics 

 because the sum of all age propor- 

 tions equals 1, indicating that the 

 estimates of age composition are not 



mutually independent. The disad- 

 vantage of the Bayesian approach 

 of Jinn et al. (1987) is that their 

 method is mathematically compli- 

 cated and does not provide explicit 

 expressions for the optimal alloca- 

 tions of length and age samples for 

 an ALK, and thus requires substan- 

 tial computing effort. 



The purpose of this paper is to 

 derive a length-based optimal sam- 

 pling design for an ALK using a 

 classic double-sampling technique. 

 The covariance of age composition 

 also is derived using the method of 

 Kimura ( 1977). This paper also pro- 

 vides answers to the question many 

 fishery scientists have asked 

 me: What is the explicit solution 

 to a length-based optimal sampling 

 design for an ALK? A discussion on 

 the general applicability of ALK in 

 the sampling program with com- 

 plexity of fishery-time-areal strati- 

 fication and tows/trips clusterization 

 is also provided. 



Methods 



I use the following notation for an 

 ALK with a general stratified-age 

 subsampling: 



N = total number of length 

 samples; 



N, = number of fish in the ith 

 length-stratum, i=l,...,L; 



1, = proportion of fish in the ith 

 length-stratum, (1, =N/N); 



n, = number of age sub-samples 

 randomly taken from the ith 

 length-stratum; 



n y = number of fish from n, as- 

 signed to the jth age-class; 



q,j = proportion of fish in the ith 

 length-stratum that fall into 

 the jth age-class (q.^n./n, ); 



A = number of age-classes; 



L = number of length-strata; 



p = proportion of population in 

 the jth age-class; 



Manuscript accepted 10 December 1992. 

 Fishery Bulletin. U.S. 92:382-388(1993). 



382 



