ASSESSMENT OF COMPOSITION OF STOCK MIXTURES 



Jerome J. Pklla' and Timothy L. Robertson^ 



ABSTRACT 



Stocks offish can occur in mixtures, and knowledge of the composition of such a mixture may be needed. 

 An estimate of the proportion of the mixture arising from each stock potentially present as well as a 

 measure of the precision of this estimate may suffice. To develop these estimators, we posit that the 

 d istnbutions of characters of individuals differ among the stocks and that rules have been developed by 

 others with which some success in stock identification of individuals can be had. We require test 

 samples of individuals from each stock included in the mixture with which to evaluate the rules; these 

 samples must be other than the learning samples used to develop the rules. The rules are also appi led to 

 a sample from the mixture. Using the numbers of individuals in each test sample and sample of the 

 mixture which are assigned to each stock, we can e.stimate the composition of the mixture and the 

 precision of this estimation. 



Approximations based on large samples underlie the estimation. Numerical studies provide some 

 idea of the sample sizes required for the approximations to be satisfactory as well as of the behavior of 

 the estimators as related to performance of rules and sample sizes. 



We note that the roles of the learning and test samples from the segregated stocks may be inter- 

 changed, allowing a repetition of the procedure. 



Stocks offish frequently occur in mixtures. When 

 these stocks are of the same species at the same life 

 stage, the stock identity of an individual may be 

 difficult or impossible to ascertain. Yet if the dis- 

 tributions of characters of individuals differ 

 among stocks, some success may be had in iden- 

 tification of individuals in a mixture by use of 

 discrmimant analysis (e.g.. Hill 1959; Fukuhara 

 etal. 1962; Anas and Murai 1969; Parsons 1972; 

 Cook and Lord 1978) or more simply by a verbal 

 key iKonovalov 1975). In most important applica- 

 tions the correct identification of individuals is not 

 of direct value. Rather the accurate determination 

 of the proportions of the mixture belonging to each 

 stock is desired. 



Critical to accurate assessment of composition 

 of a mixture are the rules of assignme,nt of indi- 

 viduals to stocks. The rules applied to a vector of 

 measurements on an individual assign the indi- 

 vidual to one stock of those possible. Among rules, 

 those with lowest error rates of assignments pro- 

 vide the most accurate assessments, of course. If 

 individuals of known stocks, either those used in 



'Northwest and Alaska Fisheries Center Auke Bay Laborato- 

 ry, National Marine Fishenes Service, NOAA. P.O. Box 155, 

 Auke Bay, AK 99821. 



-Division of Fisheries, University of Alaska, Juneau, Box 

 1 447 , Juneau . AK 99802; present address: Alaska Department of 

 Fish and Game, Division of Commercial Fisheries, 333 

 Raspberry Road, Anchorage, AK 99502. 



developing the rules or new individuals, are as- 

 signed to stocks using the rules, a measm-e of error 

 rates is provided. Although some sense of the ac- 

 curacy of the rules is obtained, this does not pro- 

 vide a satisfactory evaluation of possible errors in 

 estimates of stock proportions from new mixtures. 

 Worlund and Fredin ( 1962) began to attack this 

 problem. They developed an estimation procedure 

 for stock proportions in a mixture of an arbitrary 

 number of stocks. Further, under restrictive as- 

 sumptions concerning knowledge of the accuracy 

 of assignments, Worlund and Fredin developed an 

 approximate variance expression for the esti- 

 mates of stock proportions in the mixture when 

 only two stocks composed the mixture. We extend 

 their approach now, developing methodology to 

 estimate stock proportions in mixtures of an arbi- 

 trary number of stocks as well as the variances of 

 such estimates under less restrictive conditions. 



BACKGROUND SITUATION AND 

 SAMPLING THEORY 



We assume A' stocks are known to potentially 

 occur in the mixture. Random samples of indi- 

 viduals are taken from each stock at a time when 

 the stocks are completely segregated; these may 

 be taken before or after the mixing. A random 

 sample of individuals from the mixture is also 

 taken. 



Manu.'^cnpl acci-pttKl November 1978. 

 FISHERY BULLETIN: VOL. 77, NO. 2, 1979. 



387 



