FISHERY BULLETIN VOL. 77. NO, 2 



The sample from a stock at time of segregation is 

 partitioned into two subsamples, called the learn- 

 ing and test samples after Cook and Lord ( 1978). 

 Learning samples from the K stocks are used to 

 develop rules of assignment; the assumptions and 

 methods used are arbitrary for our purpose. The 

 realism of the assumptions forming the basis of 

 the rules is not critical; performance of the rules 

 and some knowledge of this performance is impor- 

 tant. Performance of the rules is determined by 

 their application to the test samples. The rules are 

 also applied to the sample of the mixture. Using 

 the numbers assigned to each of the stocks by 

 application of rules to the test samples from the 

 segregated stocks and the sample from the mix- 

 ture, we can estimate the composition of the mix- 

 ture and the precision of this estimation. 



A caveat concerning situations in which the 

 methodology is not appropriate is needed before 

 we begin. What follows presumes the individuals 

 of a stock in both the test sample and mixture 

 sample are drawn from a common distribution of 

 characters used in the rules. When the condition is 

 violated, performance of the rules would differ im- 

 permissibly between test samples and that of the 

 mixture. We must avoid characters on which a 

 selection process occurs between the mixture and 

 the separate stocks. 



Test Sample Theory and Analysis 



Once particular rules have been established 

 from the learning samples (e.g., using discrimi- 

 nant analysis), individuals forming each stock in 

 effect have been partitioned into K mutually ex- 

 clusive gi'oups corresponding to those assigned by 

 the rules to one of each of the K stocks. We define 

 4>i;i to be the proportion of the individuals compris- 

 ing the ^th stock which is assigned by the rules to 

 the jth stock. Also we let ?t, be the number of 

 individuals in the test sample from stock k as- 

 signed by the rules to stockj, and let T\ = (^^j, t/,.,. 

 . . . , t/^). Assuming the number of individuals in 

 the test and learning samples is small as compared 

 with the number of individuals composing the 

 stock, the probability of the occurrence of vector 

 T'j.is, to a good approximation, given by the mul- 

 tinomial probability function, i.e.,'' 



^The dot notation implies summation over the subscript. Thus 



^' ( is the size of the test sample from the ^th stock. 



Pin) 



'/; 1 'fc 2  • • '),■ K 



0fc 1 <^h2 



<t>kK 



(1) 



Because the probabilities 4>i. are usually un- 

 known, we estimate them from 7"^ by the well- 

 known maximum likelihood estimator 



'I'fe = {thiltkJk2ltu /;ck/';.-) (2) . 



corresponding to the parameter vector 't)'^, = (4>i,i, 

 t/)^.,, . . . , <^,,l- 'fcj, is unbiased and has the 

 variance-covariance matrix 



~'K = 



<>hi('^-<>i!i) 0M0;,-; 



')..■ 



t„- 



<>k2<>kl 0/;2(l-0fe2) 



ll.-- 



<>)n<>kK 



<t>k 2^k K 



<>k KC^-'h k) 



(3) 



Test samples from different stocks are statistically I 

 independent and covariance between elements of 



<1j^ and *^ 



are zero for /; ^ k' . 



Mixed Sample Theory and Anahsis 



The mixture of stocks at the time of sampling is 

 comprised of possibly as many as K stocks. Ignor- 

 ing for the moment the actual stock composition of 

 the mixture, our rules established from the learn- 

 ingsamples partition the mixture into A' mutually 

 exclusive groups again corresponding to the K 

 stocks to which individuals are assigned. We 

 define A , to be the proportion of the individuals 

 composing the entire mixture which would be as- 

 signed to the,/th stock by the rules. Also we let w, 

 be the actual number offish in the sample from the 

 mixture which are assigned to stocky. If the size of 

 the sample from the mixture is small compared 

 with the number of fish composing the mixture, 

 the probability of observing the vector ,Vf' = i/?; ,, 



;?! ^ m^\ is given by the multinomial 



probability function, i.e.. 



388 



