BERGGREN and LIEBERMAN: RELATIVE CONTRIBUTION OF STRIPED BASS 



Analysis of Spawning-Stock Specimens 



Choice of morphological characters for segrega- 

 tion of Hudson, Chesapeake, and Roanoke spawn- 

 ing stocks followed three stages of statistical 

 analysis: correlation analysis between each 

 character and fork length (FL), analysis of the 

 effects of sex and age on each character, and dis- 

 criminant analysis. Analysis involved only speci- 

 mens with observations on all counts and mor- 

 phometric and scale-annulus measurements. 



Since spawning stocks do not include immature 

 specimens which occur in the coastal waters, we 

 chose only those characters that were independent 

 (i.e., not highly correlated) offish size and could 

 therefore be used to segregate specimens from the 

 entire stock. Characters were considered to be in- 

 dependent of length when variations (r^) attribut- 

 able to length in any stock were ssO.lO. Characters 

 not independent of length were used in further 

 analysis when the distribution of character values 

 had small overlap among spawning stocks since 

 such characters help identify stock origin. 



Multivariate statistical tests were made to de- 

 termine the effect of sex and age on the characters 

 used to determine the discriminant functions, 

 since one assumption of discriminant analysis was 

 that each stock was homogeneous. Differences in 

 character values among ages for males or females 

 and between sexes within each stock were tested 

 with a procedure that combined tests of equality of 

 means and equality of covariance matrices (An- 

 derson 1958). Assuming equal covariance ma- 

 trices, rejection of the null hypotheses of equal 

 distributions indicated that one or more of the 

 character means differed among ages or between 

 sexes. 



Multivariate discriminant analysis was used to 

 gain maximum separation among stocks. Linear 

 and quadratic discriminant functions (Anderson 

 1958; Kendall and Stuart 1968) for each spawning 

 stock were determined from character values ob- 

 tained from collections of that stock. A stepwise 

 procedure on the linear function was used to indi- 

 cate the subset of characters which best separated 

 the stocks. The quadratic function based on this 

 subset was formed if the assumption of a common 

 covariance matrix among spawning stocks needed 

 for the linear function was not met. The assump- 

 tion in discriminant analysis that characters had 

 a multivariate normal distribution was investi- 

 gated with histograms. 



Ability of the discriminant functions to separate 

 stocks and accurately estimate stock proportions 

 was assessed using functions based on total 

 spawning-stock collections and functions obtained 

 from a cross-validation procedure (Mosteller and 

 Tukey 1968). In this procedure collections were 

 randomly divided in half and discriminant func- 

 tions were determined from one-half and applied 

 to each half. Percentages of correct classification 

 and estimates of stock proportion were obtained 

 for each subset and compared with those from the 

 total sample. Comparisons were also made be- 

 tween estimated and known spawning-stock per- 

 centages. 



Although these estimates of stock percentages 

 may accurately approximate true percentages in 

 spawning-stock collections, they may deviate sub- 

 stantially from stock percentages in oceanic col- 

 lections. Fukuhara et al. (1962) stated that the 

 bias in these estimates increased as stock percen- 

 tages became more disproportionate. Since stock 

 percentages in oceanic collections may be more 

 disproportionate than stock percentages in 

 spawning-stock collections (i.e., 347^ Hudson, 469c 

 Chesapeake, and 20*^ Roanoke stocks), less biased 

 estimates of stock percentages may be needed. 



Adjusting Estimates of Stock Percentages 



Two procedures were developed to obtain esti- 

 mates of stock percentages that were less biased 

 than the as-classified (i.e., classifications obtained 

 directly from discriminant functions) estimates. 

 The first procedure adjusted estimates using a 

 technique described by Worlund and Fredin 

 (1962) which generalized to the three population 

 case methodology developed in Fukuhara et al. 

 (1962). This procedure used percentages of speci- 

 mens from each spawning stock that were mis- 

 classified into other stocks to correct as-classified 

 estimates for bias due to misclassifications. When 

 adjusted estimates were negative, as-classified es- 

 timates were modified by methodology developed 

 by Schuermann and Curry. ^ 



The second procedure iteratively reclassified 

 specimens based on updated prior probabilities 

 that specimens originated from each of the spawn- 

 ing stocks. The first stage of the procedure is the 

 same as the as-classfied procedure; therefore as- 



^Schuermann, A. C, and G. L. Curry. 1973. Notes on paramet- 

 ric programming. Unpubl. manuscr. Dep. Ind. Eng. Texas A&M 

 Univ., College Station. 



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