FISHERY BULLETIN: VOL. 84, NO. 2 



knife estimator of Equation (18) and for v(Y.) see 

 Sukhatme (1954). Yj is generally subject to 

 considerable bias. 



The c.v. of total catch of bocaccio, chilipepper, and 

 widow rockfish for different categories by port-year 

 groups (Table 6) show that the estimators Yj and 

 Yj are highly efficient compared with Yj', also, Yj 

 turns out to be slightly superior to Yj since the 

 jackknife estimator v 2 (Yj ) is an underestimate and 

 does not take into account the contribution of the 

 within component of variance. Thus, the empirical 

 evidence supports strongly the use of the estimator 



Yr 



Table 6.— Coefficient of variation (in percent) of estimates of total 

 catch of bocaccio, chilipepper, and widow rockfish per cluster by 

 ports during 1978 and for different categories for the three 

 estimators 9\, Y h and V", . 



AGE-COMPOSITION: DOUBLE 

 SAMPLING 



Studies mentioned in the Introduction section 

 have shown that since aging from otoliths of each 

 individual fish in a sample is more expensive than 

 an easily measured quantity such as length, it may 

 pay 1) to choose a random subsample from the whole 

 sample of length measurements for age determina- 

 tion or 2) stratify the sample according to length 

 classes and choose a subsample from each class for 

 age determination. The technique is profitable only 

 if the correlation between length and age is fairly 

 high. 



It may be recalled that considerable bias is in- 

 troduced by applying age-length keys developed dur- 

 ing a year to subsequent years. Both Kimura (1977) 

 and Westrheim and Ricker (1978) showed that age- 

 length keys can yield most inefficient estimates of 

 numbers-at-age with substantial overlap of lengths 

 between ages. In the latter case the correlation be- 

 tween length and age will be low for the larger and 



the very small sizes. Consequently, we will need a 

 higher sampling intensity at the tails to provide 

 reliable estimates of age for such sizes. 



In the construction of length strata for selection 

 of the subsample, additional questions arise on 1) 

 number of strata to choose, 2) strata boundaries to 

 decide, and 3) the number of sampling units to be 

 allocated to each stratum for deriving maximum 

 gain from double sampling. These are discussed as 

 follows. 



Number of Strata 



The values of V(y st )/V(y) (Cochran 1977) are 

 given below as a function of L, the number of strata 

 using the linear model 



y = a + fix + £ 



(30) 



where y is the length, x the age of female widow 

 rockfish and 



ViVst) 

 V(y) 



L 2 



+ (1 - p 2 ) 



(31) 



where P is the correlation between length and age 

 in the unstratified sample and L the number of 

 strata. It can be shown for this model that when L 

 > 6 and p > 0.95, there is hardly any gain due to 

 stratification (Table 7). The improvement in 

 stratification is highest for data set 1 for which p 2 

 = 0.7004 and lowest for set 3 for which P 2 = 

 0.5278. The results for the regression model indicate 

 that unless p exceedes 0.95, little reduction in 

 variance is to be expected beyond L = 6. Data sets 

 1, 2, and 3 support this conclusion. In fact, there 

 does not seem to be any profit resulting from in- 

 crease in strata beyond L = 5. 



Strata Boundaries 



For the length-age strata on 239 females (widow 

 rockfish) landed during 1982 at San Francisco and 

 the rule based on the cumulative of \Jf{y) (Cochran 

 1977) where y denotes the length in centimeters, the 

 nearest available points for the two strata are 



Stratum 



416 



