YORK and KOZLOFF: NORTHERN FUR SEAL PUPS 



of pups as a function of numbers of breeding males 

 (Figs. 1, 2). The analyses of variance of these regres- 

 sions indicated that the quality of the fits was ex- 

 cellent and that the relationship might be used for 

 predictive purposes. No intercepts, except that for 

 1916 data, were significantly (P > 0.95) different 

 from 0. We were interested in subsampling the 

 rookeries (possibly conducting the estimation on as 

 few as four rookeries) and therefore, if a regression 

 estimator were to be used, it was desirable to reduce 

 the number of parameters as much as possible. In- 

 asmuch as the intercepts were not different from 

 0, the simpler model with no intercept was con- 

 sidered appropriate. Since the variance of the pup 

 estimates was not constant for each rookery, 

 weighting appeared necessary. The variance of the 

 estimates of pup numbers was roughly proportional 

 to the number of bulls, and in such cases (Draper 

 and Smith 1966), the best estimate of the slope of 

 regression line is the average number of pups 

 divided by the average number of bulls (equivalent 

 to the ratio of the total number of pups to the total 

 number of bulls). In this case, the total number of 

 pups on all rookeries was estimated in the follow- 

 ing manner: Let P^, . . . , P„ and Bi, . . . , B„ be as 

 above, and B a count of the total number of bulls 

 on all rookeries. Then the total number of pups on 

 all rookeries may be estimated as 



B 1 P, 



T^ = ^^^ = rB. 



(3) 





One estimate of the variance of this ratio estimator 

 is 



Var (Tj,) = ^ f\ I Var(P,) 



" \ "^ 1=1 



,,T*i^M (Cochran 1977). (4) 



When stratified random sampling was used in- 

 stead of simple random sampling, we calculated the 

 estimator in the same way since the ratio of pups 

 to breeding males did not vary significantly between 

 strata. The difference was due to the evaluation pro- 

 cedures; the number of logical sampling combina- 

 tions differed and the analysis was restricted to 

 those combinations of sample rookeries that were 

 consistent with the sampling design (e.g., one small 



rookery, one medium-sized rookery, and two large 

 rookeries). 



Another way to estimate the ratio and its variance 

 is with jackknife methods (Hosteller and Tukey 

 1977). Let r_, be the ratio of pups to breeding 

 males on all but the i^^ rookery, and r the ratio of 

 pups to breeding males on all the sample rookeries 

 (as in Equation (5)): 



I {nP - P,) 



r-i = 



1=1 



I (nB - B,) 



i=l 



(5) 



Then, the i'^ pseudovalue is r* = nr - (n - 1) 

 r_i. The jackknife estimate of the ratio, r*, is the 

 mean of the r,*'s and the variance of r* is 

 (Mosteller and Tukey 1977): 



Z (r* - r*y 



Var(r*) 



1=1 



n (n - 1) 



Thus, the jackknife estimate of the total numbers 

 of pups on all rookeries, Tj, is 



Tj = r* B, and Var(T^) = B^ Var(r*). 



The advantage of the jackknife estimate over the 

 ordinary ratio estimate is the reduction of bias and 

 a simple method of calculating the variance. 



The ordinary regression estimate (assuming that 

 the intercept is 0) of the ratio of pups to bulls is 



I P, B, 



i = l 



S = 



n 



i=l 



Thus, the regression estimator of total numbers of 

 pups is 



Tn, = sB and Var(T;j„) = B^ Var(s). 



The estimate of the variance of s is calculated from 

 the mean square residual of the regression equation. 



RESULTS 



Regressions of numbers of northern fur seal pups 



369 



