FISHERY BULLETIN: VOL. 76, N0.4 



generated by S is still relatively linear as there are 

 no terms in S containing n, k, and r and the pair- 

 wise nonlinear effects are small as discussed 

 above. Table 2 provides examples of points on this 

 three dimensional surface when n, k, and r are 

 equal in absolute values. It can be seen there, for 

 the example examined, that if the absolute values 

 of/j, /?, andr are 0.10, the sensitivity of N, 5 ranges 

 from -12 to +12. 



An empirical equation can be fitted to the sen- 

 sitivity surface (S) by fitting a linear function for 

 each factor considered independently and by de- 

 termining a nonlinear term for n and r. The gen- 

 eral form of this fitted equation is 



SAn,k,r) = (bK+b^n+b^r+b^nr) x 100 (10) 



where the 6's are constant. The exact value of the 

 b's depends on the number of years the population 

 is projected back in time. For the example consid- 

 ered here, projecting back firom 1974 to 1959, the 

 values of the b's are shown in Equation (11): 



S,-(n,/2,r) = (0.573/2+0.427 n-0.164r-0.125nr) 

 X 100. (11) 



bridled dolphin are summarized in Tables 3 

 through 6. Calculated values of the variance of A^^ 

 from Equation (9), when all of the random variables 

 are assigned a coefficient of variation of 30%, are 

 given in Table 3, over all years from 1974 to 1959. It 

 can be seen that both the variances and the 

 coefficients of variation (CV) generally decrease. 

 The reduction of the CV over time is due to the fact 

 that the major contributions to the back estimates 

 of the population size are the addition of the kills of 

 the previous years, since the reproductive rate is 

 small. The variance of a sum of independent ran- 

 dom variables is the sum of their variances. This 

 always results in a C V for the sum which is smaller 

 than the greatest C V of any of the random variables 

 when the expected values of the random variables 

 are positive (Appendix II). As a generalization, it 

 can be stated that when the net reproductive rate is 

 small the CV of the back estimate will not be larger 

 than the largest CV of any of the random variables, 

 and will usually be smaller. 



Table 4 shows the breakdown of the variances 

 calculated in Table 3 into their major components. 

 The variance of Nq is the major factor in the vari- 

 ance of these back estimates. The contribution of 



This empirical approximation [Equation (11)] de- 

 viates by <2 from the true values of Sjg for values 

 of n, k, and r <0.5. This emperical equation is 

 useful as the magnitude of the 6's provides a mea- 

 sure of the relative sensitivity of the different fac- 

 tors. Thus in Equation ( 11) it can be seen that for 

 the example considered here the 1959 abundance 

 estimate (N15) is most sensitive to bias in the esti- 

 mates of the kills. This empirical equation also 

 provides an easy way to generate approximate 

 values of S for any combination of values for n, k, 

 and r. 



Variance 



Table 3. — Calculated variance and coefficient of variation for 

 the back estimate of dolphin abundance when all random vari- 

 ables have a CV of 30%. 



Table 4. — Breakdown of the variance of AT^ into the major com- 

 ponents that contribute to the calculated variance. 



Contribution to the variance of 

 ' — /V, (  10'°) due to the variance in 



776 



