SMITH AND POLACHECK: ANALYSIS OF SIMPLE MODEL 



S^ (n, k, r) = 



N^ (n, K r)-N^ 



N, 



• 100 . (7) 



Estimation of Variance 



The variance of the backcalculated estimate of 

 A^, from Equation (4) was approximated using the 

 delta method (Sober 1973). This method is based 

 upon a Taylor series expansion for a function in 

 which quadratic and other higher order terms are 

 ignored. If/" is a function of the random variables 

 x^, X2,X2 . . . ,Xn then the expression for the vari- 

 ance of/" by the delta method is 



v(r(x^,x,,X3...,xj)= iv(xp(^]' 



+ 2S2 Gov (X.,X.) I-J^  -^1 . (8) 



i^j 



^i' v\ax. bx.i 



1 J/ 



In applying this expression to Equation (4), it is 

 necessary to be able to define which of the 

 parameters should be considered as random vari- 

 ables, and to give reasonable estimates for value of 

 the variances and covariances of these variables. 

 For the purpose of exploring the behavior of Equa- 

 tion (4), we assumed that the estimates of all the 

 parameters in Equation (4) are independent ran- 

 dom variables. The covariance terms in Equation 

 (8) are then zero. This approach provides a picture 

 of the variance of the back estimate of abundance 

 if in fact independent estimates of the kills and the 

 net reproductive rates were available for each 

 year. A generalized expression for the variance 

 using this approach is 



V(iV^) 



V(iV,) 



dN. ^2 



3A^. 



t 



+ S 



V{K.) I 



dN. 

 dN. 



+ 2 V(i?.) 

 y=i ^ r 



'dN. 



(9) 



where all parameters are defined as for the basic 

 model [Equation (4)]. For detailed expressions for 

 each of the right hand terms see Appendix I. 



As noted the method used for approximating the 

 variance of a function depends on the higher order 

 terms in the Taylor's series expansion being small. 

 The higher order terms in the delta method ex- 

 pression for the variance of A/^, are composed of the 



second and higher order derivatives of N, with 

 respect to A^n- K^, and Rf, and the higher order 

 central moments of the probability distributions of 

 the estimates of N„, K,, and R, (i.e., skewness, 

 kurtosis, etc.). The second and higher derivatives 

 with respect to A^^ andKf are zero. Thus the terms 

 involving/?, are the only higher order terms not 

 equal to zero. The higher order derivatives of Nf 

 with respect to i?, involve i?,+ j to increasing nega- 

 tive powers. The three higher order moments ofR, 

 are always decreasing since /?, is much less than 

 one. Thus each of the higher order terms in the 

 delta method expression for the variance of N, are 

 each less than the first order term in R; (iii of 

 Appendix I). The contribution of this first order 

 term in Rt to the variance ofN, is small, as shown 

 below. Thus the error induced by ignoring the 

 higher order terms in the Taylor's series appears 

 small. 



The objective in doing the variance calculations 

 was to understand the behavior of the variance of 

 the population size when estimated by the basic 

 back projection model [Equation (4)]. Thus a range 

 of variances was calculated for a range of reason- 

 able values of the variances of the estimated 

 parameters. However, in our example of bridled 

 dolphin estimates of the variance of many of the 

 parameters were not available. Many of the kill 

 estimates were not independently estimated and 

 hence have large unknown covariances (Smith 

 and Polacheck^). Estimates of net reproductive 

 rate were obtained by extrapolation from other 

 populations and fi'om assumptions about density 

 dependence. It is not clear that the uncertainty in 

 these estimates can adequately be described by 

 the notion of variance. Thus, the variances that we 

 used and that we calculated for N, should not be 

 interpreted as actual estimates of variance for this 

 population. 



RESULTS 



Bias 



The results of the sensitivity analysis of the 

 basic model will be presented by examining the 

 effects of varying each of the variables «, k, and r of 

 Equation (7), separately, and then in combina- 

 tions. 



The sensitivity of the back projected estimates 



^Smith, T. D., and T. Polacheck. 1977. Uncertainty in estimat- 

 ing historical abundance of porpoise populations. Contract Rep. 

 MM 7A C006, 39 p. Marine Mammal Commission, 1625 Eye 

 Street, Washington, DC 20006. 



773 



