LO: MORTALITY RATES OF NORTHERN ANCHOVY 



the IMRs (X(t)) of anchovy eggs and larvae. If the 

 life-stage-specific IMR is the main objective, the 

 MEM is an easy method for obtaining the estimates 

 of IMRs. The mortality curves (Equation (3)) are 

 nonlinear functions of age (t). The IMRs can be 

 estimated by either nonlinear regression (NR) or 

 linear regression (LR) after the data set (y i7 t$ is 

 transformed. The NR is based upon the assumption 

 that the errors are additive. The observed mean 

 daily production (y^ relates to the conditional sur- 

 vival probability as 



Vi = % Sife; h(t)] + «ii 



= y e- at , + e u 



k < Mi 



(4a) 



Vi = K s iik; hit)] + e 2l 



Vu,\ 



+ e 2l 



u x < ti < 20 (4b) 



where % = t h ~ 3 d old. Nonlinear regression 

 estimation procedures provided by standard statis- 

 tical packages such as BMDP statistical software 

 (Dixon et al. 1983) are then used to estimate the 

 parameters of IMRs, i.e., a and /?. 



The LR assumes that the errors are multiplicative. 

 The observed daily production (y { ) relates to the 

 conditional survival probability in the form of 



Vi = e u , SJU; Ut)) e gi for g = 1,2. 



i-i 



The logarithm of both sides of the equation yields 

 two linear functions 



ln(&) = A - ati + £ u 



t 



k < Wi 



(5a) 



Info) = B - /Jin -t + E 2i Ul <U< 20. (5b) 



u 



Equation (5a) is then fitted to data set (l n fo)> k f° r 

 k < u x ), and Equation (5b) is fitted to data set 

 (Info), \n(klu x ) for u x < k < 20 d) to estimate a and 



SINGLE-EQUATION MODEL (SEM) 



The SEM consolidates all the conditional survival 

 probabilities (S g (t)) from each mortality stanza into 

 a single equation. It not only eliminates discontin- 

 uities at transitions between life stages, but also im- 



proves the precision of overall mortality estimates 

 because of the large sample size Moreover, the SEM 

 makes it possible to estimate the IMR for life stages 

 where data are scarce. 



Based upon Equation (2), S(k) of anchovy eggs 

 and larvae is 



m = 



or 



S(k) = 



Sife) k < u, 



S^u^iti) Ui < ^ < 20 



s^tdSM k < u x 



S x (u x )S 2 (k) u x <k< 20 



where S x (u x ) = P (T > u x \T > 0) = e~ M \ S 2 (u x ) = 

 P(T> u x \T > u x ) = 1, and u x = t ys = 4.5 d. Thus 

 by creating two new independent variables x xi and 



x 2i such that 



X li - 



and 



x 2i - 



k k < u x 



u x u x < ti < 20 



u x k < u x 



k % < t { < 20 



it follows that S(k) = S x (x xi )S 2 (x 2i ) and the mortal- 

 ity curve can be written as 



E(yd = e^ x (x Xl )S 2 (x 2l ) = e,e-°*i> 



<x 



2i 



-p 



u. 



(6) 



The data set for fitting Equation (6) looks like 



u x = 4.5 d 



In order to use Equation (6) to estimate the IMRs 

 of eggs and larvae in Equation (1), a combined data 



399 



