FISHERY BULLETIN: VOL. 84, NO. 2 



set (2/ 1; x lit x 2 i), which includes all the data from 

 each life stage and the maximum ages of mortality 

 stanzas (u g 's), is important to ensure the accuracy 

 of the estimates of the IMRs. The determination of 

 u's depends primarily on the changes of the mor- 

 tality rates, which may be related to the changes in 

 morphology or behavior that affects mortality rates. 

 In the best fit of the SEM, however, the end points 

 of morphological patterns may not correspond to the 

 maximum ages. Three life stages were identified for 

 anchovy eggs and larvae, with the end point of mor- 

 tality stanza 1 being the average age of yolk-sac lar- 

 vae («! = 4.5 d). In the MEM, the hatching time (t h ) 

 was used, but, the best fit of the SEM occurred when 

 u x = 4.5 d. Two mortality stanzas were assigned to 

 three life stages of anchovy (<20 d) because from 

 the existing data, no evidence for a change in the 

 IMRs within a life stage existed although the data 

 may not have been adequate to detect such changes. 

 The regression estimates of the IMRs for the SEM 

 can be obtained by either NR or LR as described in 

 the previous section. If NR is used, Equation (6) is 

 fitted to the data set Qj it x lif and x 2 i) directly to 

 obtain estimates of parameters of X^t) and X 2 (t). 

 Because the variance of egg data is larger than that 

 of larvae, a weighted NR (WNR) would be prefer- 

 able If errors are assumed to be multiplicative, 

 taking the logarithm of both sides of the Equation (6) 

 yields 



\n(y t ) = A - ax u - pin 



' x 2i 



+ &. 



(7) 



The data set (ln(2/i)> Xj*, and \n(x 2i lu{}) is then used 

 to estimate a and (1 through linear least squares 

 regression. 



MAXIMUM LIKELIHOOD ESTIMATOR 



(MLE) 



The MLE is presented here as an alternative 

 method of estimating IMRs. Because the data used 

 for mortality estimators are grouped by age, I fol- 

 lowed the procedures described by Kulldorff (1961) 

 and McDonald and Ransom (1979) for grouped data. 

 Here, N { = Y { _ x - Y { (number of deaths between 

 ages ti_i and t£ of a single cohort are multinomial 

 variables, each with probability 



Pi = Sfo-i) - SiU). 



L{N % ,P t {z');i = 1, ...,/) ex n p&y, 



1=1 





(8) 



where z is the parameter vector in X(t). The 

 derivatives of the logarithm of likelihood function 

 with respect to the parameters z's are set equal to 

 zero. Solutions to the simultaneous equations 



31nL 



dZs 



= 



are MLEs of z, if certain conditions are satisfied 

 (Kulldorff 1961). In marine fish only the IMRs of 

 a few life stages are considered because of the lack 

 of data. It is then necessary to compute the condi- 

 tional probability 



P x = % 1 <r<i i |TGDJ 



= [S&_i) - S(tJ\/P(T 6 D) 



where D is the domain of ages of life stages con- 

 sidered. 



Because I considered only the IMRs of anchovy 

 eggs and larvae of ages >4 h (0.17 d) and <20 d, the 

 conditional probabilities are computed from a 

 truncated exponential and Pareto survival proba- 

 bility (Equation (2)) (Gross and Clark 1975, p. 

 128-132): 



P % = P(t i _ 1 <T< ti\ *i<r<20) 

 = (S(«,_i) - SitiWWd - S(20)) 



e -<-i _ e ~ ot ' 



e~ at ^ - e~ 



'20 \-" 



u. 



Pi = 



Vi 



Mi 



g-ati _ e ~au x 



(9) 

 u x < t { < 20. 



The likelihood function of N { 's for the whole life 

 cycle (i = 1, ...,/, and Yj = 0) is 



Then the likelihood function of iV/s for anchovy 

 eggs and larvae of ages <20 d is 



400 



