Roa et al,: Estimation of size at sexual maturity 



575 



percentiles of /gy,_ (Eqs. 3-11) In bootstrap, for each 

 sample size and each of the 5000 trials, we obtained 

 N 1^^^=5000 bootstrap samples. With the Monte Carlo 

 method, we resampled parameter estimate values 

 from unbounded normal distributions with Nj^^-.^ 

 5000. For completeness, 7V^^p^,^,^= 1 . 



In step 1, the deterministic size structure of the 

 population was conceived as a mixture of normal 

 probability distributions, each normal distribution 

 corresponding to an age class. The proportion of in- 

 dividuals at each size interval was characterized by 

 the following expression 



ran a 



is the random number of mature individuals 



(16) 



where the sum is over 10 age classes (0 to 9) and 41 

 size classes (0 to 40), and where f.i^ is determined by 

 a growth equation 



^, 



^R.(l-e-'-) 



(17) 



with known parameters (Table 2). Variance of size- 

 at-age (a'~) is known and constant through age (Table 

 2), and the proportion of individuals at age (P^^) is 

 given by a simple exponential mortality model 



-Ml 



-Ml 



(18) 



t=0 



where the mortality rate (M) is known and constant 

 through age (Table 2). 



Random variability came from two sources. First, 

 samples of the specified sizes were drawn, for each 

 trial, from a uniform probability distribution and 

 compared with the cumulative distribution of Equa- 

 tion 16, accumulating the scores in the respective 

 size intervals. This computation yielded a sample of 

 relative size frequencies p^^^. Next, we introduced the 

 second source of uncertainty by assessing the matu- 

 rity status (mature or immature) of individuals be- 

 longing to each size class. This random assignment 

 of maturity status came from resampling the bino- 

 mial probability distribution 



P(n = n 



rand 



U.rand 

 ^rand / 



pa 



)"-'(l-P(/)'""°'"'-""'"''),(19) 



where Pf/i was computed from the logistic model (Eq. 

 1) with known maturity parameters (Table 2) and 



out of «, 



I. rand sample ^ nl 



individuals in the size in- 

 terval /. In this way, step 1 was completed by ran- 

 domly assigning two properties to each data indi- 

 vidual: a size (continuous variable) and a maturity 

 status (dichotomous variable). With these data, step 

 2 was completed by using a nonlinear parameteriza- 

 tion of the logistic model (Eqs. 1 and 2) for obtaining 

 estimates of /3q and (i^, and their covariance matrix, 

 by means of the SIMPLEX algorithm (Press et al., 

 1992). Having this information in hand, step 3 was 

 completed by obtaining 2.5%, 50%, and 97.5% per- 

 centiles by each of the three methods. We pro- 

 grammed the MATSIMVL algorithm using Microsoft 

 FORTRAN for PowerStation 4.0 (Microsoft Corp., 

 1995). 



In the case of the Monte Carlo algorithm, we also 

 investigated the effect of the natural mortality pa- 

 rameter, by varying its level in simulation at M=0.2, 

 M=0.4, M=0.6, and M=0.8, for sample sizes of 

 N„,„^,,^=1000 and 5000 individuals. N,^,,. and N^^ 

 were both kept at 5000. 



Finally, we introduce real data to show two appli- 

 cations of the Monte Carlo method developed here. 

 First, we estimate Ip,^ (^^,^.=5000) for a single popu- 

 lation of the galatheid decapod Pleitroncodes 

 monodon. In this application, we estimate size con- 

 fidence intervals for percentages of maturity from 

 10% to 90% at steps of 10% . In this way a confidence 

 interval for the whole maturity curve is outlined. 

 Second, we compare samples of female anchovy 

 Engraulis ringcns from two localities 3^ of latitude 

 apart (N^,(^.=5000) to test the null hypothesis of equal 

 /jQ,, between them. 



Results 



The simulation analysis with MATSIMVL yielded 

 size-at-age and maturity-at-size data with the ap- 

 propriate behavior as Ng^„„i^ increased: size-fre- 

 quency distributions became smoother and maturity 

 data more closely followed a logistic curve, as shown 

 by one example output of MATSIMVL data-simulat- 

 ing routines (Fig. 1). 



A summary of the simulation results is presented 

 in Fig. 2. It shows that, under the simulation condi- 

 tions, the Monte Carlo method outperformed the 

 bootstrap and the Fieller methods in proportion of 

 success at all sample sizes and that it remained very 

 close to the nominal 95%; the bootstrap method suc- 

 ceeded 94% or less at all sample sizes, whereas the 

 Fieller method was unstable between sample sizes 

 of 500 to 5000, with a minimum of 93% success at 

 3000 (Fig. 2A). All three methods showed negligible 



