Roa et a\ : Estimation of size at sexual maturity 



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Figure 2 



Summary of results from the application of the three methods to simulated data. Open squares = Fieller's ana- 

 lytical method; open circles: bootstrap; closed circles = Monte Carlo. 



reflecting the effect of large sample size 'A^,q„, i^ = 

 4458). The Monte Carlo median /-^.^ (27.19 mm cara- 

 pace length) coincided with the MLE of /r,iy;=-MLE 

 (^„)(MLE(/3j ). The amplitude of the confidence inter- 

 val for Ip, showed an increasing trend towards ex- 

 treme values of P (Fig. 4), a reflection of the alge- 

 braic structure of Equation 3. Results of the second 

 application with two samples of female anchovies are 

 shown on Figure 5. Although different in their 

 lengths, probably due to different sample sizes, con- 

 fidence intervals from the two samples overlapped and 

 have the same upper limit. This result provides sup- 

 port to the hvpothesis of equal maturity schedules be- 

 tween female anchovies from the two localities. 



Discussion 



The logistic model is universally used as a math- 

 ematical description of the relation between body size 

 and sexual maturity. To model residuals, however, 

 two different approaches emerge: to consider them 



normally or binomially distributed, and closely re- 

 lated to this, to use the data as proportions or as 

 counts. In the first (as far as we know) formal treat- 

 ment of the problem. Leslie et al. (1945) used the 

 data as proportions, transformed to probit scores, and 

 assumed the normal distribution. Current research- 

 ers have not employed probit transformations (but 

 see Lovrich and Vinuesa, 1993) but have continued 

 using data as proportions and the normal distribu- 

 tion for residuals (Table 1). In this work however, 

 and following the arguments by Welch and Foucher 

 (1988), we emphasize the need to estimate the ma- 

 turity model using the data as counts and therefore 

 to consider residuals as binomially distributed. Un- 

 der this approach, the standard procedure for fitting 

 the maturity model is logistic regression (Hosmer and 

 Lemeshow, 1989). 



When the model has been fitted, the problem of 

 setting a confidence interval for the level of the pre- 

 dictor variable (size) that gives rise to a fixed pro- 

 portion of maturity is not trivial. We have explored 

 here three approaches: one analytical method based 



