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Fishery Bulletin 97(3), 1999 



must be transferred to this variable. The first part 

 of the problem is the selection of the maturity model. 

 The available data consist of size (normally length) 

 and maturity status, which will be assumed to take 

 only two values: mature or immature. The predictor 

 variable is continuous and the response variable is 

 dichotomous. With such variables, model errors dis- 

 tribute binomially. Welch and Foucher ( 1988) recog- 

 nized this aspect of modeling maturity and showed 

 an efficient procedure based on the principle of maxi- 

 mum likelihood that takes advantage of the binomial 

 nature of the errors. 



For dichotomous data modeled as a function of a 

 continuous variable, the following simple logistic 

 function is a consequence of the assumption of a lin- 

 ear relationship between the logit link function and 

 a single predictor variable (Shanubhogue and Gore, 

 1987; Hosmer and Lemeshow, 1989; McCullagh and 

 Nelder, 1989): 



P(l): 



a 



1 + e 



ii.*P,i 



:i) 



where P(l) = proportion mature at size /; and 

 a, Pq, and p^ = asymptote, intercept, and slope pa- 

 rameters, respectively (see also Eq. 3 ). 



The estimates of these parameters, given a data set, 

 are chosen from the point at which the product of 

 binomial mass functions of all data points (the likeli- 

 hood of the data under the model) is a maximum, or 

 equivalently when the negative of the log likelihood 



-na,pr,p,) = -^[(/!, )ln(P(/)) + (n, - /;, )ln(l- P(/))] (2) 



IS a minimum. 



these results, we may undertake the converse prob- 

 lem of estimating size at fixed P^c maturity, which 

 takes the form 



Pi 



1 



Ik 



(3) 



In Equation 3 it is assumed that the asymptote pa- 

 rameter (a) from Equation 1 is fixed at 1. This as- 

 sumption is justified on the basis of several published 

 works on size at maturity, showing that all individu- 

 als were mature above a given size during the repro- 

 ductive season (Table 1). Furthermore, if P^^ and /3j 

 are MLE of /i^ and /^^ and they are used to compute 

 Ipr-^ from Equation 3, then /^^ is also MLE. We show 

 below three procedures to perform this task and then 

 test them by generating data from Monte Carlo simu- 

 lation of the age-size structure and maturity progres- 

 sion of individuals of a hypothetical population. 



Analytical estimation 



The logistic model in Equation 1 belongs to a class of 

 generalized linear models studied by McCullagh and 

 Nelder (1989). These authors consider the problem 

 of building approximate confidence intervals for the 

 level of the predictor variable that gives rise to a fixed 

 proportion in the response variable. They suggest the 

 use of Fieller's (1944) theorem, according to which 

 the linear combination 



P,, + p,lp.-,-g(P^) = Q, 



(4) 



where Ip,^ = the value of the predictor variable for a 

 fixed proportion; and g(P^y)=ln(P^^/(l-PQ)) (the logit 

 link function) is approximately normal with mean 

 zero and analytical variance given by 



where /i - thenumber of mature individuals; and 

 n = sample size at /; 

 P(lJ = Eq. 1; and 



where a constant term that does not affect the esti- 

 mation is omitted. 



Given the nonlinear nature of normal equations, 

 the minimum is found by an iteration algorithm. The 

 parameters estimated by minimizing Equation 2 are 

 maximum likelihood estimates (MLE). In practical 

 situations, the logistic model may be modified from 

 its original form to allow more biological reality 

 (Welch and Foucher, 1988). 



The result from fitting the model (Eq. 1) to the data 

 by using the objective in Equation 2, is a vector of 

 parameter estimates and a covariance matrix, which 

 represents the uncertainty associated to them. With 



v'^ilpr.) = vaT{Po) + 2lp,; cov{Po,Pi) + Iprr, var(^i). (5) 



The lOO(l-a)'^ confidence interval is the set of val- 

 ues defined by 



Pi 



where z^^/., - a quantile of the normal distribution. 



Other link functions like probit, common in the field 

 of toxicology (Finney, 1977), are not investigated in 

 this paper. 



Bootstrap estimation 



Bootstrap is not a uniquely defined concept (Efron 

 and Tibshirani, 1993). This means that bootstrap 



I 



