Roa et al : Estimation of size at sexual maturity 



573 



samples may be obtained by conceptually different 

 resampling procedures. In the context of logistic re- 

 gression, it is possible to resample the observational 

 pair (/,,/!,), or the semiobservational pair (Z^, P(l)+£^), 

 with fj as a realization from the residual distribu- 

 tion of the logistic model. To be valid, this second 

 resampling unit needs the assumption of indepen- 

 dence between e^ and l^. As stated by Efron and 

 Tibshirani (1993), this is a strong assumption that 

 can fail even when the model P(l) is correct. These 

 authors remark that bootstrapping the obsei-vational 

 pair is less sensitive to assumptions than bootstrap- 

 ping residuals. Therefore, in our work an observa- 

 tion to be resampled with replacement is defined as 

 the pair (length, maturity status). For each and all 

 bootstrap samples, a resampled frequency distribu- 

 tion for Ipr. is obtained by fitting the maturity model 

 in Equation 1 with the objective function in Equa- 

 tion 2 and by computing Ip, with Equation 3. The 

 confidence interval is obtained by application of the 

 bias-corrected and accelerated (BCa) method, recom- 

 mended by Efron and Tibshirani (1993). 



Monte Carlo estimation 



In Monte Carlo resampling, a model is assumed for 

 the distribution of the estimator and then data are 

 generated computationally to assess the amount of 

 variation (Manly, 1997). In our case, we consider a 

 Monte Carlo resampling of maturity parameters from 

 the modeled joint probability distribution of the es- 

 timates /^Q and /Jj for computing / from Equa- 

 tion 3. In contrast to the bootstrap approach, the 

 implementation of this approach needs only one fit- 

 ting of the logistic maturity model and then uses the 

 asymptotic distribution of estimated parameters of 

 the model to generate the probability distribution of 

 the derived statistic Ip... These parameter estimates, 

 ^Q and /}j, distribute asymptotically bivariate nor- 

 mal, with mean vector equal to the population pa- 

 rameters and variance given by their covariance 

 matrix (for nonlinear least-squares: Johansen, 1984; 

 for maximum-likelihood estimates: Chambers, 1977 ). 

 The bivariate normal distribution of p^ and /3j has 

 a strong covariance component, which is the same 

 as to say that these estimates are highly correlated. 

 This also means that much of the variance in one 

 estimate is given by the variance in the other one. 

 Ignoring such correlation would lead to an overesti- 

 mation of the variance oi Ip,.. In a Monte Carlo set- 

 ting, the correlation between parameter estimates 

 may be considered in the computation by making the 

 resampling of one estimate conditional on the 

 resampling of the other one. In this work we develop 

 such a technique using the theory of least-squares 



estimates of two linearly related normal variates 

 (Draper and Smith, 1981). This approach is justified 

 by the asymptotic nature of standard errors. If /Jq 

 and j8j and are two normal random variables that 

 are linearly related, then we may write the linear 

 equation 



Pi=bo+b^Po- 



(7) 



This equation may be reversed by writing /?j, as a 

 linear function of p^ because both are random vari- 

 ables. It can be shown that (Draper and Smith, 1981) 



bi 



Ml 



(8) 



where /■ is^ the estimated linear correlation coefficient 

 between /3q and P^, and S/jq and S p^ are the respec- 

 tive standard errors. Furthermore, from Equation 7 



bo=p,-b,p,. 



(9) 



Therefore, the high correlation coefficient between 

 both maturity parameters can be accounted for by 

 free sampling from the marginal distribution of one 

 parameter estimate (for example, /Jq) in each Monte 

 Carlo trial and by computing the other by using 



Pi,=Pi-t 



A,. A 



Pi + r 



AA 



^. 



A, 



Ao 



^30+^^, 



Ao.Ai 



^0,,-A, 



A, 



Po.j 



, (10) 



which is obtained by replacing Equations 8 and 9 in 

 Equation 7. For each trial (indexed byj), a /?y value 

 is selected from the normal probability distribution 

 defined by its estimate and standard error, and 

 then the mean P^ value is computed by using Equa- 

 tion 10. 



The variance of the estimate is the p^ variance 

 due to the linear relationship with /^^ plus a residual 

 variance not explained by the relationship. The vari- 

 ance due to the relationship is directly transferred 

 from ^Q to /}j through the Monte Carlo resampling 

 of /3q and its mapping onto ^j by using Equation 10. 

 The residual variance must be added in each trial 

 with 



^p^, residual 



Ao.Ai 



(11) 



