570 



Abstract. -Size at 509c maturity is 

 commonly evaluated for wild popula- 

 tions, but the uncertainty involved in 

 such computation has been frequently 

 overlooked in the application to marine 

 fisheries. Here we evaluate three pro- 

 cedures to obtain a confidence interval 

 for size at 50% maturity, and in gen- 

 eral for P% maturity: Fieller's analyti- 

 cal method, nonparametric bootstrap, 

 and a Monte Carlo algorithm. The three 

 ' methods are compared in estimating 

 size at SC* maturity t/jf,,-! by using 

 simulated data from an age-structured 

 population, with von Bertalanffy 

 growth and constant natural mortality, 

 for sample sizes of 500 to 10,000 indi- 

 viduals. Performance was assessed by 

 using four criteria: 1 ) the proportion of 

 times that the confidence interval did 

 contain the true and known size at 50'7( 

 maturity, 2i bias in estimating /jq';- 31 

 length and 4i shape of the confidence 

 interval around /-|j,.. Judging from cri- 

 teria 2-4. the three methods performed 

 equally well, but in criterion 1, the 

 Monte Carlo method outperformed the 

 bootstrap and Fieller methods with a 

 frequency remaining very close to the 

 nominal SS'/r at all sample sizes. The 

 Monte Carlo method was also robust to 

 variations in natural mortality rate 

 (M), although with lengthier and more 

 asymmetric confidence intervals as M 

 increased. This method was applied to 

 two sets of real data. First, we used 

 data from the squat lobster Pleuron- 

 codes monodon with several levels of 

 proportion mature, so that a confidence 

 interval for the whole maturity curve 

 could be outlined. Second, we compared 

 two samples of the anchovy Engraulis 

 ringens from different localities in cen- 

 tral Chile to test the hypothesis that 

 they differed in size at 50% maturity 

 and concluded that they were not sta- 

 tisticallv different. 



Estimation of size at sexual maturity: 

 an evaluation of analytical and 

 resampling procedures 



Ruben Roa 



Departamento de Oceanografia 

 Universidad de Concepcion 

 Casilla 160-C, Concepcion, Chile 

 E-mail address rroa a udec cl 



Billy Ernst 



School of Fisheries, WH- 10 



University ol Washington, Seattle, Washington 98195 



Fabian Tapia 



Departamento de Oceanografia 

 Universidad de Concepcion 

 Casilla 160-C, Concepcion, Chile 



Manuscript accepted 28 August 1998. 

 Fi.sh. Bull. 97:570-580 11999). 



Size at 50% maturity (/jg^,) is com- 

 monly evaluated for wild popula- 

 tions as a point of biological refer- 

 ence (see Table 1 ). To estimate /gg,. , 

 a sample of organisms known to 

 have just reached sexual maturity 

 could be available and their arith- 

 metic mean size can be used as an 

 estimator. However, the sample 

 needed to obtain such a design- 

 based estimator (Smith, 1990) for 

 wild populations might be too ex- 

 pensive and would involve time-con- 

 suming histological procedures. 

 Fisheries biologists prefer to con- 

 ceive size at first maturity as the 

 average size at which 50% of the 

 individuals are mature. With this 

 conception, the estimator is not 

 based on a sampling design but on 

 a model of the relation between 

 body size and the number of indi- 

 viduals that are mature from a to- 

 tal number at each of many size in- 

 tervals. The variance of a design- 

 based estimator is determined by 

 sampling design (Thompson, 1992). 

 The variance of a model-based esti- 

 mator is not as easily obtained. A 

 sample of published works in the 

 fisheries literature provides a mea- 

 sure of the frequency with which 



statistical uncertainty of the model- 

 based /jQ,, is ignored (Table 1). In 

 this work, we show three alterna- 

 tive procedures; an analytical 

 method derived from generalized 

 linear models (McCullagh and 

 Nelder, 1989), nonparametric boot- 

 strap (Efron and Tibshirani, 1993), 

 and a Monte Carlo algorithm devel- 

 oped in our study. We show by simu- 

 lation the behavior of the three 

 methods for sample sizes of 500 to 

 10,000 individuals, concluding that 

 they are similar in terms of bias, 

 length, and shape of confidence inter- 

 vals but that the Monte Carlo method 

 outperforms the other two methods 

 in percentage of times that the confi- 

 dence interval contains the true pa- 

 rameter, which remained close to the 

 nominal 95% at all sample sizes. 



The problem 



In regression analysis, we are usu- 

 ally interested in assigning confi- 

 dence bounds to the response vari- 

 able at specified levels of the pre- 

 dictor variable. However, in matu- 

 rity modeling the attention is 

 turned to the converse problem of 



