Abstract. - For fish populations 

 in equilibrium, estimates of Z/K and 

 L^ can be obtained solely from 

 length-frequency data by using the 

 Wetherall length-based method. 

 Robustness of this method to depar- 

 tures from equilibrium conditions is 

 examined with a population simula- 

 tion model. During disequilibrium 

 conditions following either the initia- 

 tion of a fishery or a 1-year pertur- 

 bation in recruitment, estimates of 

 both parameters, especially Z/K, are 

 severely biased. Three-year averag- 

 ing of catch length-frequencies does 

 not substantially reduce this bias. A 

 test, based on the chi-square statis- 

 tic, is proposed for detecting popula- 

 tion disequilibrium. Provided that 

 the sizes of length-frequency samples 

 are sufficiently large, the test is an 

 effective way to detect population 

 disequilibrium and thereby avoid 

 biased parameter estimates. 



Robustness of the Wetherall 

 Length-based Method 

 to Population Disequilibria 



David A. Somerton 



Donald R. Kobayashi 



Honolulu Laboratory, Southwest Fisheries Science Center 



National Marine Fisheries Service. NOAA 



2570 Dole Street, Honolulu, Hawaii 96822-2396 



Manuscript accepted 31 December 1990. 

 Fishery Bulletin, U.S. 89:307-314 (1991). 



Assessments of tropical fish stocks 

 have increasingly turned to length- 

 based methods rather than the more 

 traditional age-based methods, be- 

 cause of the difficulty in ageing trop- 

 ical fishes (Pauly and Morgan 1987). 

 Although all length-based methods 

 use length-frequency data, they are 

 designed to measure different bio- 

 logical parameters and vary in their 

 requirements for additional types of 

 data. Wetherall et al. (1987) exam- 

 ined one class of length-based meth- 

 ods—those designed to estimate both 

 growth and mortality parameters 

 without the need for additional data— 

 and developed two new estimators: 

 a maximum likelihood estimator and 

 a regression estimator. The second 

 estimator, because of its simplicity, 

 has received the most widespread use 

 (Wetherall 1986, Arellano 1989, Polo- 

 vina 1989, Rawlinson 1989). 



The regression estimator of Weth- 

 erall et al. (1987), henceforth referred 

 to simply as the Wetherall method, 

 estimates two parameters. The first 

 (0) is the ratio of the instantaneous 

 total mortality rate (Z) to the rate 

 constant of the von Bertalanffy 

 growth function (K). The second 

 (L^,) is the asymptote of the von Ber- 

 talanffy growth function. These 

 parameters are estimated by regress- 

 ing the mean length (1,) of all fish 

 >1cj on l ci , a cutoff length ranging 

 from the first length category that is 

 fully selected by the fishery (L c ) to 

 the largest length category. Esti- 

 mates of Z/K and L are then calcu- 



lated as Z/K = B/(l - B) and L^ = 

 A/(l - B), where A is the intercept 

 and B is the slope of the regression. 

 Like many other length-based 

 methods, the Wetherall method re- 

 quires the population to be in equi- 

 librium, a condition often not fulfilled 

 because of variations in both the en- 

 vironment and the fishery (Csirske et 

 al. 1987, Ralston 1989). Although 

 Wetherall et al. (1987) clearly have 

 cautioned potential users that biases 

 may result if populations are in dis- 

 equilibria, the likely magnitudes of 

 such biases have not been addressed. 

 This has prompted us to investigate 

 the sensitivity of the method to two 

 common types of perturbations: (1) 

 a rapid increase in effort during the 

 fishing-up stage of a fishery, and (2) 

 a fluctuation in recruitment. In this 

 paper, we examine the temporal pat- 

 terns and magnitudes of biases asso- 

 ciated with these disequilibria and 

 present a simple statistical test that 

 can be used with catch length-fre- 

 quencies to detect population disequi- 

 libria and help minimize the conse- 

 quent potential for biased parameter 

 estimates. 



Materials and methods 



The performance of the Wetherall 

 method can be assessed by applying 

 it to catch length-frequencies 

 generated by a length-based popula- 

 tion simulation model that accounts 

 for growth, mortality, and recruit- 

 ment. The model is configured to 



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