Parrack; Estimating stock abundance from size data 



303 



The original technique to assess fish stocks from 

 size instead of age data is a stepwise double-estimation 

 procedure (see Pauly et al. 1987 for an example). Size- 

 specific catches are first transformed to age-specific 

 catches by using an inverted growth equation (Ricker 

 1975:221) or statistical estimators based on growth 

 data (Clark 1981, Bartoo and Parker 1982, Shepherd 

 1985, Hoenig and Heisey 1987, Kimura and Chikuni 

 1987) so that the stock is assumed to be composed of 

 age-specific cohorts. Size-to-age transformation meth- 

 ods that require size-frequencies only (i.e., growth data 

 are not required) are available (Macdonald and Pitcher 

 1979, Pauly 1982, Fournier et al. 1990), but Monte 

 Carlo tests have shown pronounced weaknesses in 

 these methods (Hampton and Majkowski 1987, Rosen- 

 berg and Beddington 1987, Basson et al. 1988). Vir- 

 tual population analysis (Ricker 1948, Fry 1949, Jones 

 1961, Gulland 1965, Murphy 1965) is then applied to 

 the transformed catch, but the system of cohort-spe- 

 cific catch equations is underdetermined (Agger et al. 

 1971, Doubleday 1975, Ulltang 1977, Pope and Shep- 

 herd 1982). The inclusion of auxiliary data (total fish- 

 ing effort, catch effort, or other relative abundance 

 samples) using any of several statistical procedures 

 (Laurec and Bard 1980; Paloheimo 1980; Anon. 1981b, 

 1983, 1984, 1986; Parrack 1981, 1986; Collie and 

 Sissenwine 1983; Deriso 1985; Pope and Shepherd 

 1985; Mendelssohn 1988) eliminates that problem, so 

 abundances can be estimated. If based on actual age 

 data, virtual population analysis using auxiliary infor- 

 mation does estimate stock abundances and fishing 

 mortality rates reasonably well if the natural mortal- 

 ity rate is known (Deriso 1985, Pope and Shepherd 

 1985), but if the method is used without actual age data, 

 its statistical characteristics are unknown. If the 

 population is not composed of true age-specific cohorts 

 or if the ageing of caught fish is problematic, the 

 method is not appropriate. Spawning often is too pro- 

 tracted to establish cohorts and fish cannot be aged 

 with reasonable certainty; yet because it is simple and 

 tractable, this method is used anyway. 



Several size-based abundance estimation methods do 

 not employ data auxiliary to catches (Jones 1974 and 

 1981, Brethes and Desrosiers 1981, Lai and Gallucci 

 1988). Instead of using fishing effort or relative abun- 

 dance samples to overcome the determination problem, 

 they assume that the size-frequency of the catch, and 

 thus of the stock (and recruitment magnitudes), is 

 constant (in steady state). That assumption greatly 

 restricts the usefulness of these methods. 



Three items seem important when considering stock- 

 abundance estimators. First, the data an estimator re- 

 quires often may preclude its use if such data is not 

 usually available. Next, since the likelihood procedure 

 requires one, often a sampling distribution for an ob- 



served statistic is assumed even though support for the 

 assumption cannot be offered. The resulting estimator 

 thus might be entirely based on an inappropriate prob- 

 ability expression. Last, the statistical properties of 

 an estimator are of concern. An estimator may be too 

 imprecise to be useful unless sample sizes are unrealis- 

 tically large, or its bias may be too large to ignore dur- 

 ing estimation. 



Since the method of least squares is not based on 

 probability theory, statistical characteristics of such 

 estimators are very imcertain. The likelihood procedure 

 tends to generate estimators with superior statistical 

 characteristics, but success is not guaranteed. Com- 

 monly, estimators of parameters of nonlinear models 

 are problematic. They cannot be written in closed form 

 so their expectations, which lead to bias and variance 

 expressions, cannot be derived analytically. Since the 

 estimator's performance characteristics cannot be 

 predicted, they must be established from Monte Carlo 

 studies. If such studies do not exist, the estimator's 

 usefulness is unknown. 



The first size-based procedure, a least-squares esti- 

 mator, was developed (Beddington and Cooke 1981) 

 and applied to sperm whales (Anon. 1981a, Cooke and 

 Beddington 1982, Cooke et al. 1983b, Shirakihara and 

 Tanaka 1983, de la Mare and Cooke 1984) to assess the 

 northwestern Pacific stock (Beddington et al. 1983, 

 Cooke and de la Mare 1983b, Shirakihara and Tanaka 

 1983). It is based entirely on size-specific catches and 

 assumes a known adult-progeny ratio instead of using 

 fishing effort or other auxiliary data. The statistical 

 characteristics of the estimator were established with 

 extensive Monte Carlo studies (Cooke et al. 1983a, 

 Cooke and de la Mare 1983a, Shirakihara and Tanaka 

 1984, de la Mare and Cooke 1985 and 1987, Shirakihara 

 et al. 1985, de la Mare 1988). 



The method of Fournier and Doonan (1987) was 

 derived by the likelihood method by assuming that 

 catch and effort are each lognormal random variables 

 and that the first four moments of length-frequencies 

 are normal random variables. Monte Carlo tests 

 established the estimator's ability to predict optimal 

 long-term fishing effort, but the errors of the stock- 

 abundance estimates are not described. The maximum- 

 likelihood method of Schnute et al. (1989) assumes that 

 the annual ratio of total yield to total effort is a nor- 

 mal random variable. The statistical characteristics of 

 the estimator are not described. 



The method of Sullivan et al. (1990) is a least-squares 

 estimator based on catches, but Kalman filter method- 

 ology also may be used to obtain estimates (Sullivan 

 1989). The method does not require data other than 

 catches even though it is well known that, in the case 

 of age-based (VPA) methods, the system of catch equa- 

 tions without auxiliary data is not determined (Agger 



