STOCHASTIC AGE-FREQUENCY ESTIMATION 

 USING THE VON BERTALANFFY GROWTH EQUATION 



Norman VV. Bartoo 1 and Keith R. Parker 2 



ABSTRACT 



The method of estimating age frequency from length frequency via the von Bertalanffy growth 

 equation is deterministic and yields biased results. Most of the bias can be removed by incorporating 

 a stochastic element in the von Bertalanffy relationship. The stochastic element is based on esti- 

 mated probabilities of lengths by intervals at age, the probabilities being estimated from variances 

 in lengths-at-age. Based on age-length samples from the Pacific bonito fishery the stochastic method 

 gives improved age-frequency estimates over those obtained by the deterministic method. The sto- 

 chastic application may be generalized to all growth models including discontinuous growth such as 

 in crustaceans. 



Complex population dynamics techniques rely 

 heavily on age-structure information. Frequent- 

 ly, appropriate assessment techniques for a stock 

 require an estimate of the age frequency of that 

 stock. For example yield-per-recruit analysis 

 (Ricker 1958) is computed on the dynamic rela- 

 tionship between growth and mortality: Mortal- 

 ity rates when computed via cohort analysis 

 (Murphy 1965) are based on estimated age fre- 

 quency. 



For some species accurate aging methods are 

 not available. When feasible, determining the 

 age of fish and consequently computing an age 

 frequency are most accurately accomplished by 

 visual inspection of scales, otoliths, or other struc- 

 tures (Ricker 1958). Such visual inspection is 

 time consuming and often expensive. To reduce 

 the cost and time of estimating the age structure 

 of a fisheries catch, age frequency is usually esti- 

 mated from sampled length frequency, the age- 

 length relationship being described by either an 

 age-length key or a growth curve such as the von 

 Bertalanffy growth curve (Ricker 1958). The 

 growth curve method is used when there are in- 

 sufficient data to construct an age-length key. 



Age-length keys work on the principle that age 

 can be estimated from length using information 

 contained in a previously or concurrently aged 

 sample from the population. As long as the pro- 

 portion of length-at-age remains the same for all 

 ages, then the age-length key will yield unbiased 



'Southwest Fisheries Center La Jolla Laboratory. National 

 Marine Fisheries Service, NOAA, 8604 La Jolla Shores Drive, 

 La Jolla, CA 92038. 



2 1837 Puterbaugh Street, San Diego. CA 92103. 



estimates of age for any sampled lengths from 

 that population. However, since the estimated 

 parameters of an age-length key — proportions of 

 age-at-length— are dependent on the sampled 

 population used to construct the key, the applica- 

 tion of the key to the population with altered age 

 structures can yield inaccurate results. Kimura 

 (1977) and later Westrheim and Ricker (1978) 

 demonstrated that under conditions of varying 

 year-class strength and substantial overlap of 

 lengths between ages, age-length keys can yield 

 nearly useless estimates of numbers-at-age. 



Clark (1981) effectively removes age-length 

 key bias by first proportioning numbers in length 

 intervals at age over time and then using the ma- 

 trix of these proportions standardized over time 

 to compute least-squares estimates of age fre- 

 quency from the vector of length frequency. 

 Effective applications of many stock assessment 

 growth and mortality based methods require 

 that ages are expressed in fractions of years 

 (Ricker 1958; Lenarz et al. 1974). The large num- 

 ber of aged fish required to construct a sufficient 

 key for a large number of ages is difficult and ex- 

 pensive to attain. Even with Clark's bias correc- 

 tion procedure, the construction of a sufficient 

 key can present difficulties due to data needs. 



In this paper we deal specifically with the von 

 Bertalanffy growth equation and the application 

 of stochastic methods to reduce or eliminate bi- 

 ases. However, it should be noted that the method 

 presented here may be applied to any growth 

 equation as well as to cases where no growth 

 equation has been fitted or where growth is dis- 

 continuous as in crustaceans. 



The von Bertalanffy growth equation mathe- 



Manuscript accepted June 1982. 



FISHERY BULLETIN: VOL. 81. NO. 1. 1983. 



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