FISHKRY BULLETIN: VOL. 81. NO. 1 



matically models the relationship between age 

 and length, length being the dependent variable 

 (see Equation (1)). As suggested by Gulland 

 (1973), age is estimated from length by algebrai- 

 cally rearranging the growth equation so that age 

 is the dependent variable (see Equation (2)). Re- 

 gardless of whether length or age is the depen- 

 dent variable, the von Bertalanffy relationship is 

 deterministic: There is a one-to-one correspon- 

 dence between age and length. 



For the von Bertalanffy growth equation, age 

 frequency is estimated from a length sample as 

 follows: 



1) For each length compute the corresponding 

 age. 



2) For each age interval, usually the interval 

 between midpoint ages of adjacent ages, sum 

 the number of aged fish falling within the in- 

 terval. 



3) The age frequency is then the total number 

 of aged fish falling within each age interval. 



Use of the von Bertalanffy growth equation for 

 age-frequency estimation results in several types 

 of biases, different from those inherent in age- 

 length keys. This paper documents these biases 

 and proposes a method for their resolution. 



BIASES 



When growth is modeled according to the von 

 Bertalanffy age-length relationship (Brody 1945; 

 Ricker 1958), 



Lt = L x (1 -exp[-k(t - h)]), (1) 



then age, t, can be converted to length: 



t = h + \n (\ - L t IU)l(-k) 



(2) 



where Lt 



L x 



k 



to 



length at age t 

 the asymptotic length 

 the rate at which length reaches L 

 hypothetical age at which fish 

 would have zero length. 



When computing numbers-at-age from Equa- 

 tion (2), estimation bias occurs. One bias is due to 

 L x being a fitted parameter. Thus, all numbers- 

 at-length greater than L x must either be elimi- 

 nated or arbitrarily distributed to older ages. 

 Bias also results when lengths approach L^ and 

 are mathematically allocated to ages above those 



attainable by fish within the stock. As lengths (L) 

 approach L x , Equation (2) will yield unreason- 

 ably old ages. 



Additional bias results from the deterministic 

 nature of the von Bertalanffy equation: Back cal- 

 culations of length to age, Equation (2), are on a 

 one-to-one basis. Thus, for any length there is a 

 determined age. In reality, there can be a num- 

 ber of possible ages for any given length, the 

 most probable age-at-length being that with the 

 highest relative contribution of numbers-at- 

 length. Since these back calculations are without 

 probabilistic arguments, the determined age is 

 not necessarily the most probable for the given 

 length. 



Back calculations of length to age also result in 

 a mathematical estimation bias due to the switch- 

 ing of independent and dependent variables in 

 going from Equation (1) to Equation (2). The de- 

 gree of bias is likely to be a function of the 

 amount of residual error in fitting Equation (1). 

 The bias will probably not be consistent between 

 cases and the degree of bias will have to be con- 

 sidered separately for each case. Consequently 

 this bias is not specifically dealt with in this 

 paper. 



A computer model can readily demonstrate 

 this bias. For von Bertalanffy parameters: L x = 

 90.0 units, to = 0.0 units, and k = 0.30, predeter- 

 mined numbers-at-age are arbitrarily distrib- 

 uted normally with a standard deviation equal to 

 3 units about the von Bertalanffy length-at-age, 

 Equation (1), for ages I through X. A length-fre- 

 quency vector is then generated by 1) multiply- 

 ing the number-at-age times the probability of 

 age occurring within each 0.5 unit length inter- 

 val, thus for each age generating a vector of num- 

 ber-at-length for length intervals between and 

 100 units, and 2) accumulating numbers-at- 

 length for each length interval over all ages. The 

 numbers-at-age are then deterministically esti- 

 mated from Equation (2) by accumulating num- 

 bers-at-length over the length intervals at 

 age. 



The bias from this model is illustrated in Table 

 1. Input and back-calculated numbers-at-age 

 and their differences are listed in columns 2, 3, 

 and 4, respectively. The input numbers-at-age 

 represent a sample age distribution where either 

 catchability, recruitment, mortality, or some 

 combination thereof, are age-class variant. Dif- 

 ferences, column 4, indicate a strong bias which 

 increases with overlap of length distributions at 

 age. One hundred and eleven fish were aged to be 



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