FISHERY BULLETIN: VOL. 81, NO. 1 



80 r 



LENGTH (cm) 



Figure 2.— Length frequency for the Pacific bonito, Sarda 

 chiliensis, from 1973 California landings (Campbell and Col- 

 lins 1975). 



and sample standard deviations are 



Age Length Standard deviation 



Deterministic estimates were made on lengths 

 rounded to the nearest 0.1 cm. From Equation (2) 

 the deterministic numbers-at-age are shown in 

 column 3 of Table 2 with the difference between 

 the true and estimated numbers in column 4. 

 While the estimates are reasonably close over the 

 first two ages, they become increasingly dispar- 

 ate for older ages. Thirteen fish had lengths 

 greater than those at the maximum age. Seven 

 had lengths greater than L and consequently 

 were unclassifiable. 



Quarter centimeter intervals were used to com- 

 pute the stochastic estimates of age frequency. 

 The results are shown in column 6 of Table 2, 



with the difference between true and estimated 

 numbers in column 7. For all ages, especially the 

 older ages, the stochastic estimates are closer 

 and less biased than those of the deterministic 

 method (column 3). 



Some insight into the improvement of the sto- 

 chastic age estimates over the deterministic age 

 estimates can be gained by inspection of Figure 

 1. Lengths of age I fish overlap those of age 1 1 fish 

 and vice versa. Since the deterministic cutoff 

 point for age I fish is 58.7 cm (1.5 yr), all overlap 

 is lost in the deterministic model. In contrast, for 

 the stochastic model, overlaps in lengths-at-age 

 are shared between ages, the degree of sharing 

 being relative to the probabilities of length inter- 

 vals at the respective ages. 



With increasing age, the extent of relative 

 overlap and, consequently, misaging increases 

 for the deterministic model; allocation of lengths 

 to ages becomes more sensitive. Only if the de- 

 gree of overlap between adjacent ages is equal do 

 accurate estimates of numbers-at-age result 

 from the deterministic model. In the present ex- 

 ample varying year-class strength and random 

 variability in lengths-at-age offset this sensitive 

 compensatory mechanism needed for accurate 

 estimation with the deterministic model. 



Fish lengths above 75.3 cm, the length at age 

 5.5 yr, are misclassified either as older or of in- 

 finite ages for the deterministic model. Since for 

 the stochastic model probabilities of length inter- 

 vals at age exist for all ages and lengths, even for 

 lengths above L x , fish at lengths above the 75.3 

 cm cutoff point are distributed to all ages rela- 

 tive to their respective probabilities for length 

 intervals. 



DISCUSSION 



Calculations of age from length via the von 

 Bertalanffy growth equation result in several 



Table 2.— Deterministic (col. 3) and stochastic (col. 6) estimates of numbers- 

 at-age with their respective differences from the true numbers-at-age in col- 

 umns 4 and 7 for the Pacific bonito. Sarda chiliensis. from 1973 California 

 landings (Campbell and Collins 1975). 



94 



