BARTOO and PARKER: STOCHASTIC AGE-FREQUENCY ESTIMATION 



types of bias. The degree of bias is proportional to 

 overlap in lengths-at-age and changes with weak 

 or strong year classes. When overlap increases 

 with age, age-frequency estimates will generally 

 be more biased for older than younger ages. 

 When overlap occurs, biases will always result, 

 since the numbers-at-length will be allocated to 

 unreasonably old ages. Any numbers-at-length 

 for lengths greater than L x will be undetermined 

 in age estimation, resulting in downward biases 

 for those ages contributing such lengths. 



Age estimation biases can be effectively re- 

 moved by creating a stochastic model based on a 

 matrix of length interval probabilities at age. 

 The probability matrix (P-matrix) is indepen- 

 dent of year-class strength and will effectively 

 remove all sources of estimation bias except that 

 due to random variation in length-frequency esti- 

 mation. As long as the von Bertalanffy growth 

 parameters remain the same over time, the sto- 

 chastic method based on accurate estimates of 

 variance in length-at-age will always yield un- 

 biased results. 



A probability model of the distribution of 

 length-at-age with estimated parameters is nec- 

 essary for estimating probabilities of length 

 intervals at age for the P-matrix. If age infor- 

 mation is unavailable then variances can be esti- 

 mated from visually separable length-frequency 

 modes. In the case where modes are separable for 

 the first few ages only, there will be a problem in 

 estimating variances for older ages: A model re- 

 lating the variance in length-at-age with age can 

 be used in estimating variances for these older 

 ages. Ricker (1969) proposed that while distribu- 

 tions in lengths-at-age remain normal, variances 

 increase during the first few years, stabilize, and 

 then decrease over the final years. The trend in 

 variances with age for a similar species might 

 also be substituted in cases where variances are 

 unavailable. 



The principal strengths of the stochastic meth- 

 od are that few fish are required to be aged to 

 estimate the P-matrix and that existing von Ber- 

 talanffy growth relations can be used. Accurate 

 estimates of variance in length-at-age can prob- 

 ably be achieved with as few as 20 to 30 fish/ age, 

 which is likely to be a much smaller number of 

 fish than needed to estimate accurate propor- 

 tions of age-at-length necessary to construct an 

 age-length key. 



Von Bertalanffy growth parameters have 

 been estimated for many species. Since most 

 stocks have variant year-class strength, overlaps 



in lengths-at-age, and lengths exceeding the up- 

 per bound for the last age attainable, conversion 

 to a stochastic model may be necessary, if unbi- 

 ased estimates of age frequency are desired. Re- 

 examination of age-length data used to estimate 

 the von Bertalanffy parameters may be useful in 

 estimating variances in lengths-at-age for the P- 

 matrix. Taking additional age-length samples 

 may be a cost-effective way of improving age-fre- 

 quency estimation. 



In fishery management, the overestimation of 

 maximum age by the deterministic von Berta- 

 lanffy equation may produce underestimates of 

 mortality rates which may result in overesti- 

 mates of population size and recruitment. Fur- 

 ther, the deterministic method tends to "fill in" 

 weak year classes which results in underesti- 

 mates of year-class variability and overestimates 

 of recruitment stability. In general, all of these 

 affect accuracy of a stock assessment and con- 

 tribute to improper advice for fishery manage- 

 ment. 



Application of the stochastic method shown 

 here to cover other growth equations and situa- 

 tions, such as discontinuous growth, is handled 

 by simply estimating appropriate elements in 

 the P-matrix for each case. 



ACKNOWLEDGMENTS 



We thank Douglas Chapman and Alec MacCall 

 for helping to define the problem and evaluating 

 the solution. Mark Farber, Joseph Powers, Lewis 

 J. Bledsoe, and Gary Sakagawa provided critical 

 review and comment for which the authors are 

 grateful. We are also grateful to R. A. Collins of 

 the California Department of Fish and Game for 

 providing data. 



LITERATURE CITED 



Brody, S. 



1945. Bioenergetics and growth. Reinhold Publ. N.Y.. 

 1023 p. 

 Campbell, G., and R. A. Collins. 



1975. The age and growth of the Pacific bonito, Sarda 

 chiliensis, in the eastern North Pacific. Calif. Fish 

 Game 61:181-200. 

 Clark. W. G. 



1981. Restricted-least squares estimate of age composi- 

 tion from length composition. Can. J. Fish. Aquat. Sci. 

 38:297-307. 

 Draper, N. R., and H. Smith. 



1981. Applied regression analysis. 2d ed. John Wiley 

 and Sons, Inc.. N.Y.. 709 p. 

 Gulland, J. A. 



1973. Manual of methods for fish stock assessment. Part 



95 



