540 



Fishery Bulletin 91(3), 1993 



The IALK algorithm was judged to have converged 

 when the maximum absolute deviation summed over 

 iterated proportions at age was less than 0.001 (0.1%). 

 The IALK method was applied to the annual NCDMF 

 1986-1989 length distributions separately. 



MULTIFAN MULTIFAN was introduced by Fournier 

 et al. (1990) as an extension of the methods of Schnute 

 and Fournier (1980) and Fournier and Breen (1983). 

 MULTIFAN is a likelihood-based method using the mix- 

 ture of distributions approach for the classification of 

 lengths to age, with consideration of biological con- 

 straints, to simultaneously analyze several length- 

 frequency distributions sampled at different times. The 

 major assumptions of the method include the follow- 

 ing: 1) the lengths of the animals in each age class are 

 normally distributed about the mean length at age; 



2) growth follows the von Bertalanffy function; and 



3) standard deviation of lengths about the mean length 

 at age varies as a simple function of mean length at 

 age. 



The log-likelihood function used in MULTIFAN com- 

 pares the expected probability that a fish chosen ran- 

 domly will lie in a given length interval with the ob- 

 served number of fish in that interval for the set of 

 growth parameters being tested. The formal statistical 

 basis of MULTIFAN allows for a structured and rela- 

 tively objective means of evaluating alternative inter- 

 pretations of the processes producing the observed 

 length- frequency distributions (e.g., the growth rate 

 and resulting number of significant age classes, gear 

 selectivity on younger age classes, and evidence of a 

 length dependent trend in the standard deviation of 

 length at age). The basic set of parameters of the 

 MULTIFAN model include the following: 1) the pro- 

 portions at age; 2) the mean length of the first age 

 group; 3) the mean length of the last age group; 4) the 

 von Bertalanffy growth parameter K; 5) two param- 

 eters which determine the standard deviation of length 

 at age; 6) a parameter determining the extent of selec- 

 tivity bias on the first age group; and 7) a parameter 

 determining the overall variance of the sampling er- 

 rors in the length-frequency data sets (see Fournier et 

 al. 1990 for detailed notation of the MULTIFAN model 

 parameterization ). 



For example, in the case of two suspected age classes 

 underlying a set of five length-frequency distributions, 

 the model would have a total of nine parameters. The 

 addition of one age class to the model (one age, five 

 distributions) would increase the number of param- 

 eters by five, to 14. Accounting for a length dependent 

 trend in standard deviation of length at age (LDSD) in 

 the three age-class case would increase the number of 

 parameters by one, to 15; accounting for selectivity 

 bias on the first age group would again increase the 



number of parameters estimated by one, to 16. A chi- 

 square test is used to determine the best fitting growth 

 structure. The statistical significance of increases in 

 the log-likelihood values obtained by the addition of 

 age classes are determined by the 0.90 point of the 

 chi-square random variable, to reduce the probability 

 of rejecting an additional age class when it is actually 

 present in the data. The significance of the addition of 

 other parameters is determined at the more conven- 

 tional 0.95 level (Fournier et al, 1990). 



After an initial inspection of modes apparent in 

 the length-frequency data, initial constraints for 

 MULTIFAN included reasonable ranges for 1) the num- 

 ber of expected age classes, 2) the corresponding val- 

 ues for the von Bertalanffy parameter K, and 3) a 

 range for the mean length at age of an obvious mode. 

 Initial constraints for the NCDMF 1986-1989 bluefish 

 length-frequency data were potential significant age 

 classes from 5 to 12, K range from 0.20 to 0.40, and 

 mean length range for presumed age-0 fish in the 1986, 

 1987, and 1989 samples set at 15-30 cm, and for age 1 

 in 1988 at 30-40 cm. Preliminary MULTIFAN runs 

 determined the number of significant age classes un- 

 derlying the NCDMF 1986-1989 sample length fre- 

 quencies. Additional runs with MULTIFAN were made 

 to test for evidence of selectivity bias (SEL) on the 

 first age class and for a length dependent trend in 

 standard deviation of length at age (LDSD) as means 

 of improving the fits. 



Comparison of methods 



A qualitative assessment of the utility of each method 

 was made by inspection of the estimated mean lengths 

 at age determined by the different alternatives. The 

 result of ultimate interest from the application of these 

 methods to the NCDMF length frequency distributions, 

 however, is the number of fish per age class. We as- 

 sessed the performance of the alternatives by using 

 the Kolmogorov-Smirnov (K-S) cumulative distribu- 

 tion test (Sokal and Rohlf, 1981) to compare the calcu- 

 lated annual and combined 1986-1989 proportions at 

 age from each aging approach with the NCDMF an- 

 nual and combined 1986-1989 proportions at age. 



This non-parametric method tests for differences in 

 both the shape and location of two frequency distribu- 

 tions under the null hypothesis that the samples are 

 taken from populations with the same underlying dis- 

 tribution. The proportions at age being compared are 

 considered as two samples from the commercial land- 

 ings. If the maximum unsigned difference between cu- 

 mulative frequency distributions (proportions at age) 

 exceeds some critical value (D) for a given confidence 

 level and sample size, then the null hypothesis is re- 

 jected (Sokal and Rohlf, 1981). The test provides an 



