Terceiro and Ross: Estimation of age from length data for Pomatomus saltatrix 



539 



to estimate proportions at age for comparison with the 

 NCDMF ages. It is important to note that the North 

 Carolina part of the CNN length-age data and the 

 NCDMF 1986-1989 length-age data were compiled by 

 the same age-reader and therefore were not completely 

 independent. 



NOAA (1989) length-age data Bluefish length-age 

 data collected during NEFSC autumn bottom trawl 

 research surveys, 1985-1987, formed the basis for 

 weighted, mean-calculated lengths at annulus forma- 

 tion presented in NOAA (1989). Samples were aged 

 according to the 1 January birthdate convention. These 

 data provided the von Bertalanffy growth parameters 

 of L inf = 94.6 cm, K = 0.242, and t„ = -0.128, which 

 were used in cohort slicing for estimation of bluefish 

 ages from length data in the revised yield-per-recruit 

 analysis for bluefish 3 . 



The NOAA (1989) value of/,, was adjusted to reflect 

 the difference between the 1 January (used by NEFSC) 

 and 1 June (used by NCDMF) birthdate conventions. 

 The difference of 151 days, or 0.414 years, was added 

 to the ages of fish in the raw data used to estimate the 

 parameters. Refitting the data provided a new value 

 of t - -0.542 (e.g., a 25-cm bluefish sampled on 1 

 April would typically be classified as age by NCDMF 

 staff; a 25-cm bluefish aged by cohort slicing with 

 NOAA 1989 parameters and t tl = -0.128 would be 1.140 

 years old, and classified as age 1; while a 25-cm blue- 

 fish aged by cohort slicing with NOAA 1989 param- 

 eters and t„ = -0.542 would be 0.726 years old and 

 classified as age 0). The NOAA 1989 growth param- 

 eters with the adjusted value of t„ were used in cohort 

 slicing of the NCDMF 1986-1989 annual length- 

 frequency data for comparison with the NCDMF 

 1986-1989 annual and combined age distributions. 



Statistical methods 



Iterated age-length key The iterated age-length key 

 (IALK) method was introduced by Kimura and Chikuni 

 (1987). The method combines the standard age-length 

 key (Kimura, 1977; Westrheim and Ricker, 1978) and 

 mixture of distributions approaches (Hasselblad, 1966; 

 Macdonald and Pitcher, 1979; Schnute and Fournier, 

 1980) for the resolution of lengths to ages. Application 

 of the standard age-length key method without bias 

 requires that the length-age data and length data have 

 the same underlying age composition (Kimura, 1977). 

 The IALK method has been presented as a potential 

 solution to the problem of applying age-length keys to 

 length-frequency distributions with different time and 

 space characteristics, and therefore potentially differ- 

 ent underlying age distributions (Kimura and Chikuni, 

 1987; Hoenig and Heisey, 1987). 



Assumptions for the IALK method are somewhat 

 less stringent than for standard application of the age- 

 length key. The method requires only that the length- 

 age data be adequately sampled and that the associ- 

 ated probabilities of length at age are applicable to the 

 observed length distribution to be aged, although the 

 potential for bias increases if the two data sets have 

 very different temporal and geographic characteristics. 

 The goal of the iterative procedure is to modify the 

 length distribution of the length-age data so that it 

 more closely approximates the length-frequency data 

 to be aged, until the underlying age distributions of 

 the length-age data and length-frequency data also 

 become similar, thus satisfying the assumptions of the 

 standard age-length key concept. 



Kimura and Chikuni (1987) presented the IALK 

 method in the form of an algorithm, gave conditions for 

 which the algorithm converged to a unique solution, 

 and showed the algorithm was an application of the 

 expectation-maximization (EM) algorithm (Dempster et 

 al, 1977) to mixtures of distributions. Hoenig and Heisey 

 (1987) subsequently used a similar approach for the 

 IALK as an EM algorithm, while making different as- 

 sumptions about the sampling errors of the data. The 

 steps of the IALK algorithm as implemented by Kimura 

 and Chikuni ( 1987) are the following: 



1) estimate the proportions at age, with all ages 

 initially having equal probability. Hoenig and Heisey 

 (1987) suggested that the initial estimate of propor- 

 tions at age might be set the same as those in the 

 length-age data; 



2) using observed probabilities of length at age cal- 

 culated from the length-age sample data and the cur- 

 rent estimate of proportions at age, calculate the cor- 

 responding age-length key (probabilities of age at 

 length). An intermediate result of this step is the IALK 

 estimate of the observed length distribution; 



3) apply the observed length distribution to be aged 

 to the calculated age-length key of step 2; 



4) calculate the maximum absolute deviation over 

 ages between the newly estimated proportions at age 

 of the observed length frequency and the proportions 

 at age in the previous iteration. If the deviation is less 

 than some small constant, then stop; otherwise con- 

 tinue with the next iteration of the age-length key. 



We used the CNN length-age data (Fig. 2) as the 

 source for the probabilities of length at age used in the 

 calculation of the IALK applied to the observed 

 NCDMF 1986-1989 length distributions (Fig. 1). Ini- 

 tial proportions at age were set to the same values as 

 the CNN length-age data, as suggested by Hoenig and 

 Heisey (1987), for ages to 11: 



[0.272 0.355 0.138 0.060 0.055 0.048 0.040 

 0.023 0.007 0.001 0.0001 0.0009]. 



