Goodyear: Fish age from length: an evaluation of three methods 
43 
Age-estimation methods 
For all three ageing methods evaluated, the number 
of fish in the catch at age a, N , was estimated as 
K=Cf a , 
where C, the total catch in numbers of fish, was the 
known value from the simulation, and f a was the es- 
timated fraction of catch at age a. Age frequencies 
were estimated separately for each year between 
1984 and 1994. 
With the first method, the von-Bertalanffy growth 
equation was rearranged to predict age from length, 
A = - log e (l- L/88. 24)/0.159 - 0.458, 
and the f were estimated as the ratios of the num- 
ber of sampled fish assigned age A to the total num- 
ber of fish in the sample. For this method any 
sampled fish larger than the asympotic size was dis- 
carded. 
With the second method typical age-length keys 
(Ketchen, 1950; Westrheim and Ricker, 1978) were 
constructed annually from the monthly age-fre- 
quency samples. In this case the f a were estimated 
by multiplying the observed age frequencies for each 
length stratum by the ratio of length samples in each 
length stratum to the total number of length samples 
and by summing over ages. 
The third (probabilistic) method is a proposed al- 
ternative and requires estimates of prior survival of 
year classes in the population and independent esti- 
mates of year-class strength. In this method 
j n 
SS'. 
r _ i = 1 q =0 
la 
j 
where j is the number of length samples, n is the 
number of ages, and 
_ W a R y _ a S a 
a n 
a= 1 
and 
where D a is the cumulative probability distribution 
of length for age a, L ; is the observed length of fish i, 
is the recruitment strength in year y-a, y is the 
year of observation, and S a is survival probability 
from recruitment to age a and is given by 
a-1 
s a =exp ^-(Ff+Mf), 
i = 0 
where F is the fishing mortality of the year class at 
age a when it was age i, and M ( is the natural mor- 
tality of the year class at age a when it was age i. 
Inspection of the data used in this method reveals 
that the method requires values for nearly everything 
one would wish to estimate from the age composi- 
tion of the catch and consequently seems to place 
the cart before the horse. However, in many cases 
ancillary data on year-class strength may be avail- 
able from research surveys, and estimates of natu- 
ral and fishing mortalities may be available from 
earlier assessments. In this investigation, this 
method was applied in two ways. The first assumed 
preexisting accurate knowledge of year-class 
strengths and mortality. The second application as- 
sumed knowledge of year-class strengths and natu- 
ral mortality and proceeded iteratively. In the first 
iteration, age composition was estimated with the 
assumption that there was no fishing mortality. This 
led to a set of estimates of catch at age that were 
then used through sequential population analysis to 
estimate fishing mortality at age. With the second 
iteration the resulting estimates of fishing mortal- 
ity were added and catch at age was reestimated. 
This process was repeated several additional times. 
Overall, the three methods provided 4 sets of esti- 
mates of catch at age that could be compared with 
the true values from the simulation: those from the 
growth model, those from the age-length key, those 
from the probabilistic method given knowledge of 
survival, and those from the iterated probabilistic 
method. In addition, numbers at age and fishing 
mortality for each year were estimated from the 
catches at age for each set by using sequential popu- 
lation analysis (Powers and Restrepo, 1992). For the 
purpose of this exercise, the selectivities for the ter- 
minal year of the population analysis were the known 
values from the simulation, and the tuning index was 
the known number of age-4 individuals alive at the 
beginning of the year. The methods were compared 
by correlating the known true values from the simu- 
lation to the values estimated by each method. Be- 
cause there were 31 ages in the model (0-30) and 11 
years, these provided a total maximum sample size 
of 341; however, year-age combinations where the 
true catch at age was below 100 were dropped. Thus 
sample sizes for most analyses were reduced to 331. 
In addition, scattergrams of the logs of the ratios of 
estimated to true values were constructed for each 
comparison. The r 2 values for the correlations be- 
tween true and estimated values are presented with 
