Hulson et al.: Distribution of sampling effort for age composition of multiple species 
329 
Finally, the between-length interval variance (B ay ) is 
given as 
B ay - a ly (0 l ay - §a ; y) 2 . (4) 
For fixed allocation, the formula to estimate the age- 
composition sample size in year y (A y ) is given by 
Ay 
% y CV 2 -B a>y IL : 
■ + J, 
(5) 
where F ay = the within-length interval variance (Eq. 
e ’ 2)5 
a> y = the proportion of fish at age a (Eq. 1); 
CV refers to the target CV in the age composition; 
B a y = the between-length interval variance (Eq. 4); 
L y = the number of length observations in year y; 
and 
J = the number of length intervals. 
Age-composition sample size at some level of precision 
given proportional allocation was estimated with 
^1 
(6) 
where V a y = the within-length interval variance (Eq. 3) 
and the other terms are the same as in 
Equation 5. 
For consistency across species, the length classes were 
set at 1-cm bins and were not grouped. 
We also estimated the sample size necessary to ob- 
tain some target CV under SRS. Under SRS, the age- 
composition sample size at some level of precision was 
estimated with 
(l-9 a ,y) 
9a, y CV 2 ’ 
(7) 
which is derived from the variance of a multinomial 
distribution. 
Four sampling goals to achieve some target level 
of precision in age composition were investigated that 
represented 2 general categories based on 1) a single 
age class, or 2) a group of age classes that were re- 
lated to the total number of ages in the population. 
The overall point of each of these sampling goals was 
to investigate standardized sampling goals across spe- 
cies that achieved the same level of uncertainty in the 
age-composition data. The first sampling goal was to 
achieve the target CV for the most frequently caught 
age (i.e., the age class with the largest annual propor- 
tion-at-age). The second sampling goal was to achieve 
the target CV for the age class with the maximum 
within-length interval variance (i.e., the age class with 
the least information on age from the length data). The 
third sampling goal was to achieve the target CV for 
the top 25% of age classes caught (i.e., proportionally 
the same number of age classes across species). Finally, 
the fourth sampling goal was to achieve the target CV 
for age classes with proportions-at-age that were on 
average (across time) greater than the inverse of half 
of the maximum age (i.e., greater than some propor- 
tion that is related to the longevity of the species in- 
vestigated). As an example of this final sampling goal, 
for a maximum age of 84 years for Pacific ocean perch 
(Sebastes alutus ), we would try to achieve a CV for all 
ages with proportions that were on average greater 
than 1/42 or 2.4%. It should be noted that it is often 
difficult to set a sampling goal without prior sampling 
having been completed. 
The CVs ranging from 10 to 25% were initially eval- 
uated (by 5% intervals) to estimate age-composition 
sample sizes under fixed and proportional allocation 
across the species investigated. We investigated es- 
timated sample sizes with the same age-composition 
CV across species to form a basis for comparison. The 
trends and patterns in distribution of sample size 
across species, which was our focus, were extremely 
similar across the different CVs and we present only 
the results of a target CV of 15%. The overall estimat- 
ed sample sizes and proportions of the total sample 
size presented for each species were the median across 
the years of the bottom trawl surveys needed to obtain 
the target CV in the age composition. To show the dis- 
tribution of sample size across species we calculated 
the species-specific proportion of the total sample size 
within each sampling goal (dividing the species-specific 
estimated sample size for some sampling goal by the 
sum of the estimated sample sizes across species for 
that sampling goal). 
We also investigated the use of species-specific aging 
error in the estimation of sample sizes. Reader-tester 
agreement data was compiled for all the species and 
stocks investigated from the AFSC Age and Growth 
Laboratory. Two aging error cases were investigated: 
the first was when aging error was not incorporated, 
the second was when aging error was incorporated. 
The amount of aging error (i.e., the age-reading er- 
ror standard deviation [SD] by age) was investigated 
for each species and stock according to the methods of 
Richards et al. (1992) and Hiefetz et al. (1998). In order 
to construct and implement a generalized aging error 
method for all species and stocks, a constant CV was 
used across ages for each species and stock. Aging er- 
ror was implemented into the estimates of sample size 
by multiplying the species-specific aging error matrix 
by the observed proportion of fish of length l and age 
a in year y (S^y). 
We evaluated the relationships between the distri- 
bution of age-composition sample size across species 
and life-history characteristics of species by compar- 
ing the proportion of total sample size estimates with 
4 statistics. The 4 statistics were focused on instanta- 
neous growth rates (i.e., the slope of the tangent of the 
growth curve at some age), calculated as the derivative 
of the von Bertalanffy growth curve (von Bertalanffy, 
1938). The life-history statistics investigated included 
the natural log of 1) the instantaneous growth rate at 
20% of the maximum age observed; 2) the instantaneous 
growth rate of the age at 50% of the asymptotic length 
(L m ); 3) the mean population instantaneous growth rate 
(the instantaneous growth rate at age weighted by the 
observed proportions at age); and 4) the minimum life- 
