864 
Fishery Bulletin 95(4), 1997 
has not been investigated. Jensen (1996) found that 
the CR method was less biased and more precise than 
the LS method when used to estimate mortality from 
the age structure of pooled, simulated net-hauls of 
lake whitefish, Coregonus clupeaformis . A random 
sample drawn from a known geometric distribution 
of ages will have an age distribution that varies sto- 
chastically from the true distribution. In this study, I 
evaluate the effect of sample size, mortality rate, and 
an age-frequency truncation scheme on the accuracy 
and precision of the CR (and CRt) and LS estimators 
when the sample age frequency is drawn randomly from 
a population of geometrically distributed ages. 
calculated with these data, and the mean CRt esti- 
mates of S were converted to Z. Each truncated age 
frequency was a subset of a simulation from the com- 
plete age-frequency simulations in which all fish that 
were older than the oldest age group meeting or ex- 
ceeding a threshold abundance of 5 fish were re- 
moved. Although this truncation scheme reduced the 
effective sample size within each simulation, it ac- 
curately reflected the application of a truncation 
scheme to a real sample. 
Results and discussion 
Materials and methods 
I used a stochastic model that allowed for random 
departures from the exact age distribution of the 
population to generate the simulated sample age fre- 
quencies. Under a known, constant survival rate, a 
geometric distribution function defines the cumula- 
tive probability of a fish from a fully recruited cohort 
being less than age j as 
P(age < j) = 
0 
■ y (i -s)s m - 1 
m - 1 
j< 1 
j* 1 , 
where S = the annual survival rate. 
For this simulation, age-0 fish are defined as those 
in their first year of full vulnerability to capture. I 
sampled individual aged fish from this distribution 
by choosing a random, uniform number (probabil- 
ity) within the interval from 0 to 1 and determining 
the age corresponding to this value of the cumula- 
tive distribution function. By repeating this process, 
I was able to draw randomly a specified number of 
aged fish from a known geometric distribution de- 
fined by S. Each generated sample consisted of 100- 
1,000 individuals drawn independently from geomet- 
ric distributions defined by Z values between 0.20 
and 2.00. One thousand simulations were run for 
each combination of sample size and Z. For each 
simulation, a CR estimate of S and an LS estimate 
of Z were calculated from the sample age frequency. 
Means of the Chapman-Robson estimates of S were 
converted to Z so that they could be compared to the 
means of the LS estimates of Z. 
The effect of constraining the right-hand limb of 
the sample age frequency was investigated by trun- 
cating each age-frequency distribution and recalcu- 
lating mortality. The CRt and the LS estimates were 
Simulations indicated that mean CR estimates of 
mortality for the complete age frequencies were es- 
sentially unbiased. At all Z’s and sample sizes ex- 
amined, the mean CR estimator agreed closely with 
the true value of Z. All differences between estimated 
mean Z’s and true Z’s (relative to the true Z) were 
<1% (Table 1). 
The maximum likelihood estimator developed for 
use with truncated age frequencies (CRt) showed a 
negative bias that was greatest when sample size 
was low. With a 5-fish threshold rule, the mean CRt 
estimate of instantaneous total mortality was biased 
-12% at Z = 0.2 for a random sample of 100 fish (Fig. 
1) . At sample sizes of 300 fish or more, bias was re- 
duced to less than about -4% for all Z’s (Fig. 1). 
The mean LS estimates of Z for complete age fre- 
quencies were consistently less than the true instan- 
taneous total mortality rate. This bias was greatest 
at low levels of Z when sample sizes were small (Table 
2) . At Z = 0.2, the difference between the mean esti- 
mated Z and true Z ranged from -16% for samples 
of 1,000 individuals to -37% for samples of 100. De- 
viations were much less, -4% to -8%, for all sample 
sizes when the true Z was 2.0. Bias in the LS esti- 
mator was reduced by truncating the sample age fre- 
quency. When I used a minimum threshold abundance 
of five, the negative bias was reduced to less than about 
5% at sample sizes of at least 200 fish (Fig. 1). 
Precision of the CR and CRt estimators was gen- 
erally better than that of the LS estimator, especially 
at low Z’s. Although precision improved for all esti- 
mators as sample sizes became larger, the coefficient 
of variation (CV) for the CR and CRt estimators ap- 
proached 1% for large samples at Z = 0.2, whereas 
the CV for the LS estimator approached only 6-9% 
(Fig. 2). For all given sample sizes, the precision of 
the CR and CRt estimators deteriorated as Z in- 
creased. The precision of the LS estimator changed 
little as Z increased, except when the estimator was 
based on samples of only 100 fish. In general, the 
CWs for the CR or CRt estimators were less than the 
