NOTE Murphy: Bias in Chapman-Robson and least-squares estimators of mortality rates 
867 
CV’s for the LS estimator when Z was low but were 
similar or higher when Z was high. 
When all fully recruited fish are equally available 
to a sampling gear, the CR estimator can provide a 
more accurate estimate of mortality than the LS es- 
timator can. Applying the least-squares estimator to 
these data clearly violates the linear-regression as- 
sumption of equal variances among age groups. When 
a population is subjected to a low Z, the frequency 
distribution of log-abundances for older age groups 
in a sample becomes skewed to the right because log- 
abundance reaches a lower limit at zero (log of 1; 
Fig. 3). The frequency distribution of log-abundance 
then becomes truncated (undefined) past some dis- 
tance to the left of its mean when zero abundances 
occur in the untransformed frequencies. The vari- 
ances of the log-abundances appear to be positively 
related to age until the log-abundance frequencies 
become truncated when zero abundances appear in 
the samples for older age groups. 
Empirical evidence led Chapman and Robson 
(1960) to conclude that haul data (catch rates for each 
age group) had an approximately constant variance 
when log transformed. However, the results from my 
simulations indicate that variances for the log-abun- 
dances are likely to differ among age groups. The 
assumption of constant variance is likely to be met 
only when the sampling gear operates on a few abun- 
dant age groups, in which there is no chance of only 
periodically encountering an older age group. This 
led Chapman and Robson ( 1960) to suggest that these 
data should be truncated to eliminate the age fre- 
quencies beyond the oldest age with a minimum 
abundance of five fish. Although my findings concur 
with those of Chapman and Robson, the use of this 
threshold rule to eliminate older age groups does not 
completely eliminate all bias in the LS estimator — 
bias that can be attributed to violations of the as- 
sumptions on which the linear regression is based. 
For truncated age-frequency data, both estimators 
gave biased results when small samples were drawn 
from a population of many age groups (Z=0.2). In 
these cases, truncation generally resulted in smaller 
samples that had far fewer age groups than were in 
the original complete age frequency. At high Z’s, age- 
frequency truncation reduced bias in the LS estima- 
tor to less than 5% at all sample sizes and reduced 
bias in the CRt estimator to less than 2%. 
Violations of steady-state assumptions probably 
impart the most serious biases to pooled estimators 
of mortality. By simply inspecting a plot of log-abun- 
dance versus age for evidence of concavity or for a 
trend in the linear regression residuals, one can de- 
tect gross violations to these assumptions. Subtle 
biases inherent when the assumptions required by 
linear regression are not met are more difficult to de- 
tect. Both the CR and LS methods can provide very 
accurate and precise estimates of Z for age frequencies 
that follow an exact geometric distri- 
bution (Jensen, 1985). However, the LS 
estimator is biased when sample ages 
are drawn randomly from a steady- 
state, geometrically distributed popu- 
lation, whereas the CR estimator is not. 
The LS estimator may be more robust 
when age samples are not taken ran- 
domly (Chapman and Robson, 1960). 
The CRt and LS estimators generally 
showed similar levels of bias when the 
sample age structure is truncated with 
a minimum frequency criterion of 5 
fish. In summary, the CR estimator will 
provide a more accurate and at least 
as precise an estimate of mortality as 
the LS estimator will when a random 
and complete age-frequency sample 
can be obtained from a population in 
steady-state. 
Acknowledgments 
Mike Armstrong, Gil McRae, Bob 
Muller, Judy Leiby, as wells as Lynn 
