863 
Bias in Chapman-Robson and least- 
squares estimators of mortality rates 
for steady-state populations 
Michael D. Murphy 
Florida Marine Research Institute 
Florida Department of Environmental Protection 
100 Eighth Avenue SE, St. Petersburg, Florida 33701-5095 
E-mail address: murphy_m@harpo. dep.state.fi. us 
When age-frequency data are insuf- 
ficient for fisheries scientists to es- 
timate year- or age-specific mortal- 
ity, they are often pooled to provide 
a single estimate for all fully re- 
cruited age groups. The accuracy of 
a pooled estimate depends largely 
on whether or not the sampled 
population is in a steady state, i.e. 
a state in which the rates of recruit- 
ment and mortality are relatively 
constant with respect to time and 
age. Departures from this condition 
introduce known biases to the esti- 
mate of mortality (Ricker, 1975; 
Jensen, 1984). These departures 
may be difficult to detect because 
trends in time-specific recruitment 
or time- and age-specific mortality 
can result in a population age struc- 
ture that is quite similar to that for 
a steady-state population. For in- 
stance, a long-term increasing 
trend in recruitment could result in 
a stable age frequency that would 
indicate a higher mortality rate 
than was actually occurring (for 
various scenarios see Ricker, 1975). 
Pooled-data estimation tech- 
niques that have been applied to 
age-frequency data for fish popula- 
tions include “catch curve” least- 
squares regression analysis (Seber, 
1973; Ricker, 1975) and nonre- 
gression-based methods developed 
by Heincke ( 1913), Jackson ( 1939), 
and Chapman and Robson (1960). 
Of the nonregression-based estima- 
tors, the Chapman and Robson es- 
timator is preferred because it is 
the least sensitive to sampling er- 
ror (Robson and Chapman, 1961). 
Despite the restrictive steady-state 
requirements, these techniques 
have been applied to a wide vari- 
ety of marine animals; recent ex- 
amples include Atlantic croaker, 
Micropogonias undulatus (Barbieri 
et al., 1994); blue rockfish, Sebastes 
mystinus (Adams and Howard, 
1996); red drum, Sciaenops ocel- 
latus (Ross et al., 1995); red porgy, 
Pagrus pagrus ( Pajuelo and Lorenzo, 
1996); and deep-water shrimp, 
Aristeus antennatus (Ragonese and 
Bianchini, 1996). 
The Chapman-Robson (CR) esti- 
mator is based on the probability 
density function of the geometric 
distribution and provides a unique 
minimum-variance, unbiased esti- 
mate of survival (S), 
N 
N + ^ X , ~ 1 
i=l 
where x ( = the number of years 
the ith fish is older 
than the age at full re- 
cruitment; and 
N = the total number of 
fully recruited fish. 
The underlying assumption is that 
the age of each fish sampled repre- 
sents a random, independent age 
observation from a steady-state 
population. For age-frequency data 
that have been truncated to elimi- 
nate some older age groups, a 
slightly biased maximum likelihood 
estimator of survival (CRt) that can 
be solved by iteration is 
S 
1-S 
~(K + 1) 
g(K+l) 
1 -S iK+1) 
where K + 1 = the number of fully 
recruited age groups used (Chap- 
man and Robson 1960). 
The least-squares regression (LS) 
estimator provides an unbiased es- 
timate of -log S (denoted as Z, the 
instantaneous total mortality rate) 
and is based on a linear fit to a log- 
transformed exponential decay 
model 
E (log Nj ) = log ( pNq ) - Zj, 
where AT = the number of age j 
fish in the sample; 
N = the original number of 
fish in the population; 
and 
p = the probability that a 
fish in the population is 
included in the sample 
(Seber, 1973). 
As required for linear regression, 
log-abundance data are assumed to 
be independent and normally dis- 
tributed with constant variance 
along the regression line. Concern 
about violating these assumptions 
led Chapman and Robson ( 1960) to 
recommend that when the LS 
method is used, the age-frequency 
data should be truncated to exclude 
less abundant age groups. 
Although both the CR and LS 
estimator returned very accurate 
estimates of Z for a suite of exact 
steady-state age frequencies (Jensen, 
1985), the effect of random variation 
within the sample age frequencies 
Manuscript accepted 30 May 1997. 
Fishery Bulletin 95:863-868 ( 1997). 
