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Fishery Bulletin 95( 1 ), 1997 
lution to this problem lies in a method that would 
objectively identify — and deal with — “outliers.” 
Statistical tests have been developed for identify- 
ing outliers (see Barnett and Lewis, 1994), but most 
of the straight-forward approaches can only deal with 
a few outliers in univariate analyses or in linear re- 
gression. High-breakdown robust regression meth- 
ods (Rousseeuw, 1984; Rousseeuw and Leroy, 1987) 
hold promise for addressing the issue, as suggested 
by several recent papers in fisheries literature (e.g. 
Chen et ah, 1994; Chen and Paloheimo, 1994). The 
goal of high-breakdown robust regression is to pro- 
vide model estimates that are insensitive to contami- 
nation (up to 50%) by outliers and thus will serve to 
identify outlying observations. However, most robust 
regression applications in fisheries science (Chen et 
al., 1994; Chen and Paloheimo, 1994) and in statis- 
tics literature have been developed for linear prob- 
lems (but, see Stromberg, 1993). In this study we 
seek to illustrate the application and usefulness of 
this tool by using two nonlinear examples: a 
nonequilibrium production model for North Atlantic 
swordfish, Xiphias gladius, and a sequential popu- 
lation analysis for West Atlantic bluefin tuna, 
Thunnus thynnus. Both stocks are assessed by the 
Standing Committee on Research and Statistics 
(SCRS) of the International Commission for the Con- 
servation of Atlantic Tunas (ICCAT). The analyses 
presented here are illustrative and are not intended 
to replace those of the SCRS. 
Methods 
Assessment models 
Assuming a normal (Gaussian) error structure, the 
typical tuned assessment method minimizes the 
squared deviations (residuals, r) between observed 
and predicted indices of abundance: 
m n l m 
min Xi (/ y ~4 )2 =min SS (r y 2)j (1) 
i = 1 7=1 i=l 7 = 1 
for m indices, each with n i observations. The predic- 
tion of each index, / ■ comes from a population model, 
such as a surplus production model or a sequential 
population analysis. Alternatively, the minimization 
can be made in terms of observed and predicted 
catches or in terms of observed and predicted fish- 
ing effort. Note that some maximum-likelihood ap- 
proaches do not make the normal error assumption 
(e.g. Fournier and Archibald, 1982); we focus on those 
approaches that are in a least-squares framework or 
that can be transformed to one, which include itera- 
tively reweighted least squares and some forms of 
maximum likelihood. 
In this paper, we give robust regression examples 
using two population models. A detailed explanation 
of these methods is beyond the scope of this paper 
and readers are referred to the citations given be- 
low. The surplus production model corresponds to a 
Schaeffer (logistic) form, fitted as nonequilibrium 
time series by using the continuous time method pre- 
sented by Prager (1994). This method estimates pa- 
rameters describing the carrying capacity, rate of 
intrinsic population growth, initial biomass, and 
catchability coefficients that best explain observed 
time series of relative abundance according to the 
criterion in Equation 1. The sequential population 
analysis corresponds to a tuned virtual population 
analysis method known as ADAPT, an age-structured 
assessment framework popular in the east coast of 
North America. Details on ADAPT can be found in 
Powers and Restrepo (1992, 1993), Punt (1994), and 
Gavaris. 1 ADAPT estimates age-specific fishing mor- 
tality rates in the last year of data and catchability 
coefficients that satisfy Equation 1, while forcing 
cohorts to conform to exponential survival through 
time: 
M , , = N e~ Za ' y 
• 4¥ a + l,j'+l a,y'-' ’ 
where N denotes stock size in numbers, Z denotes 
instantaneous total mortality, and a and y are sub- 
scripts for age and year. 
Data sets 
The data set used with the nonequilibrium produc- 
tion model is for North Atlantic swordfish as em- 
ployed by ICCAT in its 1994 assessment (ICCAT, 
1995). This data set consists of total landings (in 
weight) for the period 1950-93 and of a single stan- 
dardized longline series of catch per unit of effort 
(CPUE, used as a measure of relative abundance), 
spanning the period 1963-93 (Table 1). After a se- 
ries of sensitivity tests, ICCAT assumed in its “base 
case” analysis that the initial biomass in 1950 was a 
known quantity, equal to 0.875 times the stock’s car- 
rying capacity. Thus, 3 parameters were estimated: 
carrying capacity, intrinsic rate of growth, and a con- 
stant of proportionality ( q ) relating the series of rela- 
tive abundance (X) to absolute biomass units ( B ). The 
minimization of Equation 1 was done in log scale, i.e. 
Ijj = ln(X y ) and 7 y = In (qBj). 
The data for the SPA is for West Atlantic bluefin 
tuna, also as employed by ICCAT in its 1994 base 
case assessment (ICCAT, 1995). It consisted of catch 
