Restrepo and Powers: Application of robust regression to tuned stock assessment models 
151 
Table 1 
North Atlantic swordfish, Xiphias gladius, data used for the nonequilibrium production model (from ICCAT, 1995). Relative 
abundance is in Kg/1,000 standard hooks, standardized from Canadian, Japanese, Spanish, and U.S. longliners. t = metric tons. 
Year 
Landings (t) 
Relative abundance 
Year 
Landings (t) 
Relative abundance 
1950 
3,646 

1973 
6,001 
— 
1951 
2,581 
— 
1974 
6,301 
— 
1952 
2,993 
— 
1975 
8,776 
421.69 
1953 
3,303 
— 
1976 
6,587 
353.66 
1954 
3,034 
• — 
1977 
6,352 
393.92 
1955 
3,502 
— 
1978 
11,797 
649.61 
1956 
3,358 
— 
1979 
11,859 
338.57 
1957 
4,578 
— 
1980 
13,527 
430.69 
1958 
4,904 
— 
1981 
11,138 
310.18 
1959 
6,232 
— 
1982 
13,155 
356.96 
1960 
3,828 
— 
1983 
14,464 
287.88 
1961 
4,381 
— 
1984 
12,753 
286.12 
1962 
5,342 
— 
1985 
14,348 
265.94 
1963 
10,189 
1,258.10 
1986 
18,447 
255.54 
1964 
11,258 
467.29 
1987 
20,234 
217.30 
1965 
8,652 
294.86 
1988 
19,614 
207.62 
1966 
9,338 
273.50 
1989 
17,299 
196.90 
1967 
9,084 
320.22 
1990 
15,865 
199.20 
1968 
9,137 
269.55 
1991 
15,224 
194.02 
1969 
9,138 
233.95 
1992 
15,593 
182.55 
1970 
9,425 
274.25 
1993 
16,977 
172.27 
1971 
5,198 
— 
1972 
4,727 
— 
at age from 1970 to 1993 for ages 1 to 10 + (Table 2), 
and of 7 indices of relative abundance assumed to 
track different segments of the population (Table 3; 
see Fig. 4 ). A number of assumptions were made and 
these can be found in Appendix BFTW-2 of ICCAT 
(1995). The parameters estimated were 7 constants 
of proportionality relating each index of relative 
abundance to absolute biomass or numbers and 4 
fishing mortalities in 1993 (for ages 2, 4, 6, and 8). 
We reiterate that we chose the same data sets and 
model structures as those in ICCAT (1995) for illus- 
trative purposes. It may be worthwhile to investi- 
gate the results of robust regression techniques ap- 
plied to alternative data (e.g. indices obtained with 
a different standardization procedure) or to formu- 
lations (e.g. different assumptions about known 
quantities and other constraints). 
Robust regression 
Several robust minimization criteria discussed in 
Rousseeuw and Leroy (1987) have been applied to 
fisheries data (see Chen et al., 1994). In contrast with 
the method of least squares, the goal of these tech- 
niques is to moderate the influence of outliers in the 
parameter fitting process (Eq. 1). Of particular in- 
terest to us are the so-called “high-breakdown” meth- 
ods that are insensitive to up to 50% contamination 
by outliers, because they can effectively be used as 
an objective method to identify outliers. 
Two high-breakdown robust regression methods 
are least median squares (LMS) and least trimmed 
squares (LTS). LMS minimizes the median of the 
squared residuals and LTS minimizes the sum of the 
lowest xn squared residuals, where x is a fraction 
(less than 1.0 to 0.5) defined by the user. The results 
of an LMS regression and an LTS regression with a 
50% trim are essentially very similar, although the 
LTS one is statistically more efficient (Rousseeuw 
and Leroy, 1987). In our initial experimentation with 
fisheries assessment models, we found that the LTS 
minimum was somewhat easier to find (the LMS 
could sometimes not converge, indicating that a large 
number of restarts may be required). Therefore, we 
limited our investigation to the LTS minimization 
criteria discussed below. This can be either 
m nj 2+1 
LTS 1 = y min (r 2 ) (2) 
LTS 2 = min^T ^V 2 ) y:; 
i=l 1=1 
