154 
Fishery Bulletin 95(1 ), 1997 
tiple replicates and restarts so that there is some 
confidence that the solution is globally minimal. At 
this point we have no firm guidance about the 
tradeoffs between the number of replicates (s) and 
the number of restarts other than to say that repli- 
cates are probably more important than restarts. For 
example 500 replicates with 5 restarts seems prefer- 
able to 25 replicates with 100 restarts. 
Dealing with outliers 
Aside from biological or fishery considerations, sta- 
tistical outliers are data points whose residuals, 
scaled by the dispersion of errors, 
r JL 
a 
are far from the mean scaled residual. For the simple 
LS minimization (Eq. 1), the overall dispersion of the 
residuals is the mean squared error (MSE), 
o = 
*'= 1 7=1 
For the LTS regression, the dispersion is similarly 
computed as a robust measure of average dispersion 
( cr for index i in Eq. 2 or a for all data points in Eq. 3): 
a, =3.7444 
n < >. 
I>%, 
for LTSj, Eq. 2, or 
a = 3.7444 
m / 2+ 1 
for LTS 2 , Eq. 3. 
The constant 3.7444 is a correction factor used to 
achieve consistency with normal error distributions 
(Rousseeuw and Leroy, 1987). As a rule of thumb, 
Rousseeuw and Leroy (1987) suggest that absolute 
values of scaled residuals larger than 2.5 can be treated 
as statistical outliers. Owing to the small number of 
observations in some of our data series, we use a thresh- 
old based on the t - distribution with a = 0.01 and n - 1 
degrees of freedom. After obtaining the LTS estimates, 
we carried out a new least-squares minimization ex- 
cluding from the analyses any absolute scaled residu- 
als greater than the corresponding critical value. We 
refer to this final result as the “trimmed LS” solution. 
Results 
Swordfish nonequilibrium production model 
The swordfish data represent a simple example with 
a single index. Nevertheless, there are several very 
large deviations between the observed and predicted 
index values, when the traditional least-squares (LS) 
solution to the nonequilibrium production model fit 
is computed (Fig. 1). Indeed, these deviations have 
generated considerable debate (ICCAT, 1995). There- 
fore, we applied the robust regression techniques of 
the LTS algorithm and the trimmed LS method of 
outlier detection to this example. The LTS solution 
(with a 50% trim) was computed 500 times with 5 
restarts each. There did not appear to be problems 
of multiple minima with this example because vir- 
£ 0.50 
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Figure T 
Biomass index values for North Atlantic swordfish using 
least squares (LS), least trimmed squares (LTS), and 
trimmed least squares (Trimmed LS). The top panel shows 
observed and predicted index values. The bottom panel 
shows the scaled residuals from the LTS fit compared with 
the critical value for the outlier detection criterion (dashed 
lines); solid circles are those data points considered to be 
outliers and not included in the final trimmed LS solution. 
