Brooks et al. : Stock assessment of proiogynous fish 
17 
Table 3 
Summary statistics of relative error (RE) in biological reference points (BRPs) estimated by each measure of spawning biomass. 
BRPs are maximum sustainable yield ( MSY ) and the corresponding fishing mortality rate (F M SY ), spawning biomass (Sj^gy 6 ), 
and spawning potential ratio ( SPR msy )■ Statistics are 25 th quantile, 50 th quantile (median), 75 th quantile, distance covered by 
interquartile range (IQD), proportion of model runs with relative error greater than zero (RE>0), mean, and standard deviation 
(SD). Bold font designates for each BRP the median error closest to zero, mean error closest to zero, proportion of positive RE 
closest to 0.5, smallest IQD, and smallest SD. 
BRP 
Spawning biomass 
25 th quantile 
50 th quantile 
75 th quantile 
IQD 
RE>0 
Mean 
SD 
MSY 
female 
0.02 
0.09 
0.25 
0.23 
0.99 
0.19 
0.25 
male 
-0.34 
-0.23 
-0.14 
0.20 
0.05 
-0.24 
0.14 
both 
-0.15 
-0.07 
0.01 
0.16 
0.27 
-0.05 
0.16 
F MSY 
female 
0.06 
0.26 
0.68 
0.62 
0.82 
0.50 
0.63 
male 
-0.50 
-0.37 
-0.25 
0.25 
0.02 
-0.36 
0.19 
both 
-0.22 
-0.10 
0.11 
0.33 
0.3 
-0.01 
0.33 
c f 
° MSY 
female 
-0.11 
-0.05 
-0.01 
0.10 
0.12 
-0.07 
0.07 
Qtn 
° MSY 
male 
0.15 
0.33 
0.66 
0.51 
0.94 
0.54 
1.05 
qb 
° MSY 
both 
-0.09 
0.01 
0.08 
0.17 
0.54 
0.01 
0.12 
SPRmsy 
female 
0.01 
0.06 
0.18 
0.17 
0.93 
0.14 
0.19 
male 
-0.24 
-0.13 
-0.07 
0.17 
0.06 
-0.16 
0.13 
both 
-0.14 
-0.08 
-0.03 
0.11 
0.17 
-0.08 
0.11 
where iE\fm,b) indicates female, male, or both, and 
BRP represents MSY, F MSY , or SPR msy . 
When interpreting relative error, one should be aware 
that RE has no upper bound but has a lower bound of 
-1 because the BRPs and estimates are always non- 
negative. The distribution of relative errors was used to 
evaluate estimated reference points and thus to provide 
a general picture of which measure of spawning biomass 
is most robust. 
Analysis of variance (ANOVA) of relative errors was 
conducted as a form of sensitivity analysis. Factors 
that explained a significant proportion of total varia- 
tion represent biological or fishery parameters to which 
estimates were sensitive. Factors found to be important 
were then examined in greater detail. 
Results 
Primary analysis— model misspecification 
Aggregated across model runs, variability in estimation 
error, as indicated by distance covered by interquartile 
ranges and standard deviations of relative errors, was 
similar among the three measures of spawning biomass 
(Table 3). Two exceptions occurred: variability was 
relatively large when F MSY was computed from females 
only (Sf) and when S MSY was computed from males only 
(S m ). 
Estimates of BRPs were closest to the true values 
(from simulations) when the assessment model counted 
both males and females ( S b ), as indicated by mean and 
median relative error near zero (Table 3). The assess- 
ment model based on females only tended to overesti- 
mate F msy , MSY, and SPR msy , and it tended to under- 
estimate S^sy slightly. The assessment model based 
on males only showed the opposite pattern; more than 
90% of relative errors in F^ SY , MSY m , and SPR'^ sy 
were negative, and more than 90% in S'^ SY were posi- 
tive. Relative error in S'^ SY could be quite large when 
fertilization rates were independent of male availability 
(x=l). In those cases, males could be almost completely 
removed from the simulation model without detriment 
to the population’s persistence, but not from the as- 
sessment model based on males only. Consequently, the 
computation of relative error of S^ sy - (Eq. 11) included 
a denominator that approached zero, which magnified 
the relative error to values much greater than one. The 
interquartile range of relative error from S f and from 
S' n did not include the value of zero for any reference 
point, where a relative error of zero would correspond 
to a perfect estimate (Table 3). These relative errors, 
with opposite signs, were mediated when both sexes ( S h ) 
defined spawning biomass in the assessment model. 
For all measures of spawning biomass, the steep- 
ness of the fertilization function (k) explained more of 
the variation in estimated BRPs than any other model 
factor (Table 4). The slope of sex transition (/! p ) and 
steepness of the spawner-recruit function (h) explained 
much of the remaining variation. The remaining factors 
explained very little. The residual or unexplained error 
(Table 4) is attributable to interaction terms, which 
were not included in the ANOVA. 
Relative errors of estimated BRPs were further exam- 
ined by levels of k and fi p (Fig. 2, A and B). These two 
parameters were chosen for related reasons; because es- 
