18 
Fishery Bulletin 106(1) 
Table 4 
Sensitivity of relative errors in estimated biological reference points to each model factor in the primary analysis, where the 
assessment model did not account for dynamics of fertilization. For each reference point ( F MSY , S^ Y , MSY , and SPR msy ), the 
measure of spawning biomass (female, male, or both) with the smallest total model error (total SS) demonstrated the least vari- 
ability (values in italics). Table cells give the proportion of total SS explained by each factor. Values >0.1 are indicated by bold 
font and values <0.01, by dashes. The term “Residual” is variation explained by all possible interaction terms. Factors (model 
parameters) are defined in Table 1. 
Factor 
Fmsy 
of 
°MSY 
Female 
ora 
° MSY 
Male 
qb 
^ MSY 
Both 
MSY 
SPRmsy 
Female 
Male 
Both 
Female 
Male 
Both 
Female 
Male 
Both 
M 
0.03 
— 
0.02 
0.08 
— 
0.02 
0.03 
— 
— 



K 
0.42 
0.60 
0.56 
0.23 
0.17 
0.46 
0.49 
0.47 
0.64 
0.33 
0.32 
0.54 
h 
— 
0.05 
0.07 
0.07 
0.02 
0.05 
0.09 
0.28 
0.08 
0.23 
0.41 
0.06 
P P 
0.16 
0.02 
0.08 
0.16 
0.04 
0.10 
0.11 
0.03 
0.02 
0.08 
— 
— 
P g 
— 
— 
— 
— 
— 
— 
— 
— 
— 
— 
— 
— 
C g 
0.03 
0.03 
— 
— 
0.02 
— 
— 
— 
— 
— 
— 
0.02 
C s 
0.05 
0.09 
— 
— 
0.03 
— 
— 
0.04 
- 
— 
0.02 
0.02 
Residual 
0.31 
0.22 
0.27 
0.45 
0.72 
0.36 
0.27 
0.15 
0.23 
0.34 
0.23 
0.36 
Total SS 
4624 
441 
1283 
57 
12,894 
179 
728 
240 
307 
423 
183 
144 
timates were sensitive to them (Table 4) and because of 
their influences on the dynamics of fertilization (Fig. 1). 
Examining relative error by steepness of fertilization 
revealed that the most appropriate measure of spawn- 
ing biomass depended on the level of k. If male deple- 
tion had little effect on fertilization success (k in the 
range 0. 8-1.0), the conventional measure, Sf, produced 
estimates with the least error. However, as fertilization 
became more limited by male depletion (0.2 < x<0.8), er- 
ror in estimates from Sf became increasingly more vari- 
able and further from the true values. At intermediate 
values of k (~0. 4-0.7), S b produced the best estimates. 
Only for the most limiting values of k (0.2, 0.3) did S m 
appear to be appropriate. 
The influence of [3 p on fertilization success was per- 
haps more subtle than that of k. A shallower slope of 
sex transition (smaller /) ) provided a broader range of 
age classes where both males and females were pres- 
ent. This decreased the propensity for fishing-induced 
male depletion, thereby allowing sex ratio to remain in 
the range where fertilization rates were relatively high. 
Conversely, if sex transition occurred across only a few 
ages (large f} ), disproportionate fishing on males was 
more likely. The tendency for the depletion of males 
with a steeper slope of sex transition explains why the 
assessment model based on S 1 performed progressively 
worse as fi increased (Fig. 2, A and B). In general, our 
examination of relative error by slope of sex transition 
revealed that S b provided the best estimates. 
A consistent pattern in relative errors was that BRPs 
based on Sf had the opposite sign from those based on 
S m , and in most cases (fc>0.4), from those based on S b 
as well (Table 3, Fig. 2, A and B). Specifically, S MSY 
tended to be underestimated by Sf and overestimated 
by S b , and the other three reference points ( MSY , F MSY , 
and SPR msy ) tended to show the reverse. This result 
indicates that, in most cases, estimates from Sf and 
from S b could be used to bound uncertainty. 
Secondary analysis— additional misspecifications 
When the true age at 50:50 sex ratio (a ) was younger 
than the age used in the assessment model, Sf provided 
the best estimates of BRPs; when the true age used in 
the simulation model was older than the age used in 
the assessment model, S b provided the best estimates 
(Fig. 3). When it was assumed incorrectly with the 
assessment model that fecundity increased linearly with 
weight, whether too quickly or too slowly, S b generally 
provided the best results (Fig. 3). 
As was seen in the primary analysis, resilience of 
fertilization to male depletion (k) explained the most 
variation in relative errors of estimated BRPs, followed 
closely by the parameter (x p ) defining misspecifica- 
tion in the age at 50:50 sex ratio (Table 5). Steepness 
of the spawner-recruit function (h) and slope of sex 
transition (j3 ) also explained some variation. Neither 
natural mortality (M) nor the parameter (/-) defining 
misspecification of the fecundity exponent explained 
much variation. 
Discussion 
We used simulations to investigate the performance 
of three measures of spawning biomass — females only 
(Sf), males only (S m ), and both sexes combined ( S b ) — for 
their ability to estimate BRPs. Performance was quan- 
tified in terms of relative errors, which were computed 
across many sets of values of biological and fishery 
parameters. In the primary analysis, with misspecifi- 
cation in the spawner-recruit relationship only, an 
