16 
Fishery Bulletin 106(1 ) 
R(S) = 
4 hR 0 S 
R 0 <j> 0 (l-h) + (5h -1)S ’ 
( 10 ) 
where R 0 = unfished recruitment; 
li = steepness; and 
<p 0 = unfished spawning biomass per recruit. 
Because fertilization rate is not considered in the assess- 
ment model, it is assumed that x=1.0 always. Given 
the spawner-recruit relationship of Equation 10, we 
computed catch per F assuming equilibrium population 
structure. The estimate of MSY ( MSY ) was taken to be 
maximum catch, and F MSY was the F that produced that 
maximum. Also estimated were the associated spawning 
biomass (S MSY ) and SPR ( SPR msy ). These four estimates 
of BRPs were computed by using each measure of spawn- 
ing biomass and compared to the true values from the 
simvulation model. 
Scope of analyses 
These analyses were designed to quantify systemati- 
cally the magnitude and direction of error of estimated 
BRPs. Initially, only model misspecification was consid- 
ered. This part of the study is described as the primary 
analysis, because it addresses the main goal of isolating 
error associated with predicting recruits from spawning 
biomass. Subsequently, additional sources of error were 
introduced into the assessment model, described as the 
secondary analysis. Primary and secondary analyses 
are detailed below. 
Primary analysis— model misspecification 
The assessment model was misspecified in the sense that 
it did not explicitly account for dynamics of fertilization. 
Otherwise, the simulation and assessment models were 
identical, both in structure and in parameter values. 
These values were assigned according to a factorial 
design that included seven factors at various levels 
(Table 1). The factors were natural mortality rate (M, 
3 levels), steepness of spawner-recruit function (h, 3 
levels), steepness of fertilization function (k, 9 levels), 
slope of sex-transition function (/3 , 4 levels), age at 50% 
maturity in relation to its mean (c , 3 levels), slope of 
maturity function ( p g , 4 levels), and age at selection by 
the fishery in relation to maturity (c s , 3 levels). Thus, the 
simulations covered a wide array of biological and fishery 
conditions, with n = 11,664 factor-level combinations. At 
each combination, BRPs were computed with the simu- 
lation model, and then estimated with the assessment 
model by using each of the three measures of spawning 
biomass (Sf, S m , S b ). 
Table 2 
Factors (model parameters) and levels (parameter values) 
of the secondary analysis, where an incorrect value of age 
at 50:50 sex ratio was assumed in the assessment model 
or where fecundity was assumed to scale linearly with 
weight. 
Factor 
Levels 
Description 
M 
K 
h 
P P 
X p 
Xf 
10.1, 0.2, 0.3) 
|0.2, 0.3, ..., 1.01 
10.4, 0.6, 0.8) 
(0.2, 0.4, 0.8, 1.6) 
10.75, 1.0, 1.25) 
10. 75, 1.0, 1.25) 
Natural mortality rate 
Steepness of fertilization 
function (f) 
Steepness of spawner- 
recruit function 
Slope of logistic sex- 
transition function 
Multiple of age at 50:50 sex 
ratio (a p =2.3x p /M) 
Multiple of fecundity-at-age 
exponent (£ 2 =3 Xf) 
50:50 sex ratio (a ). Estimates of this parameter used 
in an assessment model may be inaccurate because of 
sampling error or adaptations in response to fishing 
mortality (Goodyear, 1980; Harris and McGovern, 1997; 
Barot et al., 2004). Sex transition in the assessment 
model remained the same (a p =2.3/M; Eq. 4) but was 
adjusted in the simulation model by a scalar multiple 
X p ( a p =2.3x p /M ). In a second subset of this analysis 
we examined violation in the assessment assumption 
that fecundity scales linearly with weight. This was 
accomplished by redefining the fecundity exponent in 
the simulation model (£ 2 =3; Eq. 3) by a scalar multiple 
Xg (£ 2 =3 x € ), without adjusting the assessment model. For 
the secondary analysis (Table 2), the remaining model 
parameter values were as in the primary analysis (Table 
1), with the following three exceptions: the slope of matu- 
ration was set to a moderate value (/3 = 0.8), age at 50% 
maturity was set to its mean (c =1), and age at selection 
was set to age at 50% maturity (c s =l). As before, the 
intent was to characterize error of estimated BRPs and 
thereby identify robust measures of spawning biomass. 
Evaluation of assessment results 
Assessment results were evaluated in terms of relative 
error, i.e., the relative difference between reference 
points known from the simulation model (Eqs. 8 and 9) 
and the corresponding estimates from the assessment 
model. At each combination of factor and level, relative 
error (RE) was computed as 
Secondary analysis— additional misspecifications 
Further analysis included additional sources of mis- 
specification. One subset of this analysis examined 
misspecification of the parameter controlling age at 
RE (BRP i ) = 
RE(SW> = 
BRP'-BRP 
BRP 
S\ 
MSY 
S\ 
■’MSY 
MSY 
( 11 ) 
