Li et al: A comparison of 4 age-structured stock assessment models 
157 
x, = first slope of the double-logistic selectivity 
function; 
X = first inflection point of the double-logistic 
selectivity function; 
B, = second slope of the double-logistic selectivity 
function; 
B,. = second inflection point of the double-logistic 
selectivity function; 
y, = first inflection point of the selectivity 
function; 
Y2 = second inflection point of the selectivity 
function; 
Pp, = distance between first inflection point and 
age at 95% selection; 
P2 = parameter to be used with p, to get first 
inflection point of the selectivity function; 
P3= second slope of the selectivity function; 
u, = first inflection point of the double-logistic 
selectivity function; 
Uy = first slope of the double-logistic selectivity 
function; 
ls = second inflection point of the double-logistic 
selectivity function; and 
li, = second slope of the double-logistic selectivity 
function. 
Spawner-recruit parameters in bias adjustment of recruitment 
In both the models in which a bias adjustment to recruit- 
ment was applied, the BAM and SS, the mean-unbiased rela- 
tionship is used for estimating MSY-based reference points. 
The Fysy from the OM was based on searching for the F 
that provides maximum equilibrium catch. The calculation 
started with computing equilibrium recruitment at F and 
involved the same equations that were used to calculate the 
initial equilibrium conditions (equation 3.4 in Supplemen- 
tary Table 1 [online only]). That step of calculation involved RO 
(Suppl. Table 3) (online only), and the use of RO in the calcula- 
tion results in differences in estimated MSY-based reference 
points depending on how bias adjustment of recruitment 
was addressed in each EM. The median-unbiased spawner- 
recruit parameters from the BAM correspond to the geo- 
metric mean curve of recruitment, and the mean-unbiased 
parameters from SS correspond to the arithmetic mean 
curve of recruitment (Table 4, Suppl. Table 3 [online only]). 
For direct comparisons, we derived a function to con- 
vert combinations of median-unbiased RO and h to mean- 
unbiased values, or to convert mean-unbiased values 
to median-unbiased values, when the Beverton—Holt 
spawner-recruit model was used. The derivation of the 
median-unbiased approach starts with calculating the 
median-unbiased RO by using the Beverton—Holt model: 
0.8ROhSO 
‘i 0.26 )RO(1—h)+(h-0.2)S0° 3 
where SO = unfished spawning biomass. 
Given that the bias adjustment component 6 equals 
eor/2, the mean-unbiased RO’ that corresponds to the 
arithmetic mean curve of recruitment is determined 
with this equation: 
‘ 60.8ROhSO’ 
~ 0.29R0(1- h)+(h-0.2)S0"" 
, 
(9) 
where SO’ = the mean-unbiased unfished SSB. 
Then, following the derivations below, the bias-adjusted 
mean-unbiased SO’ can be obtained: 
, _ b0.8ROhSO' 
0.26RO(1—h)+(h -0.2)S0’ = See. iO) 
= b0.8R0hb,, 
0.2R0(1- h) +(h- 0.2)20 = b0.8R0h, 
0 
SO’ 
0 
(h - 0.2)—— = 60.8ROh - 0.2R0(1 = h), and 
- 60.8R0hb, — 0.2R0(1 - h) oo 
: h-0.2 
When converting median-unbiased values to mean- 
unbiased values, the bias adjustment component 6 equals 
er? Inputs of the conversion function include median- 
unbiased RO, h, and $ ). The outputs of the conversion 
include adjusted unfished SSB (S0,), adjusted unfished 
recruitment (RO,), and adjusted steepness (h,;,) in mean- 
unbiased levels: 
b0.8ROh6, — 0.2)RO(1 — h) 
SO, = (11) 
h-0.2 
RO, = ony (12) 
ee b0.8ROh0.2S80, aml 8) 
0.2) RO(1 — h) + (h — 0.2)0.280, 
lon = Brew (14) 
RO, 
where R,,,,, = recruitment when the SSB is 20% of its 
unfished level. 
When converting mean-unbiased values to median- 
unbiased values, b equals e °®/”, and the inputs from the 
equation (Equation 11) need to be mean-unbiased values 
(the outputs are median-unbiased SO,, RO, and h,,). With- 
out these conversions, the difference between mean- and 
median-unbiased RO and the difference between mean- 
and median-unbiased h gradually increased when the 
true median-unbiased h was reduced and when op was 
increased (Fig. 3). 
Simulation-estimation process 
Null case (case 0) Model parameters of RO and catchabil- 
ity (q) were accurately estimated in all EMs with low bias. 
