Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 
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Table 2 
Summary of the 9 scenarios for evaluation of the sensitivity of the results 
from models of yield per recruit and spawning biomass per recruit to dif- 
ferent degrees of bias and imprecision in several parameters: natural mor- 
tality (M), von Bertalanffy growth coefficient (K), asymptotic length (L„), 
and current fishing mortality rate (.F cur ). Also the multiplicative error in 
the growth curve (cgr) and the length-weight (LW) relationship (£Rw) are 
modeled. In each scenario, the mean or standard deviation (SD) of only 
one parameter was changed. 
Scenario 
Description 
Parameter 
Range 
1 
Standard scenario 
Unchanged 
100% 
2 
Mean of M 
M 
10-1000% 
3 
SD of M 
e M 
10-1000% 
4 
Mean of K 
K 
10-1000% 
5 
Mean of 
50-200% 
6 
Error in growth curve 
£GR 
10-1000% 
7 
Error in LW relationship 
£lw 
10-1000% 
8 
Mean of F cur 
F 
1 cur 
10-1000% 
9 
SD of F cur 
f F C ur 
10-1000% 
and £gr> were generated from normal distribution with 
the means and variances listed in Table 1. These ran- 
dom errors were incorporated into the YPR and SPR 
formulae provided in Table 1, and then 4 Fbrp s (F ma x> 
Fo.i, F 3 q%, and F 50 %) were calculated. For each simula- 
tion scenario, this process was repeated 5000 times to 
produce 5000 corresponding sets of Fbrp values from 
which the empirical distributions of the 4 Frrp s were 
generated and composite risks were calculated. 
Calculation of composite risks 
Because F cur and the Frrp s are not fixed constants, we 
applied composite risk analysis that allowed for the 
incorporation of the uncertainty in both indicator and 
management reference points (Prager et al., 2003; Jiao 
et al., 2005). By the discrete approach proposed by Jiao 
et al. (2005), the composite risks were calculated as 
the expected probability of one random variable being 
larger than another. Let fix) be the empirical probabil- 
ity density function of the Fbrp of interest (e.g., F max ) 
and giy) be the empirical probability density function 
of F cur with a corresponding cumulative density func- 
tion of Giy) and then, let Ax be a small increase in 
x (i.e., the Frrp), and by summing x over its range, 
the composite risk of F cur exceeding Fbrp is calculated 
with the following equation: 
P(F cur > F brp = 1 - ]T'^ o 0O [G(x)/’(x)]Ax (3) 
culation of composite risks in discrete 
approach, refer to Jiao et al. (2005). 
Sensitivity analysis 
For the sensitivity analysis, the pa- 
rameter values in Table 1 were set as 
the standard scenario (scenario 1; Ta- 
ble 2). The parameters for which mean 
and SD values may potentially affect 
results of YPR and SPR models were 
investigated in 9 scenarios, in which 
the mean or SD of only one parameter 
was changed (Table 2): the mean of M 
(scenario 2) and its SD (£m, scenario 
3), the mean of K (scenario 4), and the 
mean of L ^ (scenario 5). In practice, 
the bias in K can result from aging er- 
ror and the use of different estimation 
approaches, and the bias in often 
results from unrepresentative sam- 
pling schemes. Therefore, we assumed 
that the sources of biases in K and 
are unrelatedand they were modeled 
independently. Because modeling the estimation errors 
in the coefficients K and can lead to underestima- 
tion of the actual uncertainty in the data (Lin et al., 
2012), the error was modeled on the growth curve (£qr) 
rather than coefficients (scenario 6 ). The remaining 
scenarios are £rw (scenario 7) and the mean and SD 
values of F cur and £p cur (scenarios 8 and 9). 
For scenarios 1-7, the mean and SD values of the 
parameters, except for L^, were decreased to 10 % with 
an increment of 5% or were increased to 1000% with 
an increment of 50% to cover the possible magnitudes 
of biological variation (as applied in Goodyear, 1993). 
Because information about L can be obtained from 
the maximum observed length in the data, that pa- 
rameter may be subject to less bias and imprecision 
and was set from 50% to 100% with an increment 
of 5% and from 100% to 200% with an increment of 
10% (Table 2). For scenarios 8 and 9, because of less 
computational load (calculation of an Fbrp is inde- 
pendent of F cur , and, therefore, we needed to compute 
only composite risks of overfishing), we used finer in- 
crements: the mean and SD of F cur were decreased to 
10 % with an increment of 1 % or increased to 1000 % 
with an increment of 10 %. 
The relative change (RC) between the mean or SD 
of one Fbrp for increment i in scenario j ( RC "p ) and 
the mean or SD of that Frrp in the standard scenario 
was used to quantify the sensitivity of a given Fbrp 
for all scenarios except 8 and 9. RC is defined with the 
following equation: 
Here, we assume a gamma distribution for F cur because 
1) our previous study (Lin and Tzeng, 2008) indicated 
that gamma distribution fitted better for the distribu- 
tion of fishing efforts, 2 ) it produces nonnegative val- 
ues, and 3) it is flexible in shape. For details in cal- 
RC'i = 100 x TS x(Tp T 1 , 
PRRP PRRP t BRP ’ 
(4) 
Where TS = the statistic (mean or SD) of the Fbrp 
Rrrp -Divr 
of interest (e.g., F max ) for increment i in 
scenario j; and 
