Erickson and Nadon: Stepwise stochastic simulation for distributions of missing life history parameter values 79 
Table 1 
List of life history parameters, with their definitions, used 
in the stepwise stochastic simulation to produce missing 
estimates of life history parameters for groupers (Serrani- 
dae), wrasses (Labridae), grunts (Haemulidae), and sharks 
from the orders Carcharhiniformes and Lamniformes. All 
lengths used for these parameters are total lengths. 
Parameter Definition 
Expected length at the oldest recorded age 
(also known as L,) 
Length at which 50% of individuals reach 
maturity 
Longest length in a growth study or 99th 
percentile of lengths in a population survey 
Asymptotic length (expected length at 
infinite age) 
Brody growth coefficient of the von Berta- 
lanffy growth function 
Theoretical age at which length equals zero 
Oldest recorded age (i.e., longevity) 
Instantaneous natural mortality rate 
are not universal but specific to taxonomic groups (e.g., 
snappers). The 4 models (A—D) of statistical relationships 
between life history parameters used in the stepwise 
approach (Fig. 1) are, sequentially, as follows: 
A) L,, = bo + biL max + € (linear function); (1) 
B) log(K) = log(b,) + bslog(L...) + € 
(power function) or (2) 
log(K) = log(b,) + b3L., + € (exponential function); (3) 
C) log(M) = log(b4) + b;log(K) + belog(Lnax) 
+ € (power function) or (4) 
log(M) = log(b4) + 65K + bgLinax 
+ € (exponential function); and (5) 
D) Linat = 07 + Ogle amax + € (linear function), (6) 
where € =a normally distributed error term (note: this 
error term can also be lognormally distributed 
for models A and D, if the residuals pattern indi- 
cates a lognormal distribution; 
b =a model coefficient, either the slope or intercept; 
and 
Liamax = the expected length at the oldest recorded age 
Glandb 
These 4 models are used in the stepwise approach to 
generate probability distributions of the parameters L.,, 
K, Lat) and M as follows. First, a local estimate of L,,,,, is 
obtained from a data set of representative lengths (note: 
we recommend using the 99th percentile of the length 
data set to avoid inclusion of an outlier L,,,, value and 
bootstrapping this data set to incorporate variability in 
Lyyax)- Second, this L,,,, value is entered into model A (also 
referred to as L,,,,~L..) to obtain the expected L., value at 
a given L,,,,- Third, a single random L,, value is sampled 
from the error distribution of L,, at that specific location 
on the linear regression model. Fourth, this random L,, 
value is used in model B to obtain the expected K value 
for this given L,, value. A random K value is then drawn 
from the error distribution at this location. Fifth, this ran- 
dom K value is entered into model C, which also includes 
the starting L,,,, value, to obtain the expected M value. 
A random M estimate is sampled at this location in the 
regression model. 
Next, the previously estimated random L.,, K, and M val- 
ues are entered into the von Bertalanffy growth function to 
obtain La max, also known as L, in Nadon and Ault (2016): 
fly oe SIb_ (oe Ny, (7) 
where Ay = the age at length zero (a fitting parameter of the 
von Bertalanffy equation that is fixed at —0.6 in 
the stepwise approach, its overall mean value). 
The parameter A,,,, is obtained by using the relation- 
ship of longevity to M presented in Hoenig (1983) and in 
Hewitt and Hoenig (2005), under the assumption that sur- 
vivorship of a cohort at the oldest recorded age (S) was 5%, 
with this formula: 
= —In(0.05)_ (8) 
Ana 
Note that when implementing the stepwise approach, 
users can calculate an M estimate under a different equa- 
tion or S value and are not bound to use an S of 0.05. The 
flexibility of natural mortality estimates is discussed fur- 
ther in the “Discussion” section. 
Finally, the La,,,, value obtained by using Equation 7 
is entered into model D to obtain the expected L,,,,; value, 
and a random L,,,, is drawn from the error distribution 
at this location on the model D curve. This stepwise pro- 
cedure is repeated several thousand times to obtain a 
multivariate probability distribution of all 4 life history 
parameters that can then be directly used in various stock 
assessment models. Uncertainty of regression model 
coefficients was included at every step by drawing a ran- 
dom set of regression coefficients from a normal multi- 
variate distribution, defined by the variance—covariance 
matrix associated with these regression coefficients. Val- 
ues of the variance—covariance matrix are presented in 
Table 2, along with model coefficients and residual error 
distributions. 
M 
Literature review and model fit for new taxa 
The previous section describes how the stepwise proce- 
dure is implemented by using predictions from a series 
of 4 previously established regression models. Regression 
models used in each step are specific to individual taxo- 
nomic groups (e.g., parrotfishes, snappers, or jacks). To 
create 4 new family-specific models, we first reviewed the 
literature to generate new data sets of life history traits. 
