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Fishery Bulletin 116(1) 
Table 1 
Parameter values used in creating the hypothetical population of a small, migratory pelagic fish to show 
bias in the estimates of fish growth. Growth was estimated by using a deterministic population dynamic 
model in which selectivity includes effects of gear and fish availability. Sensitivity analyses in this article 
provided changes in the base parameter values. CV=coefficient of variation. 
Parameter 
Base values 
Sensitivity analyses 
Maximum age 
10 
Total mortality ( Z ) 
0.5/year all ages 
Asymptotic length (L in f) 
20 cm 
Growth coefficient (K) 
0.4 
Hypothetical age when average 
length is zero (ao) 
-1 
CV of length at age 
0.15 all ages 
0.1 all ages 
Availability in sampling area 
5% at ages 0-3, 100% at 
100% at ages 0-3, 5% at 
(<4) 
ages 4 and older 
ages 4 and older 
Gear selectivity (s t ) 
Asymptotic pattern (Fig. IB) 
Domed pattern (Fig. 3B) 
can also be estimated by using random-at-length meth¬ 
ods (Hoyle and Maunder 1 ; Finer et al., 2016). Random- 
at-length estimation methods provide a comparison of 
observed and expected age distribution for a specific 
length with the assumption that ages are random with 
respect to length. Age-based processes, such as age- 
based movements (McDaniel et al., 2016) can lead to 
biased growth estimates with the use of random-at- 
length methods (Lee et al., 2017b). 
In many studies where fish growth is estimated, the 
biological (e.g., movement) and observation processes 
(e.g., selectivity) are ignored, which, if ignored, can 
lead to violations of the assumptions about random¬ 
ness (a review by Maunder and Finer, 2017). Estimat¬ 
ing growth parameters simultaneously with these pro¬ 
cesses as part of an integrated model (Fournier et al., 
1990; Methot and Wetzel, 2013) can account for these 
sources of potential bias. Proper use of the integrated 
model is based on the knowledge of biological and fish¬ 
eries processes involved in the collection of data. 
The evolution of integrated assessment modeling 
has generally lead to the inclusion of a greater number 
of factors in an attempt to reduce estimation biases. Si¬ 
multaneously estimating growth and the length-based 
effects of gear selectivity have been thought to remove 
selectivity bias (Parma and Deriso, 1990; Taylor et al., 
2005; Schueller et al., 2014; Piner et ah, 2016). How¬ 
ever in this study we show that a bias can be induced 
when estimates of mean length-at-age (random at age 
assumption) are derived by using a selectivity that 
is a combination of length-based gear and age-based 
availability. This is an approximation bias that is the 
result of approximating an age-based effect by using a 
1 Hoyle, S. D., and M. N. Maunder. 2005. Status of yellow- 
fin tuna in the eastern Pacific Ocean in 2004 and outlook 
for 2005, 102 p. Inter-Am. Trop. Tuna Comm., La Jolla, 
CA. [Available from website.] 
length-based process. The magnitude and direction of 
the bias is dependent on the spatial areas sampled and 
variability in the length-at-age relationship. 
Materials and methods 
We use a deterministic population dynamics model 
to show the effects on growth estimates of combining 
both age-based availability and length-based gear se¬ 
lectivity, into a single length-based process. Conceptu¬ 
ally the stock is distributed in two areas: one area is 
primarily a juvenile area and the other is primarily 
an adult area. The deterministic model approximates 
the spatial dynamics by using age-based availability as 
implicit areas in a single well-mixed area. Availability 
is defined as the proportion of each age class found in 
an area. In our study, the life history and fishery char¬ 
acteristics of a small migratory pelagic fish are used 
to create the hypothetical population (Table 1). To fur¬ 
ther simplify the example, we assume that all fishing 
takes place at the same time each year and therefore 
length-at-age can be calculated without the additional 
complication of within year growth. 
The mean length-at-age from fishery data collected 
in the adult area is calculated in three ways: 1) true 
(used to generate population numbers at age/length), 2) 
observed (does not account for length-based selectivity 
or age-based availability), and 3) estimated (which ac¬ 
counts for the observed length-based selection, which is 
a combination of length-based gear selectivity and an 
approximation of age-based availability). Sensitivity of 
the estimates of mean length-at-age to changes in life 
history and fishery characteristics are illustrated as 
single-instance changes to the example given in Table 
1. The equations governing the simulated population 
are given below. 
The value for population proportions-at-age is given by 
