An empirical weight-at-age approach reduces estimation bias compared to modeling parametric growth in integrated, statistical stock assessment models when growth is time varying
Introduction
Mean size at age in stock assessments is generally assumed constant through time and is typically characterized by estimating a somatic growth curve. However, assuming time-invariant somatic growth may be overly simplistic because growth can vary due to temporal variability in a number of factors such as temperature (Taylor, 1958), food availability (Jones, 1986), salinity (Boeuf and Payan, 2001), and light (Boeuf and Le Bail, 1999). Estimating time-varying growth curves from observed size at age is difficult, as changes can also be due to other factors, such as changes in selectivity or length-weight relationships. Selectivity can vary temporally due to changes in fishery or survey gear types (Myers and Hoenig, 1997), potentially confounding the estimation of time-varying growth parameters (Thorson and Simpfendorfer, 2009). Additionally, fishing gear can be size-selective and disproportionately sample old, large fish and young, quick-growing fish (Schnute and Fournier, 1980) resulting in biased estimates of size at age if unaccounted for (Goodyear, 1995). Thus, estimating time-varying or even time-invariant somatic growth can be difficult and subject to bias.
As such, most stock assessments in the U.S. account for size-at-age by either estimating a time-invariant growth relationship or incorporating empirical weight-at-age (EWAA) information. Two exceptions are a previous U.S./Canadian Pacific hake (Merluccius productus) assessment, which estimated von Bertalanffy growth parameters, length at age infinity (L∞) and growth rate (K), in blocks (Helser et al., 2006), and the California and Oregon chilipepper rockfish (Sebastes goodei) assessment, which estimates K in five blocks corresponding to periods in the Pacific Decadal Oscillation (Field, 2007). The chilipepper rockfish assessment explicitly mentions that additional data and conditional age-at-length data would improve methods of estimating time-varying growth (Field, 2007). Conditional age-at-length data are paired age-and-length observations that improve estimates of variation around mean length at age (He et al., 2016, Monnahan et al., 2016), but only a few stock age-structured assessment frameworks such as Stock Synthesis (SS, Methot and Wetzel, 2013) and CASAL (Bull et al., 2005) can incorporate these data, although virtual population analysis and cohort analysis use EWAA as the standard approach as well.
Within integrated, statistical age-structured stock assessments, somatic growth and length–weight relationships facilitate the conversion of numbers at age to biomass—the unit of catches and management quantities that forms the scientific foundation for management decisions such as maximum sustainable yield (MSY). Somatic growth is often represented as a parametric function where the parameters are either estimated internally to the assessment model (e.g., Taylor and Wetzel, 2011) or fixed at values estimated externally to the assessment model (e.g., Hamel and Ono, 2011).
The EWAA approach is an alternative method of accounting for changes in size at age that foregoes growth estimation completely and directly translates numbers to biomass at age. The EWAA approach is currently used in assessments of Bering Sea walleye pollock (Theragra chalcogramma, Ianelli et al., 2014), Pacific halibut (Hippoglossus stenolepis, Stewart and Martell, 2014), and Pacific hake (Taylor et al., 2014), among others. These assessments calculate mean EWAA from paired age and weight observations for each year of fishery and survey catches external to the assessment and typically include fishery-dependent and -independent age compositions, catches, and biomass indices (Taylor et al., 2014). The EWAA approach may not be appropriate for all fisheries due to high data requirements both in terms of numbers of fish sampled, as well as complete coverage of the time-series of weight at age. For example, the recent U.S. and Canadian Pacific hake assessment used 160,000 age–weight combinations. Note that the EWAA values are assumed to be known without error and are not strictly data because there is no likelihood associated with them—rather, the EWAA values are assumed to be known conversion factors. This contrasts with observed mean weight-at-age data, which would have an associated likelihood and can be used to estimate a parametric growth curve.
The EWAA approach captures the variability in the age–weight relationship among years, without requiring parametric relationships (Helser et al., 2006). Not requiring a parametric relationship can be particularly advantageous when analysts fail to identify relationships between growth and environmental processes needed for time-varying parametric functions (e.g. Field, 2007) and given that the risk of finding spurious environmental correlations with biological data is high (Walters and Collie, 1988). Although the EWAA approach eliminates growth estimation as a source of bias, biases may be introduced due to changes in selectivity. As a result, the EWAA approach necessitates the assumption that observed weights at age are not biased due to changes in selectivity (Taylor et al., 2014). Furthermore, assessments that use the EWAA approach may be particularly sensitive to data quantities and qualities. Missing values for years, or ages within a specific year, must be interpolated, and the methods of filling values may influence biomass-based management reference points.
Here, we used simulation testing to compare the performance of an integrated age-structured stock assessment model that estimates a time-invariant parametric growth curve from age-and-length data, an assessment model that estimates parametric growth curves in time blocks, and an assessment model that foregoes estimation of a parametric growth curve and uses the EWAA approach. We compared these approaches for two life histories, across a range of data quality and quantity. The goals of this study were to (1) investigate biases that arise from assuming time-invariant somatic growth when the underlying somatic growth pattern is time-varying and (2) identify the qualities and quantities of data necessary to precisely and accurately characterize population abundances under time-invariant and time-varying somatic growth.
Section snippets
Model description
We conducted our simulation analysis using ss3sim (Anderson et al., 2014a, Anderson et al., 2014b)—an open source package for the statistical software environment R (R Core Team, 2015), which is built around SS, version 3.24o (Methot and Wetzel, 2013). Simulations in ss3sim have two components: (1) an OM which is conditioned on a real stock assessment model setup and generates the “true” (i.e., simulated) population dynamics from which data are sampled, and (2) an estimation model (EM) that
Results
The age-and-length and EWAA approaches were unbiased when growth was time-invariant, although EWAA configurations were less precise in all data scenarios (Figs. 2a–c and g–i, 3a–c and g–i). For both the hake-like and rockfish-like life histories, MARE values in EWAA configurations were more than twice as large as those in the age-and-length configurations. Additionally, the range in minimum and maximum median relative errors was higher in EWAA configurations (Figs. 2g–i and 3g–i). Precision in
Discussion
We used simulations to quantify differences in the performance of integrated statistical catch-at-age stock assessment models that estimate a time-invariant growth relationship, estimate a time-varying growth relationship, or use the EWAA approach across data situations and life histories. With time-invariant growth, the EWAA approach results in slightly more biased and less precise estimates of SSB than those from the age-and-length approach. If a stock assessment analyst has strong reasons to
Acknowledgements
This growth research was supported through the Center for the Advancement of Population Assessment Methodology (CAPAM) in La Jolla CA, USA, as part of the good practices in stock assessment modeling program. This publication is partially funded by the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement No. NA10OAR4320148, Contribution No. 2450. PTK, CCM and CCS were partially supported for this work by Sea Grant/NOAA Fisheries Population Dynamics
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