Best practices for modeling time-varying growth in state-space stock assessments

Giancarlo M. Correa\(^{1,2}\), Cole Monnahan\(^3\), Jane Sullivan\(^4\), James T. Thorson\(^3\), Andre E. Punt\(^2\)

\(^1\)AZTI, Marine Research, Sukarrieta, Spain. \(^2\)School of Aquatic and Fishery Sciences, University of Washington, Seattle, WA, USA. \(^3\)Alaska Fisheries Science Center, NOAA, Seattle, WA, USA. \(^4\)Alaska Fisheries Science Center, NOAA, Juneau, AK, USA

Somatic growth

Somatic growth

  • Definition: the increase in size or weight of a fish.
  • In stock assessments: at the stock level.
  • Growth and recruitment are contributors to the stock biomass.
  • Time-variable:
    • By year, cohort, or age.
    • Affected by internal and external factors.

Somatic growth in assessment models

  • Not explicitly modeled in surplus production models.
  • Accounted for in VPA and SCAA using empirical weight-at-age.
  • Explicitly modeled in some integrated models (commonly assumed to be time-invariant).
  • Ignoring temporal variability may lead to biased results.

State-space models

State-space models

Process equation: \(E[z_t \mid z_{t-1}] = h(z_{t-1},\theta)\)

Observation equation: \(E[y_t \mid z_t]= g(z_t,\theta)\)

\(\theta\): vector of all unknown model parameters (fixed effects).

Auger-Méthé et al. (2021)

The Woods Hole Assessment Model (WHAM, Stock and Miller et al. 2021)

  • Fully state-space age-structured model
  • Data: catch, indices of abundance, age compositions, empirical weight-at-age, environmental covariates (Ecov)
  • Separability of total catch and proportions-at-age, as well as annual F and selectivity
  • Random effects in selectivity, M, NAA, Q, Ecov
  • Written in TMB and R (user friendly!) ( see R package ).

Growth in state-space models

Growth in state-space models

  • Goal: implement a flexible framework to model population mean length or weight at age in WHAM.
  • Data:
    • Length compositions
    • Conditional age at length (CAAL)
    • Ageing error
    • Observed mean weight at age
  • Parameters:
    • Population mean length at age (LAA)
    • Length-weight (LW) relationship
    • Population mean weight at age (WAA)

Growth modeling overview

Correa et al. (2023)

LAA parametric approach

  1. von Bertalanffy (\(k\), \(L_{\infty}\), \(L_{\tilde{a}}\))
  2. Richards (\(k\), \(L_{\infty}\), \(L_{\tilde{a}}\), \(\gamma\))

Length-at-age variability incorporated through two parameters (\(SD_\tilde{a}\) and \(SD_A\)) and a transition matrix (\(\varphi_{y,l,a}\)).

Predicting random effects:

\[log(G_{t}) = \mu_{G} + \delta_{t}\]

\(G\) is a growth parameter, \(t\) represents year or cohort, \(\delta\) are random effects (\(iid\) or \(AR1\) structure).

LAA nonparametric approach

Population mean length at age (\(L_{a}\)) assumed to be fixed effects. \(SD_\tilde{a}\) and \(SD_A\) still needed.

Time variability can be modeled by predicting random effects:

\[log(\hat{L}_{y,a}) = \mu_{L_{a}} + \delta_{y,a}\]

\(\delta_{y,a}\) can be \(iid\), \(2dAR1\), or \(3dGMRF\).

LAA semiparametric approach

  1. Use parametric approach (without random effects) to calculate \(L_{y,a}\).
  1. Predict random effects on \(L_{y,a}\):

\[log(\hat{L}_{y,a}) = \mu_{L_{y,a}} + \delta_{y,a}\]

\(\delta_{y,a}\) can be \(iid\), \(2dAR1\), or \(3dGMRF\).

WAA parametric

Use the LW relationship:

\[w_l = \Omega_1 l^{\Omega_2}\]

Random effects on \(\Omega_1\) and \(\Omega_2\) can also be predicted.

Use transition matrix to calculate population mean weight at age:

\[\hat{w}_{y,a} = \sum_l \varphi_{y,l,a}w_l\]

\(\hat{w}_{y,a}\) can also be fitted to \(\bar{w}_{y,a}\) (observed mean weight at age)

WAA nonparametric

Like the LAA nonparametric. Population mean length at age (\(w_{a}\)) assumed to be fixed effects.

Time variability can be modeled by predicting random effects:

\[log(\hat{w}_{y,a}) = \mu_{w_{a}} + \delta_{y,a}\]

\(\delta_{y,a}\) can be \(iid\), \(2dAR1\), or \(3dGMRF\).

More new features


Originally, only selectivity-at-age functions were available.

New functions added:

  • Age double normal (6 parameters).

  • Length logistic (2 parameters).

  • Length decreasing logistic (2 parameters).

  • Length double normal (6 parameters).

More new features

Environmental covariates

New growth-related parameters can be linked to an environmental covariate. For example:

\[P_t = P exp(\beta_1 X_t)\]

\(P\) is the base state (parameter) value. Other links are also available (polynomials). Lags can be modeled.


Methods applied to three stocks in Alaska:

  1. Gulf of Alaska Walleye pollock: age data, observed mean weight at age, WAA nonparametric.
  2. Gulf of Alaska Pacific cod: length and CAAL data, LAA parametric.
  3. Eastern Bering Sea Pacific cod: length data, LAA parametric with time-varying \(L_\tilde{a}\).

See Correa et al. (2023) Modeling time-varying growth in state-space stock assessments. ICES Journal of Marine Sciences.

Good practices

Strategies that have been shown through research and evaluation to be effective and/or efficient, and to reliably lead to a desired result.

Simulation experiment

Goal: provide guidelines for growth modeling in state-space assessment models under diverse scenarios.

  • Data type: age compositions vs length compositions vs CAAL (random vs stratified sampling).
  • Data source: fishery vs survey.
  • Data quality: data rich vs data poor.
  • Modeling approach: parametric vs nonparametric vs semiparametric vs Ecov.
  • Time-varying parameter: changes in \(k\), \(L_{\infty}\) or \(L_{\tilde{a}}\).

Simulation experiment


  1. Operating model: simulates the true population dynamics. Changes in growth by varying \(k\), \(L_{\infty}\) or \(L_{\tilde{a}}\).

  2. Sample data from operating model.

  3. Estimation model uses sampled data with assumptions on the population dynamics.

Preliminary results


  • Expansion of the applicability of state-space assessment models.
  • Implementation of a flexible framework to model time-varying growth in state-space assessment models.
  • Recommendations to model time-varying growth under diverse scenarios.


Tim Miller, Brian Stock, Jim Ianelli, Steve Barbeaux, Peter Hulson


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