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A spatiotemporal Richards–Schnute growth model and its estimation when data are collected through length-stratified sampling

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Abstract

We propose a spatiotemporal generalized von Bertalanffy (vonB) growth model that also includes between-individual (BI) variation and male/female correlation. The generalized vonB model includes the effect of maturation on growth. The model and the methodology are applied to a long time-series of survey observations of age and length for American plaice on the Grand Bank off the northeast coast of Canada. The bias in age-length data due to size selectivity of the survey gear is accounted for. The survey design includes length-stratified age sampling which is a type of response selective sampling design for growth model estimation. We propose and implement a conditional empirical proportion likelihood approach for these data. Neglecting this sampling scheme can lead to seriously biased estimation results. We found that a 6-parameter growth model is necessary for capturing the biphasic growth patterns of the American plaice on the Grand Bank, and the survey gear selectivity and BI variation are important for a good model fit. We proposed an empirically optimal BI variation model for this data. Our estimation results indicate that there are substantial differences in size-at-age for male and female American plaice, and this changes over time and between regions.

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Acknowledgements

We thank all of the people involved in the collection and processing of these data. Research funding to NZ and NC was provided by the Ocean Frontier Institute, through an award from the Canada First Research Excellence Fund. Research funding to NC was also provided by the Ocean Choice International Industry Research Chair program at the Fisheries and Marine Institute of Memorial University of Newfoundland. Many thanks for the comments from the two anonymous reviewers that greatly improved this manuscript.

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Correspondence to Nan Zheng.

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Appendices

Appendices

1.1 Appendix A: Stationary results

If the AR(1) models in (6) are stationary, the marginal distributions of \(\log (L_{\infty ,c,d,g})\) and \(\log (K_{c,d,g})\) are

$$\begin{aligned} \log (L_{\infty ,c,d,g})&\sim \mathrm {N}(\mu _{\infty ,d,g},\omega _{\infty ,d,g}^2), \nonumber \\ \log (K_{c,d,g})&\sim \mathrm {N}(\mu _{k,d,g},\omega _{k,d,g}^2) \end{aligned}$$
(29)

with \(\omega _{\infty ,d,g}^2 = \delta _{\infty ,d,g}^2/(1-\varphi _{\infty ,d,g}^2)\) and \(\omega _{k,d,g}^2 = \delta _{k,d,g}^2/(1-\varphi _{k,d,g}^2)\), and the distributions of \(\log (L_{\infty ,1,d,g})\) and \(\log (K_{1,d,g})\) are given by (29).

Based on the stationary assumption, the marginal sex and division correlations are

$$\begin{aligned} \mathrm {corr}(\log (L_{\infty ,c,d,g}),\log (L_{\infty ,c,d',g'}))&= \dfrac{\sqrt{1-\varphi _{\infty ,d,g}^2}\,\sqrt{1-\varphi _{\infty ,d',g'}^2}}{1-\varphi _{\infty ,d,g}\,\varphi _{\infty ,d',g'}}\,\rho _{\infty ,dg,d'g'},\nonumber \\ \mathrm {corr}(\log (K_{c,d,g}),\log (K_{c,d',g'}))&= \dfrac{\sqrt{1-\varphi _{k,d,g}^2}\,\sqrt{1-\varphi _{k,d',g'}^2}}{1-\varphi _{k,d,g}\,\varphi _{k,d',g'}}\,\rho _{k,dg,d'g'}. \end{aligned}$$
(30)

1.2 Appendix B: Conditional distribution of a captured fish

The conditional distribution of a captured fish (i.e. \(C=1\)) is

$$\begin{aligned} f(y\,|\, C=1, a; \pmb {\theta })&= \dfrac{\mathrm {Pr}\{ Y=y, C=1 \,|\, A=a;\pmb {\theta } \}}{ \mathrm {Pr}\{ C=1 \,|\, A=a;\pmb {\theta } \} }\\&= \dfrac{\mathrm {Pr}\{ C=1 \,|\, Y=y, A=a;\pmb {\theta } \}\, \mathrm {Pr}\{ Y=y \,|\, A=a;\pmb {\theta } \} }{ \mathrm {Pr}\{ C=1 \,|\, A=a;\pmb {\theta } \} }\\&= \dfrac{s(y)\, f( y \,|\, a;\pmb {\theta } ) }{ \mathrm {Pr}\{ C=1 \,|\, A=a;\pmb {\theta } \} } \end{aligned}$$

In the last step we applied \(\mathrm {Pr}\{ C=1 \,|\, Y=y, A=a;\pmb {\theta } \} = \mathrm {Pr}\{ C=1 \,|\, Y= y \}\); that is, the probability of capture does not depend on age given the size of a fish.

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Zheng, N., Cadigan, N. & Morgan, M.J. A spatiotemporal Richards–Schnute growth model and its estimation when data are collected through length-stratified sampling. Environ Ecol Stat 27, 415–446 (2020). https://doi.org/10.1007/s10651-020-00450-8

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