Methods

\begin{align} logit(p_{d,i}) = \alpha_1 + \beta_1d_i + \beta_2d_i^2 + \beta_3t_i + \beta_4d_i & \times t_i + \beta_5d_i^2 \times t_i + \epsilon_{\alpha,1} + \epsilon_{\beta_1,1} + \epsilon_{\beta_2,1} + \epsilon_{\beta_3,1} \tag{1} \\ \epsilon_{\alpha,1} \sim N&(0, \sigma_{\alpha,1}) \nonumber \\ \epsilon_{\beta_1,1} \sim N&(0, \sigma_{\beta_1,1}) \nonumber \\ \epsilon_{\beta_2,1} \sim N&(0, \sigma_{\beta_2,1}) \nonumber \\ \epsilon_{\beta_3,1} \sim N&(0, \sigma_{\beta_3,1}) \nonumber \end{align}

where \(\alpha_1\) is the intercept, the \(\beta\) parameters are the slopes associated with day-of-year as a main effect (\(d\)) and quadratic effect (\(d^2\)), a transportation index (\(t\); presence/absence of transportation represented by 1/0), and their interaction effects for each individual \(i\). The random effects (\(\epsilon_{\alpha,1}\), \(\epsilon_{\beta_1,1}\), \(\epsilon_{\beta_2,1}\), \(\epsilon_{\beta_3,1}\)) are associated with \(\alpha_1\), \(\beta_1\), \(\beta_2\), and \(\beta_3\), respectively.

\begin{align} logit(p_{\theta,i}) = \alpha_2 + \beta_6\theta_i + \beta_7\theta_i^2 + \beta_8t_i + \beta_9\theta_i & \times t_i + \beta_{10}\theta_i^2 \times t_i + \epsilon_{\alpha,2} + \epsilon_{\beta_1,2} + \epsilon_{\beta_2,2} + \epsilon_{\beta_3,2} \tag{2} \\ \epsilon_{\alpha,2} \sim N&(0, \sigma_{\alpha,2}) \nonumber \\ \epsilon_{\beta_1,2} \sim N&(0, \sigma_{\beta_1,2}) \nonumber \\ \epsilon_{\beta_2,2} \sim N&(0, \sigma_{\beta_2,2}) \nonumber \\ \epsilon_{\beta_3,2} \sim N&(0, \sigma_{\beta_3,2}) \nonumber \end{align}

where the parameters are analogous to those in Eq. 1, except that day-of-year (\(d\)) is replaced by river temperature (\(\theta\)). A separate model was determined for each species (spring/summer Chinook Salmon or Steelhead) and rearing type (wild or hatchery) combination.

For outmigration years with incomplete adult returns (i.e., outmigration years without adults through age 4-ocean [or 6 years total]), out-of-sample predictions were determined by taking 3,000 random draws from the posteriors that included random effects from all historical years analyzed.

We applied the brm function from the brms R package (Version 2.23.0; Bürkner 2017). We simulated 2,000 draws for each of the three randomly initiated chains, and discarded the first 1,000 draws, resulting in a total of 3,000 draws saved. We assess convergence by the traceplots, Gelman-Ruben statistic of \(\hat{R}\) < 1.05, and absence of any divergent transitions.

References

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