Fraser River sockeye salmon: data synthesis and cumulative impacts

(Equation A3.5-3) provided a better fit than the Larkin model, which allows for densitydependence among cohorts of the same stock and requires 3 extra parameters for each stock(Equation A3.5-4). Based on their findings and in order to minimize the number of parametersin the model we assumed a Ricker stock recruitment model structure for all models in ouranalysis. All models used a stock specific b term (representing the within stock densitydependence)based on the wide range of estimates observed for different stocks (Peterman andDorner, 2011). Stock is represented by the subscript i in the following equations and year isrepresented by the subscript t.ln( = + (Equation A3.5-3)Ri, t/ Si,t) a biSi,tln( R (Equation A3.5-4)i, t/ Si,t) = a + biSi,t+ b1iSi,t− 1+ b2,iSi,t−2+ b3,iSi,t−3Two types of covariate variables were included: random effects and fixed effects. Year and stockwere incorporated as random effects as we are not interested in predicting the results for a givenyear or stock but rather in understanding the variability among stocks or years. All modelsinclude year and stock as crossed random effects (i.e., each year has multiple stocks, and eachstock has multiple years so they are crossed rather than nested random effects (Bates, 2010). Asingle parameter is used to represent the variability among years, R 1,t ~N(0,σ years ), and stocks, R 2,i~N(0,σ stocks ). Models differ only in the number and composition of stressor variables. These arealways incorporated into the model as fixed effects: F 1,t,i …F N-1,t,i . The final component is theerror term ε t,i ~ N(0, σ ε ), which represents the remaining error that is unexplained by the model.⎛ Rt, i ⎞ln⎜iStiFt iNFS⎟ = β0+ β1,,+ β2 1, ,+ ... + β+⎝ t,i ⎠N − 1, t,i+ R1,t+ R2,iεt,i(Equation A3.5-5)All analyses were completed using the statistical software package, R (R Development CoreTeam, 2010). Table A3.5-2 illustrates the process we used to document the model structure foreach model set. We then used a custom-built function in R to read the data provided in TableA3.5-5 and write the necessary R code to run the analyses and summarize all results. The actualregression analysis was completed using the lme4 package in R. While restricted maximumlikelihood (REML) estimates are generally preferred to Maximum likelihood estimates (MLE),we used the MLE approach as it is recommended for the purpose of comparing among models(Bates 2010, p. 8). We then extracted the AICs and calculated the AICc (small sample sizecorrected AIC) and the AIC weights for all models within a model set, as per Burnham andAnderson (1998). These along with the estimates for all fixed effects in each model werecompiled and are reported in Appendix 4.219