Fraser River sockeye salmon: data synthesis and cumulative impacts

Multiple regression can be used to determine the relative importance of each covariate forexplaining the variability in sockeye productivity. Non-linear relationships between covariatesand sockeye productivity can be explored. Covariates that are hypothesized to have an additivecumulative impact on sockeye productivity (i.e., each factor on its own may have an insignificantbiological impact but when encountered together the sum of the effects may be biologicallyimportant) can be analyzed in groups rather than one at a time. Regression can also be used totest hypothesized interactions between covariates (i.e., multiplicative cumulative impacts).Multiple regression is valuable tool for addressing the primary objective of this analysis (i.e.,understanding the cumulative and relative impact of all of the stressors).Regression analysis is used to understand how different variables relate to one another. Typicallythere is one response variable (i.e., dependent variable) of interest and one or more predictorvariables (i.e., independent variables or covariates). In this case the dependent variable is anannual stock specific index of total productivity (ln (R/S), the natural logarithm ofrecruits/spawner 6 ). The independent variables are all factors identified as likely to be importantby each of the other Cohen Commission Technical Reports (e.g., sea surface temperature).Regression analysis entails specifying a mathematical model that describes the functional formof the relationship between the covariates and the response variable and using the observed datato estimate the parameters in the model. The model parameters provide information on thedirection and strength of the relationship between the covariates and the response variable.Many different models are possible. For example, models may include different covariates,linear and non-linear covariates, and/or interactions among different covariates. As long as thereare sufficient data, parameters for any model can be estimated, but this does not mean that themodel is sensible. Not surprisingly there is a vast amount of literature dedicated to the subject ofmodel selection and comparison. We use the Burnham and Anderson (1998) hypothesis-drivenapproach to model selection and inference. In hypothesis-driven analyses, the only factors thatwould be allowed to enter the analyses would be those that are connected to a logical, and in thiscase, biologically justified hypothesis. This reduces the potential that some variables will emergeas significant simply by chance and not as a result of any underlying mechanism, which is quitelikely to happen in a project where there are large numbers of covariates and hence potentialmodels. Standard practice is to select multiple feasible candidate models, fit each model (i.e.,estimate the parameters), and then compare the performance of each model. There are manyapproaches for comparing model performance; we used the small sample size corrected Akaike’sinformation criterion (AIC c ) (Burnham and Anderson, 1998).6 Using the natural logarithm of (R/S) transforms the Ricker spawner-recruit model into a linear form which makesit easier to apply regression analysis.25