David L. Strayer, Michael L. Pace, Nina F. Caraco, Jonathan J. Cole, and Stuart E. G. Findlay. 2008. Hydrology and grazing jointly control a large-river food web. Ecology 89:12–18.

Appendix A. Notes on the calculation of scope.

The goal of this calculation is to estimate how much the variation in each independent variable has affected each dependent variable over the period of study (“scope”) and thereby allow direct comparisons of the effects of flow and grazing. We demonstrate the derivation and logic of our procedure for S_Q; the procedure for S_Z is analogous. First, we calculate expected value of a response variable at the flow and grazing levels that actually occurred in a given year as

The value of the response variable that would be expected if the mean value of flow had occurred in that year is

mQ + Zobsmz + Zobsmint.

Taking the difference between these two expressions as a measure of how much the deviation from mean flow in that year affected the response variable, we obtain

which can be simplified to

We repeat this calculation for each year of the study, giving us estimates of the how much the deviation in flow each year away from the mean flow affected the response variable each year. Scope (S_Q) is then simply estimated as the range in these estimates. This estimate of S_Q tells us how much the variation in flow affected a particular response variable over the period of record (i.e., given the conditions that occurred in the river over the study period).

Note that S_Q depends on variation in both zebra mussel occurrence and flow, as a result of the interaction between flow and grazing.

We note that our formulation of “scope” is similar to the expression recommended by Harrell (2001, pp. 14–15) for evaluating the effect of a unit increase in one variable when that variable interacts with another variable. Modifying Harrell’s notation to conform to ours, we estimate the effect of a one-unit increase in flow at any given value of zebra mussel grazing (Z) to be

(i.e., Eq. 2.15 of Harrell). Because we want to compare the effects of flow and grazing, we aren’t interested in the effects of a one-unit increase, but rather the effects of the full observed range of variation in flow (i.e., Q_max – Q_min). For a given value of Z, the effects of the full observed range of variation in flow would then be

But for our analysis, we’re not interested in a particular value of Z (no single value of Z would provide an adequate estimate of interactions between Q and Z), but rather the entire range of Z actually observed during our period of study. Hence the year-by-year evaluation that we adopted.

LITERATURE CITED

Harrell, F. E. 2001. Regression modeling strategies with applications to linear models, logistic regression, and survival analysis. Springer-Verlag, New York, New York, USA.