Ecological Archives E096-132-A2

Heather M. Cayton, Nick L. Haddad, Kevin Gross, Sarah E. Diamond, and Leslie Ries. 2015. Do growing degree days predict phenology across butterfly species? Ecology 96:1473–1479. http://dx.doi.org/10.1890/15-0131.1

Appendix B. Species trait analysis to determine the predictability of growing degree days.

As a diagnostic on the degree of collinearity present among the butterfly species’ traits (Table B1), we computed the generalized variance-inflation factors for linear models of emergence and peak abundance as functions of the four traits, larval host plant class, larval host plant diversity (natural log transformed), overwintering stage, and voltinism. We also examined the generalized variance-inflation factors adjusted for degrees of freedom of the traits (Fox and Monette 1992). The variance inflation factors represent the degree of inflation in variance for the coefficient(s) of the term in comparison with the variance for orthogonal data, i.e., in the case where the moderator of interest is uncorrelated with the other moderators. All generalized variance-inflation factors (adjusted for degrees of freedom) for the traits we examined were less than 5, the recommended cutoff for collinearity; see Table B2.

Table B1. Species’ traits used in analysis.

Table B2. Generalized variance inflation factors (GVIF), degrees of freedom (df), and GVIFs adjusted for the degrees of freedom via GVIF1/(2*df).

We performed model selection analysis to examine the influence of butterfly species’ traits (larval host plant class, larval host plant diversity, overwintering stage and voltinism) on the predictability of emergence and peak abundance based on GDD (Table B3). Higher values of predictability indicate GDD was a better predictor of emergence or peak abundance compared with ordinal date. The global models for emergence and peak abundance contained main effects only for each of the four butterfly species’ traits. We used AICc (AIC corrected for small sample size) to identify a subset of best-fitting models (ΔAICc < 10 sensu Grueber et al. 2011) for each of the two responses. Model fit was calculated as , and . Relative importance values for a given trait are the sum of the model weights for each model in which that particular trait occurs. Relative importance values range from 0 to 1, with values closer to 1 being more important traits influencing the predictability of emergence and peak abundance based on GDD.

Table B3. Model selection results: the best-fitting model subset (ΔAICc < 10) for predictability of emergence and peak abundance from GDD are presented.

To account for uncertainty among the best-fitting models for the predictability of emergence based on GDD, we performed model averaging using the model.avg function in the R library MuMIn (Barton 2014), and present the model averaged parameter estimates and unconditional standard errors (sensu Buckland et al. 1997; Burnham and Anderson 2002). Because a single best-fitting model was identified for peak abundance, we present the non-averaged model containing the two terms with relative importance values of 1 (Table B4).

Table B4. Model averaging results for the predictability of emergence based on GDD, and the single best-fitting model for the predictability of peak abundance based on GDD.

* Reference level set as “Univoltine”; ** Reference level set as “Larva”

Literature cited

Barton, K. 2014. MuMIn: Multi-model inference. R package version 1.10.5. http://CRAN.R-project.org/package=MuMIn

Buckland, S. T., Burnham, K. P., and Augustin, N. H. 1997. Model selection: an integral part of inference. Biometrics, 603–618.

Burnham, K. P., and Anderson, D. R. 2002. Model selection and multimodel inference: a practical information-theoretic approach. Springer.

Fox, J., and G. Monette. 1992. Generalized collinearity diagnostics. Journal of the American Statistical Association 87:178–183.

Grueber, C. E., Nakagawa, S., Laws, R. J., and Jamieson, I. G. 2011. Multimodel inference in ecology and evolution: challenges and solutions. Journal of Evolutionary Biology 24(4):699–711.

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