Extended Data Fig. 8: Assessing evidence for nonlinear phenology–temperature relationships.

Comparison of R² values obtained using 10-fold cross-validation of models of flowering DOY versus TMEAN_Normal and TMEAN_Anomaly obtained from (a) linear regressions assuming linear relationships between phenology and temperature or (b) generalized additive models (GAMs) accounting for potential nonlinear relationships. The shaded region in each panel represents the among-species kernel distribution of cross-validated R² values obtained using each model type (linear regression or GAM). The mean and SD of R² values each are presented as text insets in each panel. The model that generated the sensitivity estimates presented in the main text assumed linear relationships between flowering dates and TMEAN_Normal and TMEAN_Anomaly. To verify whether such an assumption was warranted for our data, we compared the predictive ability of single-species models assuming linear relationships between phenology and temperature (fitted using linear regression) and models accounting for possible nonlinear relationships (fitted using Generalized Additive Models). We reasoned that if omitted nonlinear relationships between flowering time and temperature were pervasive in our data and potentially biased our results, then models accounting for nonlinear relationships would tend to perform better than linear regressions among species in our data. We used 10-fold cross-validation to compare the out-of-sample performance (quantified through R² values) of linear regressions and GAMs. For each model type (linear regression or GAM), this procedure randomly split the observations for each species into 10 groups, each of which was omitted from a model estimated from the remaining 9 groups. The performance of each of these models was then assessed against the observations omitted in fitting the model, generating 10 out-of-sample R² values for each model type (linear or GAM) per species. We then compared the distribution of mean cross-validated R² values obtained from linear models and GAMs to assess whether nonlinear models explained additional variance.