Shift in phenology
We analyzed phenological shift with the following linear mixed effects model:
\(\frac{Y_{i}=a+X_{i}b+\ Zu+\ \varepsilon_{i}}{\sqrt{w_{i}}}\)
Where Y is the day of year that species i was observed,a is the intercept (average day of year for observing speciesi ), X is the year when the observation occurred (continuous, centered variable), b is the effect of Year on day of year(Y ), Z is a design matrix for the site-level random effects u , and ɛ are the residuals weighted by the abundance of species i on the observation day. The model was fitted separately for each species, and resulted in a slope coefficient describing the mean annual shift in adult flight timing over the study period.