Models for hypothesis testing
To test the degree to which Lepidoptera use range shifts and/or phenology shifts as a response to climate change, the effect of the responses and their combinations on population trends, and the role that life-history traits may play for identifying species able to capitalize on the responses, we use the PGLS-models described above. For the purpose of statistical analyses, these categorical variables were converted into dummy variables, and the trait body size was square-root transformed.
First, to measure average response across species, we conducted intercept-only PGLS-models on both continuous responses and the population trend. For this purpose, we used the unscaled versions of the variables to allow inference directly on the measured scale (days, kilometers). The model is defined as:
\begin{equation} Y=a+\ \varepsilon\nonumber \\ \end{equation}
where Y stands for the continuous dependent variable, a is the intercept (average response), and ɛ are the residuals with covariance matrix C , which is optimized based on the phylogeny.
Second, to test our main hypotheses (Fig. 1), we fitted five separate models. Hypothesis 1 suggests that either phenology advance or a northwards shift of NRB would be mirrored in positive population trends (Fig. 1). Thus, we tested the effect of direction of shifts in a) phenology and b) NRB on population trend. Hypothesis 2 postulates that an advance in phenology would increase the probability that species can shift their NRB northwards. Thus, we tested the effect of c) the direction of shift in phenology on shift in NRB. To test the effect of different combinations of the responses on population trends, and which hypothesis provides a more viable strategy for species (as opposed to which strategy is the more common), we tested the effect of the d) combined responses and e) hypothesis-wise responses on population trend.
Finally, to test the effect of life-history traits on the responses and population trend, we fitted three PGLS-models, one on each continuous and scaled response and population trend as response variables, with all four life-history traits as explanatory variables. We also applied PGLS-models to test for a potential difference in the responses and population trends of the major taxonomic groups (moths and butterflies).
Models a)-e), and models on the effect of traits and taxonomic groups are structured as follows:
\begin{equation} Y=a+\ Xb+\ \varepsilon\nonumber \\ \end{equation}
where Y stands for the continuous dependent variable, a is the intercept (average response), X is a 289*k-1dimensional design matrix indicating the independent factorial variables, b are the effects of the independent variables, and ɛ are the residuals with covariance matrix C , which is optimized based on phylogenetic signal (Symonds & Blomberg 2014).
The assumptions of the models were verified by visually observing plots of the residuals against each response and explanatory variable validation, and the distribution (histogram) and QQ plots of the residuals. All patterns indicated normally distributed residuals with no signs of heteroscedasticity nor extreme outliers. All data management and analyses were conducted in R studio (R version 3.5.3; R Core Team 2019).