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).