Structural Equation Models: We observed a strong association between changes in flight period and
changes in abundance (see Results). This correlation could be the result
of various causal factors. We explored two of these factors using
structural equation models (SEMs). SEMs can be used to analyze the
effect size and directionality of causal pathways among the variables
(Grace et al., 2015). Breed et al. (2013) had previously found that
species were more likely to be increasing if they were multivoltine or
if our study area (Massachusetts) was north of the center of their
range. Our a priori model had a direct effect of range type
(northern, central, or southern) on voltinism and abundance trends;
voltinism had a direct effect on flight period trends; and range type
had a direct effect on flight period (See Appendix S5). We scaled
each variable in the SEM to x̄ = 0 and sd = 1 such that each per unit
change in the response is with respect to 1 change standard deviation of
the predictor. This scaling facilitates comparisons between variables
with different units.
SEM pathways were built and evaluated using the ‘lavaan’ package in R
(Rosseel et al., 2017). After fitting our a priori model, we
explored model fit using the modindices() function to look for
correlations that were not included in our initial hypothesis (following
Grace et al., 2015). We used this exploratory tool to add additional
ecologically plausible relationships among voltinism, range type, and
trends in flight period and abundance. We also interpreted the original
and updated models to understand the causal relationships between trends
in flight period and trends in abundance.