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.