2.4 Factors influencing local spur length
The factors affecting spur length in A. buergeriana var.buergeriana were estimated by a linear mixed model (LMM) analysis. In this analysis, average spur length of each population was used as the objective variable, and average visitor size (all visitors), average visitor size (only bumblebees), plant height, number of flowers per individual, and altitude of each population were used as explanatory variables. Here, plant height and the number of flowers per individual were used as indicators of the nutritional status of the plant, and altitude was used as a representative indicator of non-biotic environmental factors (e.g. meteorological changes). Population was added to the model as a random effect.
Before conducting the LMM analysis, the variance inflation factor (VIF) statistic was calculated to check for correlation (multicollinearity) between variables, using VIF = 0.5 as the threshold (Neter et al., 1996). For all variables, VIF was less than 0.25, confirming that no multicollinearity existed. Next, we conducted a likelihood ratio test using the parametric bootstrap method (Hoel et al., 1971) to select the effective variables. In this test, for each variable, the difference in deviance, obtained by 1000 bootstrap calculations, between the global model with all variables and a model lacking that variable was determined. No variables were removed as a result of this analysis.
For appropriate model selection, we first prepared a global model that included all of the following variables: average spur length of the population, average visitor size (all visitors) of the population, average visitor size (only bumblebees) of the population, plant height, number of flowers per individual, and altitude. Then, using the ”dredge” function of the R package ”MuMIn”, we compared the global model to simple models with fewer explanatory variables. Then, we adopted the model with the lowest Akaike Information Criterion (AIC) was adopted. We excluded the 9 ON-1000 population, which was not visited by bumblebees, from this analysis, treating it as a missing value.