Materials, methods, and analysis
The data analyzed here were obtained from Mitsuishi’s data (3–6) published in Zenkoku Hotaru Kenkyukai-shi (Proceedings of the Japan Association for Fireflies Research). These articles are freely available on the website of the Japan Association for Fireflies Research (http://zenhoken-std.sakura.ne.jp/). Mitsuishi (3–6) recorded his measurement data as the mean values of body length (length from anterior margin of pronotum to posterior margin of elytra), body width (width across both elytra covering mesonotum) and pronotum width at 41 sites in Nagano Prefecture.
In the present study, cluster analysis was first performed to classify the measurement data into two groups using Ward’s method with Euclidean distance. Then, allometric equations were applied to the two groups using standardized major axis regression with the smatr package in R version 4.2.2 (14). The data were log10 transformed and thereby the log-log relationship between pronotum width (x ) and body length (y ), the log10-transformed allometric equation was expressed as:
\begin{equation} \operatorname{}\mathbf{y}\mathbf{=}\operatorname{}{\mathbf{a+b}\operatorname{}\mathbf{x}}\nonumber \\ \end{equation}
where a and b are constants. The constant b is the slope of the line, also known as the allometric coefficient.
This allometric equation was fitted to each of the two groups identified by the cluster analysis. As shown in Figure 1, the two allometric lines representing the two groups did not differ significantly in slope (χ 2 = 0.077, df = 1, p = 0.78), but did differ significantly in elevation (χ 2 = 229.6, df = 1, p< 0.001). The two allometric lines were found to be parallel and share a common slope of 0.34.