Re-analysis on geometric energy

In many experimental systems, signals, for example the immunohistochemical fluorescent intensities measured in our study, often take continuous values and thus cannot be dichotomized without an artificial threshold. Although dichotomization may benefit some analyses, we would note that dichotomizing methods also neglect signal intensities. In particular, if there are a lot of cells near the threshold value, any subtle change in the threshold employed may cause an unexpected iceberg effect. Thus, in our recent paper (Makino et al. 2016), we devised two new parameters to tame this statistical variation in continuous variables: the geometric energy, Eg, and the geometric entropy, Hg. In this context, the question posed by Davies et al. about whether we would ‘‘have reached similar conclusions if neuron intensities were dichotomized into low and high expression values’’ seems to be out of focus; note that, even if the conclusions are inconsistent, this fact might merely indicate that previous methods using arbitral thresholds led to erroneous conclusions. In the case of our paper, however, the conclusions are consistent both with and without dichotomization (Fig. 1a). Specifically, we separated all neurons into immunochemically positive and negative cells at three arbitrary thresholds of mean, mean ? 2 9 SD, and mean ? 5 9 SD of their fluorescence intensities, and labelled them ‘1’ and ‘0’, respectively. Irrespective of the thresholds used, the Eg values of the real data were consistently higher than those of the corresponding surrogate data, providing robust evidence of spatial clustering in the artificially dichotomized data. Compared to the original comparison (Figure 3C in Makino et al. 2016), the difference is small. This is due simply to the use of ‘1’ for positive cells. If the positive cells are assigned a larger number, such as 10 or 100, the difference becomes larger.