Directions Old and New: Palaeomagnetism and Fisher (1953) Meet Modern Statistics

Most modern articles in the palaeomagnetism literature are based on statistics developed by Fisher's 1953 paper ‘Dispersion on a sphere’, which assumes independent and identically distributed (iid) spherical data. However, palaeomagnetic sample designs are usually hierarchical, where specimens are collected within sites and the data are then combined across sites to calculate an overall mean direction for a geological formation. The specimens within sites are typically more similar than specimens between different sites, and so the iid assumptions fail. This article has three principal goals. The first is to review, contrast and compare both the statistics and geophysics literature on the topic of analysis methods for clustered data on spheres. The second is to present a new hierarchical parametric model, which avoids the unrealistic assumption of rotational symmetry in Fisher's 1953 paper ‘Dispersion on a sphere’ and may be broadly useful in the analysis of many palaeomagnetic datasets. To help develop the model, we use publicly available data as a case study collected from the Golan Heights volcanic plateau. The third goal is to explore different methods for constructing confidence regions for the overall mean direction based on clustered data. Two bootstrap confidence regions that we propose perform well and will be especially useful to geophysics practitioners.