Statistical models for point-counting data

Point-counting data are a mainstay of petrography, micropalaeontology and palynology. Conventional statistical analysis of such data is fraught with problems. Commonly used statistics such as the arithmetic mean and standard deviation may produce nonsensical results when applied to point-counting data. This paper makes the case that point-counts represent a distinct class of data that requires different treatment. Point-counts are affected by a combination of (1) true compositional variability and (2) multinomial counting uncertainties. The relative magnitude of these two sources of dispersion can be assessed by a chi-square statistic and test. For datasets that pass the chi-square test for homogeneity, the 'pooled' composition is shown to represent the optimal estimate for the underlying population. It is obtained by simply adding together the counts of all samples and normalising the resulting values to unity. However, more often than not, point-counting datasets fail the chi-square test. The overdispersion of such datasets can be captured by a random effects model that combines a logistic normal population with the usual multinomial counting uncertainties. This gives rise to the concept of a 'central' composition as a more appropriate way to average overdispersed data. Two- or three-component datasets can be displayed on radial plots and ternary diagrams, respectively. Higher dimensional datasets may be visualised and interpreted by Correspondence Analysis (CA). This is a multivariate ordination technique that is similar in purpose to Principal Component Analysis (PCA). CA and PCA are both shown to be special cases of Multidimensional Scaling (MDS). Generalising this insight to multiple datasets allows point-counting data to be combined with other data types such as chemical compositions by means of 3-way MDS. All the techniques introduced in this paper have been implemented in the provenance R-package, which is available from http://provenance.london-geochron.com.