The role of spurious correlation in the development of a komatiite alteration model

Current procedures for assessing the degree of alteration in komatiites stress the construction of variation diagrams in which ratios of molecular proportions of the oxides are the axes of reference. For example, it has been argued that unaltered komatiites related to each other by olivine fractionation will display a linear variation with a slope of 0.5 in the space defined by [SiO2/TiO2] and [(MgO+FeO)/TiO2]. Extensive metasomatism is expected to destroy such a consistent pattern. Previous workers have tended to make use of ratios that have a common denominator. It has been known for a long time that ratios formed from uncorrelated variables will be correlated (a so-called spurious correlation) if both ratios have a common denominator. The magnitude of this spurious correlation is a function of the coefficients of variation of the measured amounts of the variables. If the denominator component has a coefficient of variation that is larger than those of the numerator components, the spurious correlation will be close to unity; that is, there will be nearly a straight-line relationship. As a demonstration, a fictitious data set has been simulated so that the means and variances of SiO2, TiO2, and (MgO + FeO) match those of an observed data set but the components themselves are uncorrelated. A plot of (SiO2/TiO2) versus [(MgO + FeO)/TiO2] of these simulated data produces a distribution of points that appears every bit as convincing an illustration of the lack of significant metasomatism as does the plot of the observed data. The assessment of the strength of linear association is a test of the observed correlation against an expected value (the null value) of zero. When a spurious correlation arises as a result of the formulation of ratios with a common denominator, zero is clearly an inappropriate choice as the null. It can be argued that the spurious correlation is, in fact, a more suitable null value. An analysis of komatiites from Gorgona Island and the Barberton suite reveals that the strong linear association could have been produced by forming ratios from uncorrelated starting chemical components. Ratios without parts in common are to be preferred in the construction of petrogenetic models.