Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocks based on compositional statistics is introduced. It improves and generalizes the common joins-count statistics known from map analysis in geographic information systems. Assigning phases independently to objects in RD is modelled by a single-trial multinomial random function Z(x), where the probabilities of phases add to one and are explicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistencies of the tests based on the conventional joins{count statistics and their possibly contradictory interpretations are avoided. In practical applications we assume that the probabilities of phases do not depend on the location but are identical everywhere in the domain of de nition. Thus, the model involves the sum of r independent identical multinomial distributed 1-trial random variables which is an r-trial multinomial distributed random variable. The probabilities of the distribution of the r counts can be considered as a composition in the Q-part simplex SQ. They span the so called Hardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This is a generalisation of the well-known Hardy-Weinberg law of genetics. If the assignment of phases accounts for some kind of spatial dependence, then the r-trial probabilities do not remain on H. This suggests the use of the Aitchison distance between observed probabilities to H to test dependence. Moreover, when there is a spatial uctuation of the multinomial probabilities, the observed r-trial probabilities move on H. This shift can be used as to check for these uctuations. A practical procedure and an algorithm to perform the test have been developed. Some cases applied to simulated and real data are presented. Key words: Spatial distribution of crystals in rocks, spatial distribution of phases, joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinberg manifold, Aitchison geometry

Intrinsic test of independence in contingency tables

A condition needed for testing nested hypotheses from a Bayesian viewpoint is that the prior for the alternative model concentrates mass around the small, or null, model. For testing independence in contingency tables, the intrinsic priors satisfy this requirement. Further, the degree of concentration of the priors is controlled by a discrete parameter m, the training sample size, which plays an important role in the resulting answer regardless of the sample size. In this paper we study robustness of the tests of independence in contingency tables with respect to the intrinsic priors with different degree of concentration around the null, and compare with other “robust” results by Good and Crook. Consistency of the intrinsic Bayesian tests is established. We also discuss conditioning issues and sampling schemes, and argue that conditioning should be on either one margin or the table total, but not on both margins. Examples using real are simulated data are given

Incongruence between test statistics and P values in medical papers

Given an observed test statistic and its degrees of freedom, one may compute the observed P value with most statistical packages. It is unknown to what extent test statistics and P values are congruent in published medical papers. Methods: We checked the congruence of statistical results reported in all the papers of volumes 409–412 of Nature (2001) and a random sample of 63 results from volumes 322–323 of BMJ (2001). We also tested whether the frequencies of the last digit of a sample of 610 test statistics deviated from a uniform distribution (i.e., equally probable digits).Results: 11.6% (21 of 181) and 11.1% (7 of 63) of the statistical results published in Nature and BMJ respectively during 2001 were incongruent, probably mostly due to rounding, transcription, or type-setting errors. At least one such error appeared in 38% and 25% of the papers of Nature and BMJ, respectively. In 12% of the cases, the significance level might change one or more orders of magnitude. The frequencies of the last digit of statistics deviated from the uniform distribution and suggested digit preference in rounding and reporting.Conclusions: this incongruence of test statistics and P values is another example that statistical practice is generally poor, even in the most renowned scientific journals, and that quality of papers should be more controlled and valued