Mathematical bounds on Shannon entropy given the abundance of the ith most abundant taxon

Mathematical bounds on r2 and the effect size in case-control genome-wide association studies

Case-control genome-wide association studies (GWAS) are often used to find associations between genetic variants and diseases. When case-control GWAS are conducted, researchers must make decisions regarding how many cases and how many controls to include in the study. Connections between variants and diseases are made using association statistics, including χ2. Previous work in population genetics has shown that LD statistics, including r2, are bounded by the allele frequencies in the population being studied. Since varying the case fraction changes sample allele frequencies, we use the known bounds on r2 to explore how the fraction of cases included in a study can affect statistical power to detect associations. We analyze a simple mathematical model and use simulations to study a quantity proportional to the χ2 noncentrality parameter, which is closely related to r2, under various conditions. Varying the case fraction changes the χ2 noncentrality parameter, and by extension the statistical power, with effects depending on the dominance, penetrance, and frequency of the risk allele. Our framework explains previously observed results, such as asymmetries in power to detect risk vs. protective alleles, and the fact that a balanced sample of cases and controls does not always give the best power to detect associations, particularly for highly penetrant minor risk alleles that are either dominant or recessive. We show by simulation that our results can be used as a rough guide to statistical power for association tests other than χ2 tests of independence.

Using mathematical constraints to explain narrow ranges for allele-sharing dissimilarities

Allele-sharing dissimilarity (ASD) statistics are measures of genetic differentiation for pairs of individuals or populations. Given the allele-frequency distributions of two populations-possibly the same population-the expected value of an ASD statistic is computed by evaluating the expectation of the pairwise dissimilarity between two individuals drawn at random, each from its associated allele-frequency distribution. For each of two ASD statistics, which we termD 1 andD 2 , we investigate the extent to which the expected ASD is constrained by allele frequencies in the two populations; in other words, how is the magnitude of the measure bounded as a function of the frequency of the most frequent allelic type? We first consider dissimilarity of a population with itself, obtaining bounds on expected ASD in terms of the frequency of the most frequent allelic type in the population. We then examine pairs of populations that might or might not possess the same most frequent allelic type. Across the unit interval for the frequency of the most frequent allelic type, the expected allele-sharing dissimilarity has a range that is more restricted than the [0, 1] interval. The mathematical constraints on expected ASD assist in explaining a pattern observed empirically in human populations, namely that when averaging across loci, allele-sharing dissimilarities between pairs of individuals often tend to vary only within a relatively narrow range.

Mathematical constraints on a family of biodiversity measures via connections with Rényi entropy

The Hill numbers are statistics for biodiversity measurement in ecological studies, closely related to the Rényi and Shannon entropies from information theory. Recent developments in the mathematics of diversity in the setting of population genetics have produced mathematical constraints that characterize how standard measures depend on the highest-frequency class in a discrete probability distribution. Here, we apply these constraints to diversity statistics in ecology, focusing on the Hill numbers and the Rényi and Shannon entropies. The mathematical bounds can shift perspectives on the diversities of communities, in that when upper and lower bounds on Hill numbers are evaluated in a classic butterfly example, Hill numbers that are initially larger in one community switch positions-so that associated normalized Hill numbers are instead smaller than those of the other community. The new bounds hence add to the tools available for interpreting a commonly used family of statistics for ecological data.