Combining Independent, Weighted P-Values: Achieving Computational Stability by a Systematic Expansion with Controllable Accuracy

Given the expanding availability of scientific data and tools to analyze them, combining different assessments of the same piece of information has become increasingly important for social, biological, and even physical sciences. This task demands, to begin with, a method-independent standard, such as the -value, that can be used to assess the reliability of a piece of information. Good's formula and Fisher's method combine independent -values with respectively unequal and equal weights. Both approaches may be regarded as limiting instances of a general case of combining -values from groups; -values within each group are weighted equally, while weight varies by group. When some of the weights become nearly degenerate, as cautioned by Good, numeric instability occurs in computation of the combined -values. We deal explicitly with this difficulty by deriving a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics. We illustrate the utility of this systematic approach with a few examples. In addition, we also provide here an alternative derivation for the probability distribution function of the general case and show how the analytic formula obtained reduces to both Good's and Fisher's methods as special cases. A C++ program, which computes the combined -values with equal numerical stability regardless of whether weights are (nearly) degenerate or not, is available for download at our group website http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads/CoinedPValues.html

Elevational Gradients in Bird Diversity in the Eastern Himalaya: An Evaluation of Distribution Patterns and Their Underlying Mechanisms

BACKGROUND: Understanding diversity patterns and the mechanisms underlying those patterns along elevational gradients is critically important for conservation efforts in montane ecosystems, especially those that are biodiversity hotspots. Despite recent advances, consensus on the underlying causes, or even the relative influence of a suite of factors on elevational diversity patterns has remained elusive. METHODS AND PRINCIPAL FINDINGS: We examined patterns of species richness, density and range size distribution of birds, and the suite of biotic and abiotic factors (primary productivity, habitat variables, climatic factors and geometric constraints) that governs diversity along a 4500-m elevational gradient in the Eastern Himalayan region, a biodiversity hotspot within the world's tallest mountains. We used point count methods for sampling birds and quadrats for estimating vegetation at 22 sites along the elevational gradient. We found that species richness increased to approximately 2000 m, then declined. We found no evidence that geometric constraints influenced this pattern, whereas actual evapotranspiration (a surrogate for primary productivity) and various habitat variables (plant species richness, shrub density and basal area of trees) accounted for most of the variation in bird species richness. We also observed that ranges of most bird species were narrow along the elevation gradient. We find little evidence to support Rapoport's rule for the birds of Sikkim region of the Himalaya. CONCLUSIONS AND SIGNIFICANCE: This study in the Eastern Himalaya indicates that species richness of birds is highest at intermediate elevations along one of the most extensive elevational gradients ever examined. Additionally, primary productivity and factors associated with habitat accounted for most of the variation in avian species richness. The diversity peak at intermediate elevations and the narrow elevational ranges of most species suggest important conservation implications: not only should mid-elevation areas be conserved, but the entire gradient requires equal conservation attention