Small number

Abstract: Children’s sense of numbers before formal education is thought to rely on an approximate number system based on logarithmically compressed analog magnitudes that increases in resolution throughout childhood. School-age children performing a numerical estimation task have been shown to increasingly rely on a formally appropriate, linear representation and decrease their use of an intuitive, logarithmic one. We investigated the development of numerical estimation in a younger population (3.5- to 6.5-year-olds) using 0–100 and 2 novel sets of 1–10 and 1–20 number lines. Children’s estimates shifted from logarithmic to linear in the small number range, whereas they became more accurate but increasingly logarithmic on the larger interval. Estimation accuracy was correlated with knowledge of Arabic numerals and numerical order. These results suggest that the development of numerical estimation is built on a logarithmic coding of numbers—the hallmark of the approximate number system—and is subsequently shaped by the acquisition of cultural practices with numbers.

Abstract: This paper describes the theoretical background, algorithm and validation of a recently developed novel method of ranking based on the sum of ranking differences [TrAC Trends Anal. Chem. 2010; 29: 101–109]. The ranking is intended to compare models, methods, analytical techniques, panel members, etc. and it is entirely general. First, the objects to be ranked are arranged in the rows and the variables (for example model results) in the columns of an input matrix. Then, the results of each model for each object are ranked in the order of increasing magnitude. The difference between the rank of the model results and the rank of the known, reference or standard results is then computed. (If the golden standard ranking is known the rank differences can be completed easily.) In the end, the absolute values of the differences are summed together for all models to be compared. The sum of ranking differences (SRD) arranges the models in a unique and unambiguous way. The closer the SRD value to zero (i.e. the closer the ranking to the golden standard), the better is the model. The proximity of SRD values shows similarity of the models, whereas large variation will imply dissimilarity. Generally, the average can be accepted as the golden standard in the absence of known or reference results, even if bias is also present in the model results in addition to random error. Validation of the SRD method can be carried out by using simulated random numbers for comparison (permutation test). A recursive algorithm calculates the discrete distribution for a small number of objects (n < 14), whereas the normal distribution is used as a reasonable approximation if the number of objects is large. The theoretical distribution is visualized for random numbers and can be used to identify SRD values for models that are far from being random. The ranking and validation procedures are called Sum of Ranking differences (SRD) and Comparison of Ranks by Random Numbers (CRNN), respectively. Copyright © 2010 John Wiley & Sons, Ltd.

Abstract: It is now clear that the genetic basis of adaptation does not resemble that assumed by the infinitesimal model. Instead, adaptation often involves a modest number of factors of large effect and a greater number of factors of smaller effect. After reviewing relevant experimental studies, I consider recent theoretical attempts to predict the genetic architecture of adaptation from first principles. In particular, I review the history of work on Fisher's geometric model of adaptation, including recent studies which suggest that adaptation should be characterized by exponential distributions of gene effects. I also present the results of new simulation studies that test the robustness of this finding. I explore the effects of changes in the distribution of mutational effects (absolute versus relative) as well as in the nature of the character studied (total phenotypic effect versus single characters). The results show that adaptation towards a fixed optimum is generally characterized by an exponential effects trend.The situation to which these studies point is not one of a large number of genes all with more or less equal effect. It seems, rather, that a small number of genes with large effects are responsible for most of the response, the remainder of the response being due to a larger number of loci with small effects.D. S. Falconer (1981)

Abstract: Previous work has shown that it is often essential to account for the variation in rates at different sites in phylogenetic models in order to avoid phylogenetic artifacts such as long branch attraction. In most current models, the gamma distribution is used for the rates-across-sites distributions and is implemented as an equal-probability discrete gamma. In this article, we introduce discrete distribution estimates with large numbers of equally spaced rate categories allowing us to investigate the appropriateness of the gamma model. With large numbers of rate categories, these discrete estimates are flexible enough to approximate the shape of almost any distribution. Likelihood ratio statistical tests and a nonparametric bootstrap confidence-bound estimation procedure based on the discrete estimates are presented that can be used to test the fit of a parametric family. We applied the methodology to several different protein data sets, and found that although the gamma model often provides a good parametric model for this type of data, rate estimates from an equal-probability discrete gamma model with a small number of categories will tend to underestimate the largest rates. In cases when the gamma model assumption is in doubt, rate estimates coming from the discrete rate distribution estimate with a large number of rate categories provide a robust alternative to gamma estimates. An alternative implementation of the gamma distribution is proposed that, for equal numbers of rate categories, is computationally more efficient during optimization than the standard gamma implementation and can provide more accurate estimates of site rates.