Why do we differ in number sense? Evidence from a genetically sensitive investigation

Basic intellectual abilities of quantity and numerosity estimation have been detected across animal species. Such abilities are referred to as ‘number sense’. For human species, individual differences in number sense are detectable early in life, persist in later development, and relate to general intelligence. The origins of these individual differences are unknown. To address this question, we conducted the first large-scale genetically sensitive investigation of number sense, assessing numerosity discrimination abilities in 837 pairs of monozygotic and 1422 pairs of dizygotic 16-year-old twin pairs. Univariate genetic analysis of the twin data revealed that number sense is modestly heritable (32%), with individual differences being largely explained by non-shared environmental influences (68%) and no contribution from shared environmental factors. Sex-Limitation model fitting revealed no differences between males and females in the etiology of individual differences in number sense abilities. We also carried out Genome-wide Complex Trait Analysis (GCTA) that estimates the population variance explained by additive effects of DNA differences among unrelated individuals. For 1118 unrelated individuals in our sample with genotyping information on 1.7 million DNA markers, GCTA estimated zero heritability for number sense, unlike other cognitive abilities in the same twin study where the GCTA heritability estimates were about 25%. The low heritability of number sense, observed in this study, is consistent with the directional selection explanation whereby additive genetic variance for evolutionary important traits is reduced

The algebra of Bonferroni bounds: discrete tubes and extensions

Bonferroni, or inclusion-exclusion, bounds and identities have a rich history. They concern the indicator function, and hence the probability content, of the union of sets. In previous work, the authors defined a discrete tube which yields upper and lower bounds which are at least as tight as the standard bounds obtained by truncating inclusion-exclusion identities at particular depths. Here, some connections to other fields are made, based particularly on the algebra of indicator functions. These leads to the consideration of the complexity of more general Boolean statements

Improved inclusion-exclusion inequalities for simplex and orthant arrangements

Improved inclusion-exclusion inequalities for unions of sets are available wherein terms usually included in the alternating sum formula can be left out. This is the case when a key abstract tube condition, can be shown to hold. Since the abstract tube concept was introduced and refined by the authors, several examples have been identified, and key properties of abstract tubes have been described. In particular, associated with an abstract tube is an inclusion-exclusion identity which can be truncated to give an inequality that is guaranteed to be at least as sharp as the inequality obtained by truncating the classical inclusion-exclusion identity. We present an abstract tube corresponding to an orthant arrangement where the inclusion-exclusion formula terms are obtained from the incidence structure of the boundary of the union of orthants. Thus, the construction of the abstract tube is similar to a construction for Euclidean balls using a Voronoi diagram. However, the proof of the abstract tube property is a bit more subtle and involves consideration of abstract tubes for arrangements of simplicies, and an intricate geometric arguments based on their Voronoi diagrams

Grobner bases, abstract tubes, and inclusion-exclusion reliability bounds

There is a close mathematical relationship between integer grids of a particular echelon form and coherent systems in reliability in the case of states coded as integer grid points. This paper shows that such an integer representation is the link between abstract tube theory, which gives improved inclusion-exclusion bounds, and an algebraic method, Grobner bases, based on the polynomial ideal of the failure even

Construction of exact simultaneous confidence bands for a simple linear regression model

A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava &amp; Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working &amp; Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 ?? level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation. <br/

Study of atmospheric pollution scavenging. Eighteenth progress report

The analysis of aerosol samples obtained in rural east-central Illinois reveals a seasonal maximum in SO/sub 4/ during May to July and a similar pattern for NH/sub 4/. The annual median SO/sub 4/ is about 1 to 1.5 ..mu..g/m/sup 3/. In contrast to these ions, NO/sub 3/ displays highest values in the cold season. Soil-related species (Ca, K) seem to maximize in relation to farm tillage and harvesting practices. The NO/sub 3/ in recent precipitation samples over the northeast US increased between 1 and 2 times the values observed in the mid-1950's. A case study from SCORE-78 suggests that all ion concentrations analyzed from sequentially collected samples decreased from the onset of rain to a minimum corresponding to the heaviest rain rates. Four groups of elements in 10 event rain samples were identified using factor analysis. The groups include soluble and insoluble crustal elements, soluble pollutant metals and sulfate, and insoluble pollutant metals. Utilizing the factor analysis approach, the St. Louis METROMEX precipitation chemistry data showed that the SO/sub 4/ deposition patterns group consistently with those of other soluble pollutants. Additional factor analysis efforts on the St. Louis rainwater data set revealed that soluble and insoluble concentrations of a given element have different deposition patterns suggesting that scavenging and/or precipitation formation processes dictate the patterns. An approach to managing the vast data base of rain chemistry used in the above studies is described. The software also examines the data for certain aspects of quality assurance. The procedures used to analyze ambient air filter samples are discussed

Multiple comparisons of several homoscedastic multivariate populations

Multiple comparisons, Bonferroni-type inequality, Maximum of correlated Hotelling’s T 2 statistics, Multivariate central nonsingular or singular chi-square distribution,