On the volume of 4-cycle trades

A 4-cycle trade of volume t corresponds to a simple graph G without isolated vertices, where the edge set can be partitioned into t 4-cycles in at least two different ways such that the two collections of 4-cycles have no 4-cycles in common. The foundation of the trade is v = \V(G)\. This paper determines for which values oft and a there exists a 4-cycle trade of volume t and foundation v.

Fitting genetic models to twin data with binary and ordered categorical responses: A comparison of structural equation modelling and Bayesian hierarchical models

We compare Bayesian methodology utilizing free-ware BUGS (Bayesian Inference Using Gibbs Sampling) with the traditional structural equation modelling approach based on another free-ware package, Mx. Dichotomous and ordinal (three category) twin data were simulated according to different additive genetic and common environment models for phenotypic variation. Practical issues are discussed in using Gibbs sampling as implemented by BUGS to fit subject-specific Bayesian generalized linear models, where the components of variation may be estimated directly. The simulation study (based on 2000 twin pairs) indicated that there is a consistent advantage in using the Bayesian method to detect a correct model under certain specifications of additive genetics and common environmental effects. For binary data, both methods had difficulty in detecting the correct model when the additive genetic effect was low (between 10 and 20%) or of moderate range (between 20 and 40%). Furthermore, neither method could adequately detect a correct model that included a modest common environmental effect (20%) even when the additive genetic effect was large (50%). Power was significantly improved with ordinal data for most scenarios, except for the case of low heritability under a true ACE model. We illustrate and compare both methods using data from 1239 twin pairs over the age of 50 years, who were registered with the Australian National Health and Medical Research Council Twin Registry (ATR) and presented symptoms associated with osteoarthritis occurring in joints of the hand.

Quantitative trait loci for agronomic and seed quality traits in an F-2 and F-4 : 6 soybean population

Molecular breeding is becoming more practical as better technology emerges. The use of molecular markers in plant breeding for indirect selection of important traits can favorably impact breeding efficiency. The purpose of this research is to identify quantitative trait loci (QTL) on molecular linkage groups (MLG) which are associated with seed protein concentration, seed oil concentration, seed size, plant height, lodging, and maturity, in a population from a cross between the soybean cultivars 'Essex' and 'Williams.' DNA was extracted from F-2 generation soybean leaves and amplified via polymerase chain reaction (PCR) using simple sequence repeat (SSR) markers. Markers that were polymorphic between the parents were analyzed against phenotypic trait data from the F-2 and F-4:6 generation. For the F-2 population, significant additive QTL were Satt540 (MLG M, maturity, r(2)=0.11; height, r(2)=0.04, seed size, r(2)=0.061, Satt373 (MLG L, seed size, r(2)=0.04; height, r(2)=0.14), Satt50 (MLG A1, maturity r(2)=0.07), Satt14 (MLG D2, oil, r(2)=0.05), and Satt251 (protein r(2)=0.03, oil, r(2)=0.04). Significant dominant QTL for the F-2 population were Satt540 (MLG M, height, r(2)=0.04; seed size, r(2)=0.06) and Satt14 (MLG D2, oil, r(2)=0.05). In the F-4:6 generation significant additive QTL were Satt239 (MLG I, height, r(2)=0.02 at Knoxville, TN and r(2)=0.03 at Springfield, TN), Satt14 (MLG D2, seed size, r(2)=0.14 at Knoxville, TN), Satt373 (MLG L, protein, r(2)=0.04 at Knoxville, TN) and Satt251 (MLG B I, lodging r(2)=0.04 at Springfield, TN). Averaged over both environments in the F-4:6 generation, significant additive QTL were identified as Satt251 (MLG B 1, protein, r(2)=0.03), and Satt239 (MLG 1, height, r(2)=0.03). The results found in this study indicate that selections based solely on these QTL would produce limited gains (based on low r(2) values). Few QTL were detected to be stable across environments. Further research to identify stable QTL over environments is needed to make marker-assisted approaches more widely adopted by soybean breeders.

Bridging the gap between different statistical approaches: An integrated framework for modelling

This paper proposes a template for modelling complex datasets that integrates traditional statistical modelling approaches with more recent advances in statistics and modelling through an exploratory framework. Our approach builds on the well-known and long standing traditional idea of 'good practice in statistics' by establishing a comprehensive framework for modelling that focuses on exploration, prediction, interpretation and reliability assessment, a relatively new idea that allows individual assessment of predictions. The integrated framework we present comprises two stages. The first involves the use of exploratory methods to help visually understand the data and identify a parsimonious set of explanatory variables. The second encompasses a two step modelling process, where the use of non-parametric methods such as decision trees and generalized additive models are promoted to identify important variables and their modelling relationship with the response before a final predictive model is considered. We focus on fitting the predictive model using parametric, non-parametric and Bayesian approaches. This paper is motivated by a medical problem where interest focuses on developing a risk stratification system for morbidity of 1,710 cardiac patients given a suite of demographic, clinical and preoperative variables. Although the methods we use are applied specifically to this case study, these methods can be applied across any field, irrespective of the type of response.

The spectrum of minimal defining sets of some Steiner systems

For a design D, define spec(D) = {\M\ \ M is a minimal defining set of D} to be the spectrum of minimal defining sets of D. In this note we give bounds on the size of an element in spec(D) when D is a Steiner system. We also show that the spectrum of minimal defining sets of the Steiner triple system given by the points and lines of PG(3,2) equals {16,17,18,19,20,21,22}, and point out some open questions concerning the Steiner triple systems associated with PG(n, 2) in general. (C) 2002 Elsevier Science B.V. All rights reserved.

More on exact bicoverings of 12 points

The minimum number of incomplete blocks required to cover, exactly lambda times, all t-element subsets from a set V of cardinality v (v > t) is denoted by y(lambda, t; v). The value of g(2, 2; v) is known for v = 3, 4,..., 11. It was previously known that 14 less than or equal to g(2, 2; 12) less than or equal to 16. We prove that g(2, 2; 12) greater than or equal to 15.

Factorizations of the complete graph into C-5-factors and 1-factors

In this paper, we show that K-10n can be factored into alpha C-5-factors and beta 1-factors for all non-negative integers alpha and beta satisfying 2alpha + beta = 10(n) - 1.

Skolem-type difference sets for cycle systems

Cyclic m-cycle systems of order v are constructed for all m greater than or equal to 3, and all v = 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle system of K-v - F for all m greater than or equal to 3, and all v = 2(mod 2m).

Special twin environments, genetic influences and their effects on the handedness of twins and their siblings

It has been suggested that twinning may influence handedness through the effects of birth order, intra-uterine crowding and mirror imaging. The influence of these effects on handedness (for writing and throwing) was examined in 3657 Monozygotic (MZ) and 3762 Dizygotic (DZ) twin pairs (born 1893-1992). Maximum likelihood analyses revealed no effects of birth order on the incidence of left-handedness. Twins were no more likely to be left-handed than their singleton siblings (n = 1757), and there were no differences between the DZ co-twin and sibling-twin covariances, suggesting that neither intra-uterine crowding nor the experience of being a twin affects handedness. There was no evidence of mirror imaging; the co-twin correlations of monochorionic and dichorionic MZ twins did not differ. Univariate genetic analyses revealed common environmental factors to be the most parsimonious explanation of familial aggregation for the writing-hand measure, while additive genetic influences provided a better interpretation of the throwing hand data.

The role of melanocortin-1 receptor polymorphism in skin cancer risk phenotypes

We have examined melanocortin-1 receptor (MC1R) variant allele frequencies in the general population and in a collection of adolescent dizygotic and monozygotic twins to determine statistical associations of pigmentation phenotypes with increased skin cancer risk. This included hair and skin color, freckling, mole count and sun exposed skin reflectance. Nine variants were studied and designated as either strong R (OR = 63; 95% CI 32-140) or weak r (OR = 5; 95% CI 3-11) red hair alleles. Penetrance of each MC1R variant allele was consistent with an allelic model where effects were multiplicative for red hair but additive for skin reflectance. To assess the interaction of the brown eye color gene BEY2/OCA2 on the phenotypic effects of variant MC1R alleles we imputed OCA2 genotype in the twin collection. A modifying effect of OCA2 on MC1R variant alleles was seen on constitutive skin color, freckling and mole count. In order to study the individual effects of these variants on pigmentation phenotype we have established a series of human primary melanocyte strains genotyped for the MC1R receptor. These include strains which are MC1R wild-type consensus, variant heterozygotes, and homozygotes for strong R alleles Arg151Cys and Arg160Trp. Ultrastructural analysis demonstrated that only consensus strains contained stage III and IV melanosomes in their terminal dendrites whereas Arg151Cys and Arg160Trp homozygous strains contained only immature stage I and II melanosomes. Such genetic association studies combined with the functional analysis of MC1R variant alleles in melanocytic cells should provide a link in understanding the association between pigmentary phototypes and skin cancer risk.

Genetic and Environmental Influences on Migraine: A Twin Study Across Six Countries

Migraine is a common neurovascular brain disorder that is manifested in recurrent episodes of disabling headache. The aim of the present study was to compare the prevalence and heritability of migraine across six of the countries that participate in GenomEutwin project including a total number of 29,717 twin pairs. Migraine was assessed by questionnaires that differed between most countries. It was most prevalent in Danish and Dutch females (32% and 34%, respectively), whereas the lowest prevalence was found in the younger and older Finnish cohorts (13% and 10%, respectively). The estimated genetic variance (heritability) was significant and the same between sexes in all countries. Heritability ranged from 34% to 57%, with lowest estimates in Australia, and highest estimates in the older cohort of Finland, the Netherlands, and Denmark. There was some indication that part of the genetic variance was non-additive, but this was significant in Sweden only. In addition to genetic factors, environmental effects that are non-shared between members of a twin pair contributed to the liability of migraine. After migraine definitions are homogenized among the participating countries, the GenomEUtwin project will provide a powerful resource to identify the genes involved in migraine.

Sex differences in heritability of BMI: A comparative study of results from twin studies in eight countries

Body mass index (BMI), a simple anthropometric measure, is the most frequently used measure of adiposity and has been instrumental in documenting the worldwide increase in the prevalence of obesity witnessed during the last decades. Although this increase in overweight and obesity is thought to be mainly due to environmental changes, i.e., sedentary lifestyles and high caloric diets, consistent evidence from twin studies demonstrates high heritability and the importance of genetic differences for normal variation in BMI. We analysed self-reported data on BMI from approximately 37,000 complete twin pairs (including opposite sex pairs) aged 20-29 and 30-39 from eight different twin registries participating in the GenomEUtwin project. Quantitative genetic analyses were conducted and sex differences were explored. Variation in BMI was greater for women than for men, and in both sexes was primarily explained by additive genetic variance in all countries. Sex differences in the variance components were consistently significant. Results from analyses of opposite sex pairs also showed evidence of sex-specific genetic effects suggesting there may be some differences between men and women in the genetic factors that influence variation in BMI. These results encourage the continued search for genes of importance to the body composition and the development of obesity. Furthermore, they suggest that strategies to identify predisposing genes may benefit from taking into account potential sex specific effects.

Heritability of adult body height: A comparative study of twin cohorts in eight countries

A major component of variation in body height is due to genetic differences, but environmental factors have a substantial contributory effect. In this study we aimed to analyse whether the genetic architecture of body height varies between affluent western societies. We analysed twin data from eight countries comprising 30,111 complete twin pairs by using the univariate genetic model of the Mx statistical package. Body height and zygosity were self-reported in seven populations and measured directly in one population. We found that there was substantial variation in mean body height between countries; body height was least in Italy (177 cm in men and 163 cm in women) and greatest in the Netherlands (184 cm and 171 cm, respectively). In men there was no corresponding variation in heritability of body height, heritability estimates ranging from 0.87 to 0.93 in populations under an additive genes/unique environment (AE) model. Among women the heritability estimates were generally lower than among men with greater variation between countries, ranging from 0.68 to 0.84 when an additive genes/shared environment/unique environment (ACE) model was used. In four populations where an AE model fit equally well or better, heritability ranged from 0.89 to 0.93. This difference between the sexes was mainly due to the effect of the shared environmental component of variance, which appears to be more important among women than among men in our study populations. Our results indicate that, in general, there are only minor differences in the genetic architecture of height between affluent Caucasian populations, especially among men.