Family Size Distribution Research Articles

Assessing the use of GEE methods for analyzing binary outcomes in family studies: the Strong Heart Family Study

ABSTRACT The generalized estimating equations method (GEE) is commonly applied to analyze data obtained from family studies. GEE is well known for its robustness on misspecification of correlation structure. However, the unbalanced distribution of family sizes and complicated genetic relatedness structure within each family may challenge GEE performance. We focused our research on binary outcomes. To evaluate the performance of GEE, we conducted a series of simulations, on data generated adopting the kinship matrix (correlation structure within each family) from the Strong Heart Family Study (SHFS). We performed a fivefold cross-validation to further evaluate the GEE predictive power on data from the SHFS. A Bayesian modeling approach, with direct integration of the kinship matrix, was also included to contrast with GEE. Our simulation studies revealed that GEE performs well on a binary outcome from families having a relatively simple kinship structure. However, data with a binary outcome generated from families with complex kinship structures, especially with a large genetic variance, can challenge the performance of GEE.

Estimation of the contemporary effective population size from SNP data while accounting for mating structure.

A new method is developed to estimate the contemporary effective population size (Ne ) from linkage disequilibrium (LD) between SNPs without information on their location, which is the usual scenario in non-model species. The general theory of linkage disequilibrium is extended to include the contribution of full-sibs to the measure of LD, leading naturally to the estimation of Ne in monogamous and polygamous mating systems, as well as in multiparous species, and with non-random distributions of full-sib family size due to selection or other causes. Prediction of confidence intervals for Ne estimates was solved using a small artificial neural network trained on a dataset of over 105 simulation results. The method, implemented in a user-friendly and fast software (currentNe), is able to estimate Ne even in problematic scenarios with large population sizes or small sample sizes and provides confidence intervals that are more consistent than resampling methods.

A dearth of small members in the Haumea family revealed by OSSOS

While collisional families are common in the asteroid belt, only one is known in the Kuiper belt, linked to the dwarf planet Haumea. The characterization of Haumea's family helps to constrain its origin and, more generally, the collisional history of the Kuiper belt. However, the size distribution of the Haumea family is difficult to constrain from the known sample, which is affected by discovery biases. Here, we use the Outer Solar System Origins Survey (OSSOS) Ensemble to look for Haumea family members. In this OSSOS XVI study we report the detection of three candidates with small ejection velocities relative to the family formation centre. The largest discovery, 2013 UQ15, is conclusively a Haumea family member, with a low ejection velocity and neutral surface colours. Although the OSSOS Ensemble is sensitive to Haumea family members to a limiting absolute magnitude (Hr) of 9.5 (inferred diameter of ~90 km), the smallest candidate is significantly larger, Hr = 7.9. The Haumea family members larger than approximately 20 km in diameter must be characterized by a shallow H-distribution slope in order to produce only these three large detections. This shallow size distribution suggests that the family formed in a graze-and-merge scenario, not a catastrophic collision.

Family-size variability grows with collapse rate in a birth-death-catastrophe model.

Forest-fire and avalanche models support the notion that frequent catastrophes prevent the growth of very large populations and as such, prevent rare large-scale catastrophes. We show that this notion is not universal. A new model class leads to a paradigm shift in the influence of catastrophes on the family-size distribution of subpopulations. We study a simple population dynamics model where individuals, as well as a whole family, may die with a constant probability, accompanied by a logistic population growth model. We compute the characteristics of the family-size distribution in steady state and the phase diagram of the steady-state distribution and show that the family and catastrophe size variances increase with the catastrophe frequency, which is the opposite of common intuition. Frequent catastrophes are balanced by a larger net-growth rate in surviving families, leading to the exponential growth of these families. When the catastrophe rate is further increased, a second phase transition to extinction occurs when the rate of new family creations is lower than their destruction rate by catastrophes.

Coagulation and universal scaling limits for critical Galton–Watson processes

AbstractThe basis of this paper is the elementary observation that then-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs becomelinear and purely dilatationalwhen expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence ofuniversalcritical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generateallpossible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

Sibship size and mental illness.

1. This study is based on a sample of 6,795 psychiatric referrals to the North-Eastern Region of Scotland and of 3,425 subjects from the general population. 2. In both the psychiatric and the control sample a marked and significant relationship was demonstrated between year of birth and family size. 3. In the psychiatric sample an equally marked and significant relationship was demonstrated between social class of parent and family size. 4. The family size distributions of the psychiatric and control samples were strikingly similar, suggesting that there is no relationship between mental illness and family size. 5. When allowance was made for differences in age distribution among diagnostic groups the diagnostic distribution within each family size group did not differ from the expected.

Interpretations of family size distributions: The Datura example

Young asteroid families are unique sources of information about fragmentation physics and the structure of their parent bodies, since their physical properties have not changed much since their birth. Families have different properties such as age, size, taxonomy, collision severity and others, and understanding the effect of those properties on our observations of the size-frequency distribution (SFD) of family fragments can give us important insights into the hypervelocity collision processes at scales we cannot achieve in our laboratories.Here we take as an example the very young Datura family, with a small 8-km parent body, and compare its size distribution to other families, with both large and small parent bodies, and created by both catastrophic and cratering formation events. We conclude that most likely explanation for the shallower size distribution compared to larger families is a more pronounced observational bias because of its small size. Its size distribution is perfectly normal when its parent body size is taken into account. We also discuss some other possibilities. In addition, we study another common feature: an offset or “bump” in the distribution occurring for a few of the larger elements. We hypothesize that it can be explained by a newly described regime of cratering, “spall cratering”, which controls the majority of impact craters on the surface of small asteroids like Datura.

Size distribution of function-based human gene sets and the split–merge model

The sizes of paralogues—gene families produced by ancestral duplication—are known to follow a power-law distribution. We examine the size distribution of gene sets or gene families where genes are grouped by a similar function or share a common property. The size distribution of Human Gene Nomenclature Committee (HGNC) gene sets deviate from the power-law, and can be fitted much better by a beta rank function. We propose a simple mechanism to break a power-law size distribution by a combination of splitting and merging operations. The largest gene sets are split into two to account for the subfunctional categories, and a small proportion of other gene sets are merged into larger sets as new common themes might be realized. These operations are not uncommon for a curator of gene sets. A simulation shows that iteration of these operations changes the size distribution of Ensembl paralogues and could lead to a distribution fitted by a rank beta function. We further illustrate application of beta rank function by the example of distribution of transcription factors and drug target genes among HGNC gene families.

Population Genetic Consequences of the Allee Effect and the Role of Offspring-Number Variation

A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. But many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbor on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. It is also conceivable, however, that an Allee effect itself leads to an increase in offspring-number variation. In this case, successful populations with an Allee effect can exhibit less genetic diversity despite growing faster at small population sizes. Unlike in many classical population genetics models, the role of offspring-number variation for the population genetic consequences of the Allee effect cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.

All for One? Family Size and Children's Educational Distribution under Credit Constraints

This paper examines the possibility that a child's years of schooling could increase in the number of siblings, instead of being diminished by competition for parents' resources: if unable to finance the education of their younger children, parents may do so through their older children's labor income. We examine this possibility in a model combining convex returns to education and credit constraints. Our model predicts correlations among family size, years of schooling and birth order, which would not exist when either of these two elements is absent. Empirical patterns shown in the United States, Mexico, and South Korea support the model predictions.

Word Formation Is Aware of Morpheme Family Size

Words are built from smaller meaning bearing parts, called morphemes. As one word can contain multiple morphemes, one morpheme can be present in different words. The number of distinct words a morpheme can be found in is its family size. Here we used Birth-Death-Innovation Models (BDIMs) to analyze the distribution of morpheme family sizes in English and German vocabulary over the last 200 years. Rather than just fitting to a probability distribution, these mechanistic models allow for the direct interpretation of identified parameters. Despite the complexity of language change, we indeed found that a specific variant of this pure stochastic model, the second order linear balanced BDIM, significantly fitted the observed distributions. In this model, birth and death rates are increased for smaller morpheme families. This finding indicates an influence of morpheme family sizes on vocabulary changes. This could be an effect of word formation, perception or both. On a more general level, we give an example on how mechanistic models can enable the identification of statistical trends in language change usually hidden by cultural influences.

SIZE DISTRIBUTION OF GENE FAMILIES IN A GENOME

We consider a probabilistic model of genome evolution. We are interested in size distribution of gene families. The model is based on three fundamental evolutionary events: gene loss, duplication and accumulated change. We assume that the probability of gene loss and duplication is constant and the probability of gene mutation mi depends on the size i of a family. We prove that size distribution of paralogous gene families in a genome converges to the equilibrium as time goes to infinity. Moreover, we show how this equilibrium depends on the sequence (mi). Theoretical results are compared with the available genomic data.

Sequence and structure space model of protein divergence driven by point mutations

New folds of protein structures emerge in evolution as a result of insertions, deletions or shuffling of fragments of underlying gene sequences, and from aggregated effects of point mutations. The result of these evolutionary processes is a rich and complex universe of protein sequences and structures, with characteristic features such as heavy-tailed distribution of fold occurrences, and a distinct shape of relationship between sequence identity and structure similarity. Better understanding of how the protein universe evolved to its present form can be achieved by creating models of protein structure evolution. Here we introduce a stochastic model of evolution that involves residue substitutions as the sole source of structure innovation, and is nonetheless able to reproduce the diversity of the protein domains repertoire, its cluster structure with heavy-tailed distribution of family sizes, and presence of the twilight zone populated with remote homologs.

Subsistence economics, family size, and the emergence of social complexity in hunter–gatherer systems in eastern North America

A consideration of ethnographic data, computational model results, and archaeological data suggest that changes in family-level economics coincident with subsistence intensification contributed to the emergence of social complexity among prehistoric hunter–gatherers in eastern North America by creating the conditions for a “rich get richer” scenario. Ethnographic data are used to construct a general computational model representing key person- and family-level behaviors, constraints, and decisions affecting the size and composition of hunter–gatherer families. Results from model experiments suggest that lowering the age at which children make a significant contribution to subsistence (e.g., through the broadening of the diet to include the kinds of mass-harvested, “low quality” foods that were increasingly exploited during the Archaic and Woodland periods) relaxes constraints on family size and makes large, polygynous families economically viable. Positive feedbacks between the productive and reproductive potentials of larger families produce right-tailed distributions of family size and “wealth” when the productive age of children is low and polygyny is incentivized. Size data from over 800 prehistoric residential structures suggest right-tailed distributions of family size were present during the Late Archaic through Middle Woodland periods. These distributions would have provided variability in family-based status that permitted the emergence of hereditary social distinctions.

Social contacts and receipt of help among older people in England: are there benefits of having more children?

To investigate whether number of children and, among parents, having a daughter is associated with older people's likelihood of at least weekly face-to-face social contact and later receipt of help if needed. Multivariate analysis of data from Waves 1 and 2 of the English Longitudinal Study of Ageing (ELSA). Older parents in England had higher chances of at least weekly face-to-face social contact than their childless counterparts but larger family size had only a slight additional effect. For parents, having at least one daughter was more important than number of children. Larger family size was positively associated with receipt of help from a child by parents with activities of daily living (ADL) or instrumental activities of daily living (IADL) limitations. Childless women were more likely than mothers to receive help from friends but even so had lower odds of receiving help from any informal source. Contact with a child in 2002 predicted receipt of help 2 years later. These results show some advantages for older parents compared with childless individuals in terms of social contact and receipt of help and, among parents, an additional effect of having a daughter. Changes in family size distributions have implications for the support of older people and for planners of formal services.

Constraining asteroid dynamical models using GAIA data

GAIA is expected to greatly enhance our knowledge on the orbital, spin-state, spectral and size distribution of individual asteroids and asteroid families. Dynamical models, incorporating resonant phenomena and thermal (Yarkovsky/YORP) effects, can be used to understand the observations, in particular the orbital, size-frequency and spin-axis distributions in families and groups. Conversely, the GAIA data can be used to appropriately modify/constrain our models. Such work will necessarily involve an optimization procedure, for which a large number of dynamical experiments will typically be needed; this is clearly a limiting factor. We present here a preliminary version of a statistical model that describes asteroid transport due to gravitational perturbations in the 3-D space of proper elements, coupled with Yarkovsky drift effects. Computationally, the model is very efficient for studying the long-term evolution of large ensembles of asteroids (e.g. families). Thus, given a “final” distribution of orbits, spins and sizes, our model could be used in an iterative fashion, to derive e.g. an optimal guess of the initial ejection velocity field in a family-forming event.

A comparison between rubble-pile and monolithic targets in impact simulations: Application to asteroid satellites and family size distributions

Collisions are a fundamental process in the creation of asteroid families and in satellite formation. For this reason, understanding the outcome of impacts is fundamental to the accurate modeling of the formation and evolution of such systems. Smoothed-Particle Hydrodynamics/N-body codes have become the techniques of choice to study large-scale impact outcomes, including both the fragmentation of the parent body and the gravitational interactions between fragments. It is now possible to apply this technique to targets with either monolithic or rubble-pile internal structures. In this paper we apply these numerical techniques to rubble-pile targets, extending previous investigations by Durda et al. (Durda, D.D., Bottke, W.F., Enke, B.L., Merline, W.J., Asphaug, E., Richardson, D.C., Leinhardt, Z.M. [2004]. Icarus 170, 243–257; Durda, D.D., Bottke, W.F., Nesvorný, D., Enke, B.L., Merline, W.J., Asphaug, E., Richardson, D.C. [2007]. Icarus 186, 498–516). The goals are to study asteroid–satellite formation and the morphology of the size–frequency distributions (SFDs) from 175 impact simulations covering a range of collision speeds, impact angles, and impactor sizes. Our results show that low-energy impacts into rubble-pile and monolithic targets produce different features in the resulting SFDs and that these are potentially diagnostic of the initial conditions for the impact and the internal structure of the parent bodies of asteroid families. In contrast, super-catastrophic events (i.e., high-energy impacts with large specific impact energy) result in SFDs that are similar to each other. We also find that rubble-pile targets are less efficient in producing satellites than their monolithic counterparts. However, some features, such as the secondary-to-primary diameter ratio and the relative separation of components in binary systems, are similar for these two different internal structures of parent bodies.

Origin and evolution of gene families in Bacteria and Archaea

BackgroundComparison of complete genomes of Bacteria and Archaea shows that gene content varies considerably and that genomes evolve quite rapidly via gene duplication and deletion and horizontal gene transfer. We analyze a diverse set of 92 Bacteria and 79 Archaea in order to investigate the processes governing the origin and evolution of families of related genes within genomes.ResultsGenes were clustered into related groups using similarity criteria derived from BLAST. Most clusters contained genes from only one or a small number of genomes, and relatively few core clusters were found that spanned all genomes. Gene clusters found in larger numbers of genomes tended to have larger numbers of genes per genome; however, clusters with unusually large numbers of genes per genome were found among both narrowly and widely distributed clusters. Larger genomes were found to have larger mean gene family sizes and a greater proportion of families of very large size. We used a model of birth, death, and innovation to predict the distribution of gene family sizes. The key parameter is r, the ratio of duplications to deletions. It was found that the model can give a good fit to the observed distribution only if there are several classes of genes with different values of r. The preferred model in most cases had three classes of genes.ConclusionsThere appears to be a rapid rate of origination of new gene families within individual genomes. Most of these gene families are deleted before they spread to large numbers of genomes, which suggests that they may not be generally beneficial to the organisms. The family size distribution is best described by a large fraction of families that tend to have only one or two genes and a small fraction of families of multi-copy genes that are highly prone to duplication. Larger families occur more frequently in larger genomes, indicating higher r in these genomes, possibly due to a greater tolerance for non-beneficial gene duplicates. The smallest genomes contain very few multi-copy families, suggesting a high rate of deletion of all but the most beneficial genes in these genomes.

Joint scaling laws in functional and evolutionary categories in prokaryotic genomes

We propose and study a class-expansion/innovation/loss model of genome evolution taking into account biological roles of genes and their constituent domains. In our model, numbers of genes in different functional categories are coupled to each other. For example, an increase in the number of metabolic enzymes in a genome is usually accompanied by addition of new transcription factors regulating these enzymes. Such coupling can be thought of as a proportional ‘recipe’ for genome composition of the type ‘a spoonful of sugar for each egg yolk’. The model jointly reproduces two known empirical laws: the distribution of family sizes and the non-linear scaling of the number of genes in certain functional categories (e.g. transcription factors) with genome size. In addition, it allows us to derive a novel relation between the exponents characterizing these two scaling laws, establishing a direct quantitative connection between evolutionary and functional categories. It predicts that functional categories that grow faster-than-linearly with genome size to be characterized by flatter-than-average family size distributions. This relation is confirmed by our bioinformatics analysis of prokaryotic genomes. This proves that the joint quantitative trends of functional and evolutionary classes can be understood in terms of evolutionary growth with proportional recipes.

Are There Laws of Genome Evolution?

Research in quantitative evolutionary genomics and systems biology led to the discovery of several universal regularities connecting genomic and molecular phenomic variables. These universals include the log-normal distribution of the evolutionary rates of orthologous genes; the power law–like distributions of paralogous family size and node degree in various biological networks; the negative correlation between a gene's sequence evolution rate and expression level; and differential scaling of functional classes of genes with genome size. The universals of genome evolution can be accounted for by simple mathematical models similar to those used in statistical physics, such as the birth-death-innovation model. These models do not explicitly incorporate selection; therefore, the observed universal regularities do not appear to be shaped by selection but rather are emergent properties of gene ensembles. Although a complete physical theory of evolutionary biology is inconceivable, the universals of genome evolution might qualify as “laws of evolutionary genomics” in the same sense “law” is understood in modern physics.