We describe the trajectory of the human sex ratio from conception to birth by analyzing data from (i) 3-to 6-d-old embryos, (ii) induced abortions, (iii) chorionic villus sampling, (iv) amniocentesis, and (v) fetal deaths and live births. Our dataset is the most comprehensive and largest ever assembled to estimate the sex ratio at conception and the sex ratio trajectory and is the first, to our knowledge, to include all of these types of data. Our estimate of the sex ratio at conception is 0.5 (proportion male), which contradicts the common claim that the sex ratio at conception is malebiased. The sex ratio among abnormal embryos is male-biased, and the sex ratio among normal embryos is female-biased. These biases are associated with the abnormal/normal state of the sex chromosomes and of chromosomes 15 and 17. The sex ratio may decrease in the first week or so after conception (due to excess male mortality); it then increases for at least 10-15 wk (due to excess female mortality), levels off after ∼20 wk, and declines slowly from 28 to 35 wk (due to excess male mortality). Total female mortality during pregnancy exceeds total male mortality. The unbiased sex ratio at conception, the increase in the sex ratio during the first trimester, and total mortality during pregnancy being greater for females are fundamental insights into early human development.he sex ratio at conception in humans is unknown, despite hundreds of years of speculation and research. Investigations of the sex ratio date back at least as far as Graunt (1) who described a net excess of male births (2). By the late 1800s, it was clear that more males than females die during later pregnancy (3). Beyond these facts, the demographic and genetic dynamics of the sex ratio from conception to birth are poorly resolved.The claim that the conception or primary sex ratio (PSR) is more male-biased than the birth sex ratio appears often in textbooks (4, 5) and in the scientific literature (e.g., refs. 6-11), usually with little or no description of evidence. Estimates of the PSR in these studies are typically 0.56 (proportion males) or greater. Many fewer researchers have claimed that the PSR is unbiased or slightly male-biased (12-16). A handful of researchers has claimed or implied that the PSR is female-biased (17-19) or claimed that the PSR cannot be estimated due to lack of appropriate data and/or methodological problems (20)(21)(22).Previous estimates of the PSR have no meaningful basis in data from the time of conception (or within at least a month of it). At best, the PSR has been estimated via backward extrapolation from data on induced or spontaneous abortions, fetal deaths, or live births; most of the non-live-birth data stems from the second or third trimester of pregnancy. In addition, even if one ignores the fallibility of extrapolation, biased estimates of the PSR based on spontaneous abortions and fetal deaths have usually been regarded as arising from unbiased samples of a population of embryos or fetuses having a biased PSR. The alternativ...

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285-1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at infinity is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at infinity is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.Comment: Published in at http://dx.doi.org/10.1214/10-AOP623 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. We extend some results on the convergence of one-dimensional diffusions killed at the boundary, conditioned on extended survival, to the case of general killing on the interior. We show, under fairly general conditions, that a diffusion conditioned on long survival either runs off to infinity almost surely, or almost surely converges to a quasistationary distribution given by the lowest eigenfunction of the generator. In the absence of internal killing, only a sufficiently strong inward drift can keep the process close to the origin, to allow convergence in distribution. An alternative, that arises when general killing is allowed, is that the conditioned process is held near the origin by a high rate of killing near ∞. We also extend, to the case of general killing, the standard result on convergence to a quasistationary distribution of a diffusion on a compact interval. MotivationAmong the puzzles that occupy mathematical demographers, one of the most tantalizing is the phenomenon sometimes called "mortality plateaus". While the knowledge that human mortality rates increase with age during the years of maturity remounts to the primitive past of statistical science, only in recent years has it become apparent that this accelerating decrepitude slows in extreme old age, and may even stop [Vau97]. Analogous flattening of the mortality curves for fruit flies [CLOV92] is well established.In their attempts to explain this widespread, and possibly near-universal phenomenon, mathematical demographers have turned repeatedly to Markov models of mortality. These are stochastic processes where "death" is identified with a random stopping time, which arises from a typically unobserved Markov process. Some models, such as the "cascading failures" model of H. Le Bras [Le 76] (described in section 6), were originally introduced to model the classical exponentially increasing "Gompertz" mortality curve, and were only later shown [GG91] to converge to a constant plateau mortality rate. Others, such as the series-parallel model of L. Gavrilov and N. Gavrilova [GG91], and the drifting Brownian motion model of J. Weitz and H. Fraser [WF01], were introduced explicitly with mortality plateaus in mind. (The drifting Brownian motion, though, it should be pointed out, was first

A probability model is presented for the dynamics of mutation-selection balance in a haploid infinite-population infinite-sites setting sufficiently general to cover mutation-driven changes in full age-specific demographic schedules. The model accommodates epistatic as well as additive selective costs. Closed form characterizations are obtained for solutions in finite time, along with proofs of convergence to stationary distributions and a proof of the uniqueness of solutions in a restricted case. Examples are given of applications to the biodemography of aging.

celebrated formula for the age-specific force of natural selection furnishes predictions for senescent mortality due to mutation accumulation, at the price of reliance on a linear approximation. Applying to Hamilton's setting the full nonlinear demographic model for mutation accumulation recently developed by Evans, Steinsaltz, and Wachter, we find surprising differences. Nonlinear interactions cause the collapse of Hamilton-style predictions in the most commonly studied case, refine predictions in other cases, and allow walls of death at ages before the end of reproduction. Haldane's principle for genetic load has an exact but unfamiliar generalization.biodemography | hazard functions | senescence T he best-known formula at the intersection of genetics and demography is doubtless W. D. Hamilton's "age-specific force of natural selection," the starting point for the models in ref. 1 applied in this paper. Hamilton (2, 3) differentiated a measure of fitness, Lotka's intrinsic rate of natural increase, with respect to an increment to age-specific mortality at an age a. Thus, he obtained a linear approximation for loss in fitness due to any deleterious mutations that raised mortality at an age a. The greater the loss in fitness, the faster should mutant alleles be selected out of a population, and the fewer should be found at equilibrium as recurring mutations balance natural selection.By this route, Sir Peter Medawar's concept of mutation accumulation as an evolutionary reason for senescence takes on mathematical form. As in refs. 4-6, richly developed in ref. 7, the idea involves genetic load produced by large numbers of mildly deleterious mutations occurring at widely separated loci, each with some small age-specific effect on vital schedules.Hamilton's work has been assessed and extended by Baudisch (8). Sophisticated genetic models of mutation-selection balance are available (9). Demographers mainly put up with less sophisticated models of the genome in return for more refined treatments of agespecific structure, as we do here. Age-specific predictions for vital schedules may be robust to details of genetic specification, in line with a principle of Haldane (10), which equates the population loss in fitness from genetic load to the total mutation rate, independent of the form of action of mutations.Current interest has been stimulated by the expansion of biodemography, reviewed in refs. 11-14, and by the appreciation of two widely occurring cross-species commonalities in graphs of mortality rates as functions of age: exponential increase at adult ages (the "Gompertz-Makeham" mortality pattern) and plateaulike shapes at older ages. Working with an appealingly simple specification, Brian Charlesworth (15) gave an elegant demonstration that Gompertz-Makeham mortality could be predicted exactly by the Hamilton-based linear approximate model. He also proposed an optional fix that would lead to plateaus at extreme ages. In Charlesworth's setting, a "wall of death" with infinite mortality rates and zero survivorship ...