The beta-geometric distribution applied to comparative fecundability studies

A convenient measure of fecundability is time (number of menstrual cycles) required to achieve pregnancy. Couples attempting pregnancy are heterogeneous in their per-cycle probability of success. If success probabilities vary among couples according to a beta distribution, then cycles to pregnancy will have a beta-geometric distribution. Under this model, the inverse of the cycle-specific conception rate is a linear function of time. Data on cycles to pregnancy can be used to estimate the beta parameters by maximum likelihood in a straightforward manner with a package such as GLIM. The likelihood ratio test can thus be employed in studies of exposures that may impair fecundability. Covariates are incorporated in a natural way. The model is illustrated by applying it to data on cycles to pregnancy in smokers and nonsmokers, with adjustment for covariates. For a cross-sectional study, when length-biased sampling is taken into account, the pre-interview attempt time is shown to follow a beta-geometric distribution, so that the same methods of analysis can be applied even though all of the available data are right-censored. For a cohort followed prospectively, there will be some couples enrolled whose fecundability is effectively 0, and for such applications, the beta could be considered to be contaminated by a distribution degenerate at 0. The mixing parameter (proportion sterile) can be estimated by application of the expectation-maximization (EM) algorithm. This, too, can be carried out using GLIM.     

Time to Pregnancy: A Computational Method for Using the Duration of Non-Conception for Predicting Conception

An important problem in reproductive medicine is deciding when people who have failed to become pregnant without medical assistance should begin investigation and treatment. This study describes a computational approach to determining what can be deduced about a couple''s future chances of pregnancy from the number of menstrual cycles over which they have been trying to conceive. The starting point is that a couple''s fertility is inherently uncertain. This uncertainty is modelled as a probability distribution for the chance of conceiving in each menstrual cycle. We have developed a general numerical computational method, which uses Bayes'' theorem to generate a posterior distribution for a couple''s chance of conceiving in each cycle, conditional on the number of previous cycles of attempted conception. When various metrics of a couple''s expected chances of pregnancy were computed as a function of the number of cycles over which they had been trying to conceive, we found good fits to observed data on time to pregnancy for different populations. The commonly-used standard of 12 cycles of non-conception as an indicator of subfertility was found to be reasonably robust, though a larger or smaller number of cycles may be more appropriate depending on the population from which a couple is drawn and the precise subfertility metric which is most relevant, for example the probability of conception in the next cycle or the next 12 cycles. We have also applied our computational method to model the impact of female reproductive ageing. Results indicate that, for women over the age of 35, it may be appropriate to start investigation and treatment more quickly than for younger women. Ignoring reproductive decline during the period of attempted conception added up to two cycles to the computed number of cycles before reaching a metric of subfertility.     

Gender heterogeneity in self-reported hypertension

We investigate the gender gap in hypertension misreporting using the French Constances cohort. We show that false negative reporting of hypertension is more frequent among men than among women, even after conditioning on a series of individual characteristics. As a second step, we investigate the causes of the gender gap in hypertension misreporting. We show that women go to the doctor more often than men do and that they have better knowledge of their family medical history. Once these differences are taken into account, the gender gap in false negative reporting of hypertension is reversed. This suggests that information acquisition and healthcare utilisation are crucial ingredients in fighting undiagnosed male hypertension.     

Women's estradiol predicts preference for facial cues of men's testosterone

A growing body of research has shown that women express stronger attraction to more masculine traits when they are tested near ovulation than when tested during other times in the menstrual cycle. Although these effects have been interpreted as increased preferences for markers of elevated testosterone during times in the cycle when conception is most likely, no previous studies have directly demonstrated that women express stronger attraction to higher testosterone men at different times in the cycle. In addition, little research has addressed which hormonal or other physiological mechanisms may regulate temporal shifts in women's attractiveness judgments. In this research, we demonstrate that women with higher estradiol concentrations exhibit stronger preferences for the faces of men with higher testosterone concentrations, and that women's testosterone preference and estradiol curves track one another across days of the cycle. The findings are the first direct demonstration in humans that hormone concentrations in one sex are associated with attraction to cues of hormonal status in the opposite sex. The results support a functional role for estradiol in calibrating women's mating psychology to indices of their current fertility, analogous to similar processes that have been documented in nonhuman species. A strong correlation between estradiol and testosterone preference specifically during the luteal phase further suggests that women's mate preferences may track their fertility between different cycles in addition to being calibrated to the timing of ovulation within individual cycles.     

THE BETA-BINOMIAL MODEL: ACCOUNTING FOR INTER-TRIAL VARIATION IN REPLICATED DIFFERENCE AND PREFERENCE TESTS

Binomial tests are commonly used in sensory difference and preference testing under the assumptions that choices are independent and choice probabilities do not vary from trial to trial. This paper addresses violations of the latter assumption (often referred to as overdispersion) and accounts for variation in inter-trial choice probabilities following the Beta distribution. Such variation could arise as a result of differences in test substrate from trial to trial, differences in sensory acuity among subjects or the existence of latent preference segments. In fact, it is likely that overdispersion occurs ubiquitously in product testing. Using the Binomial model for data in which there is inter-trial variation may lead to seriously misleading conclusions from a sensory difference or preference test. A simulation study in this paper based on product testing experience showed that when using a Binomial model for overdispersed Binomial data, Type I error may be 0.44 for a Binomial test specification corresponding to a level of 0.05. Underestimation of Type I error using the Binomial model may seriously undermine legal claims of product superiority in situations where overdispersion occurs. The Beta-Binomial (BB) model, an extension of the Binomial distribution, was developed to fit overdispersed Binomial data. Procedures for estimating and testing the parameters as well as testing for goodness of fit are discussed. Procedures for determining sample size and for calculating estimate precision and test power based on the BB model are given. Numerical examples and simulation results are also given in the paper. The BB model should improve the validity of sensory difference and preference testing.     

Modeling menstrual cycle length using a mixture distribution

In reproductive health studies, epidemiologists are often interested in examining the effects of covariates on menstrual cycle length which is a convenient, noninvasive measure of women's ovarian and reproductive function. Previous literature (Harlow and Zeger, 1991) suggests that the distribution of cycle length is a mixture of a major symmetric distribution and a component featuring a long right tail. Motivated by the shape of this marginal distribution, we propose a mixture distribution for cycle length, representing standard cycles from a Normal distribution and nonstandard cycles from a shifted Weibull distribution. The parameters are estimated using an estimating equation derived from the score function of an independence working model. The fitted mixture distribution agrees well with the distribution estimated using nonparametric approaches. We propose two measures to help determine whether a cycle is standard or nonstandard, developing tools necessary to identify characteristics of the menstrual cycles that are biologically indicative of ovarian dysfunction. We model the effect of a woman's age on the mean and variation of both standard and nonstandard cycle lengths using multiple measurements of women.     

Disease effects on reproduction can cause population cycles in seasonal environments

1. Recent studies of rodent populations have demonstrated that certain parasites can cause juveniles to delay maturation until the next reproductive season. Furthermore, a variety of parasites may share the same host, and evidence is beginning to accumulate showing nonindependent effects of different infections. 2. We investigated the consequences for host population dynamics of a disease-induced period of no reproduction, and a chronic reduction in fecundity following recovery from infection (such as may be induced by secondary infections) using a modified SIR (susceptible, infected, recovered) model. We also included a seasonally varying birth rate as recent studies have demonstrated that seasonally varying parameters can have important effects on long-term host-parasite dynamics. We investigated the model predictions using parameters derived from five different cyclic rodent populations. 3. Delayed and reduced fecundity following recovery from infection have no effect on the ability of the disease to regulate the host population in the model as they have no effect on the basic reproductive rate. However, these factors can influence the long-term dynamics including whether or not they exhibit multiyear cycles. 4. The model predicts disease-induced multiyear cycles for a wide range of realistic parameter values. Host populations that recover relatively slowly following a disease-induced population crash are more likely to show multiyear cycles. Diseases for which the period of infection is brief, but full recovery of reproductive function is relatively slow, could generate large amplitude multiyear cycles of several years in length. Chronically reduced fecundity following recovery can also induce multiyear cycles, in support of previous theoretical studies. 5. When parameterized for cowpox virus in the cyclic field vole populations (Microtus agrestis) of Kielder Forest (northern England), the model predicts that the disease must chronically reduce host fecundity by more than 70%, following recovery from infection, for it to induce multiyear cycles. When the model predicts quasi-periodic multiyear cycles it also predicts that seroprevalence and the effective date of onset of the reproductive season are delayed density-dependent, two phenomena that have been recorded in the field.     

Sources of error in recording the blood pressure of patients with hypertension in general practice.

Sources of error in recording the blood pressures of patients with hypertension in general practice were examined for 1072 patients over two and a half years. A highly significant preference for terminal digit 0 was shown, but terminal digit preference operates in a different manner around important thresholds. Doctors did not adhere consistently to the protocol that they accepted. These sources of bias affect the calculations of mean blood pressure and have considerable implications for decisions about treatment for many patients.     

Digit ratio varies with sex, egg order and strength of mate preference in zebra finches

The steroid environment encountered by developing vertebrates has important organizational effects on physiology and behaviour that persist throughout an organism's lifetime. Optimal allocation of maternal steroids to zygotes may be difficult to achieve because of the sexually antagonistic effects of steroids; thus, for example, a hormone environment beneficial to a developing male may be much less beneficial to a developing female. Research into the important topic of how mothers might adaptively adjust steroid titres experienced by particular young has been constrained by the difficulty of measuring the steroid environment experienced by the embryo at critical times in development. A potential approach to this problem has been suggested by research on variation in digit ratios in humans, where the ratio of the length of the second and fourth digits reflects the steroid environment experienced by the foetus; notably, digit 4 lengthens in response to androgens. In light of the conservative nature of homeobox genes regulating early development in tetrapods, we questioned whether a sex difference in digit ratio exists in a passerine bird, the zebra finch, Taeniopygia guttata castanotis, and whether observed variation in the ratio is consistent with the previously reported pattern that androgen allocation to zebra finch egg yolk declines across laying order. We established an aviary population of outbred, wild-type zebra finches, and allowed them to breed freely. Hatchlings were marked to correspond to their egg order, and their digit ratios were measured after birds reached adulthood. We found that digit ratio increased across egg order, which is consistent with a pattern of decreasing androgen allocation. Moreover, digit ratios differed between the sexes. We also investigated whether variation in digit ratio among adult females predicted variation in their performance in mate-choice tests. Digit ratio accounted for almost 50% of the variance in strength of female preference for an attractive male trait: specifically, females with higher (presumably less 'androgenized') ratios had stronger preferences for attractive males. Digit ratio may prove to be an extremely useful tool for addressing a wide range of questions about vertebrate differentiation and behaviour.     

Clustering in large networks does not promote upstream reciprocity

Upstream reciprocity (also called generalized reciprocity) is a putative mechanism for cooperation in social dilemma situations with which players help others when they are helped by somebody else. It is a type of indirect reciprocity. Although upstream reciprocity is often observed in experiments, most theories suggest that it is operative only when players form short cycles such as triangles, implying a small population size, or when it is combined with other mechanisms that promote cooperation on their own. An expectation is that real social networks, which are known to be full of triangles and other short cycles, may accommodate upstream reciprocity. In this study, I extend the upstream reciprocity game proposed for a directed cycle by Boyd and Richerson to the case of general networks. The model is not evolutionary and concerns the conditions under which the unanimity of cooperative players is a Nash equilibrium. I show that an abundance of triangles or other short cycles in a network does little to promote upstream reciprocity. Cooperation is less likely for a larger population size even if triangles are abundant in the network. In addition, in contrast to the results for evolutionary social dilemma games on networks, scale-free networks lead to less cooperation than networks with a homogeneous degree distribution.     

Grasping kinematics from the perspective of the individual digits: a modelling study

Grasping is a prototype of human motor coordination. Nevertheless, it is not known what determines the typical movement patterns of grasping. One way to approach this issue is by building models. We developed a model based on the movements of the individual digits. In our model the following objectives were taken into account for each digit: move smoothly to the preselected goal position on the object without hitting other surfaces, arrive at about the same time as the other digit and never move too far from the other digit. These objectives were implemented by regarding the tips of the digits as point masses with a spring between them, each attracted to its goal position and repelled from objects' surfaces. Their movements were damped. Using a single set of parameters, our model can reproduce a wider variety of experimental findings than any previous model of grasping. Apart from reproducing known effects (even the angles under which digits approach trapezoidal objects' surfaces, which no other model can explain), our model predicted that the increase in maximum grip aperture with object size should be greater for blocks than for cylinders. A survey of the literature shows that this is indeed how humans behave. The model can also adequately predict how single digit pointing movements are made. This supports the idea that grasping kinematics follow from the movements of the individual digits.     

Selective breeding for a behavioral trait changes digit ratio

The ratio of the length of the second digit (index finger) divided by the fourth digit (ring finger) tends to be lower in men than in women. This 2D:4D digit ratio is often used as a proxy for prenatal androgen exposure in studies of human health and behavior. For example, 2D:4D ratio is lower (i.e. more "masculinized") in both men and women of greater physical fitness and/or sporting ability. Lab mice have also shown variation in 2D:4D as a function of uterine environment, and mouse digit ratios seem also to correlate with behavioral traits, including daily activity levels. Selective breeding for increased rates of voluntary exercise (wheel running) in four lines of mice has caused correlated increases in aerobic exercise capacity, circulating corticosterone level, and predatory aggression. Here, we show that this selection regime has also increased 2D:4D. This apparent "feminization" in mice is opposite to the relationship seen between 2D:4D and physical fitness in human beings. The present results are difficult to reconcile with the notion that 2D:4D is an effective proxy for prenatal androgen exposure; instead, it may more accurately reflect effects of glucocorticoids, or other factors that regulate any of many genes.     

Maximum likelihood estimation of the parameters of the prior distributions of three variables that strongly influence reproductive performance in cows

Ovulation detection rate, pregnancy rate, and embryo loss rate greatly affect the reproductive performance of cows. A previous model described the separate effects of these variables on the resulting calving patterns and assumed that the variables have the same value for all cows belonging to the same herd. This is not a realistic biological assumption, so the beta distribution is used to introduce "between-cow" variation in the three variables. Two approaches are used to find maximum likelihood estimates of the parameters of these prior beta distributions. The first considers sequences of ovulations, artificial inseminations, and pregnancies, separately. For both ovulation detection rate and pregnancy rate this approach considers the number of "successes" of each event for a particular cow (e.g., in the case of an ovulation, a success is a detection), and conditions on the total number of occurrences of that event in the cow, so that beta-binomial distributions are considered. However, for embryo loss rate the number of pregnancies required until a particular cow calves is considered, so that a beta-geometric distribution results. If the cow is removed before she calves, a censored sequence will result. The second approach considers the sequences of ovulations, artificial inseminations, pregnancies, and embryo losses, together, which will stop only when the cow calves. Otherwise, if she is removed before that time, a censored sequence will result. In this case, a joint distribution, with three independent prior beta distributions, is considered. The results of the analysis of data from 22 herds are discussed.     

The evolutionary significance of sexual selection

A model for the joint evolution of a secondary sexual male trait Z and a female mating preference Y is discussed. Recurrence relations for the moments of (Z, Y) are given under the assumption that the traits are binormally distributed. It is shown that female preference for a male character can lead to an equilibrium distribution of the male trait with non-zero variances. The conditions under which the distribution is stable, are given. Unstable situations, in which a continued exaggeration of the male trait occurs, are described. It is demonstrated that the effect of sexual selection on the evolution of the male trait depends on the intensity of natural selection, i.e. the effect of the sexual selection increases when the intensity of natural selection is reduced. The effect of the female preference on the male trait also increases with increasing availability of males. This provides a link to several ecological conditions which have generally been known to be correlated with the degree of sexual selection. Furthermore, it is demonstrated that perturbations away from the equilibrium may cause rapid evolution of the male character, eventually leading to speciation.     

Computational principles underlying the recognition of acoustic signals in insects

Many animals produce pulse-like signals during acoustic communication. These signals exhibit structure on two time scales: they consist of trains of pulses that are often broadcast in packets—so called chirps. Temporal parameters of the pulse and of the chirp are decisive for female preference. Despite these signals being produced by animals from many different taxa (e.g. frogs, grasshoppers, crickets, bushcrickets, flies), a general framework for their evaluation is still lacking. We propose such a framework, based on a simple and physiologically plausible model. The model consists of feature detectors, whose time-varying output is averaged over the signal and then linearly combined to yield the behavioral preference. We fitted this model to large data sets collected in two species of crickets and found that Gabor filters—known from visual and auditory physiology—explain the preference functions in these two species very well. We further explored the properties of Gabor filters and found a systematic relationship between parameters of the filters and the shape of preference functions. Although these Gabor filters were relatively short, they were also able to explain aspects of the preference for signal parameters on the longer time scale due to the integration step in our model. Our framework explains a wide range of phenomena associated with female preference for a widespread class of signals in an intuitive and physiologically plausible fashion. This approach thus constitutes a valuable tool to understand the functioning and evolution of communication systems in many species.     

A model for evaluating the effect of son or daughter preference on population size

This paper develops a micro probabilistic model to describe the family extension process. The parameters are: the probability that a newborn is a boy (P), the number of desired boys (B), the number of desired girls (G), and the maximum possible number of children (N). This maximum is a stopping rule rather than a biological maximum. The variables are the ultimate number of boys and girls. According to this model, each couple determines B, G, and N, at the beginning of the reproductive period and continues to reproduce until at least B and at least G are achieved, or until the total number of children reaches N. The probability distribution of the ultimate number of boys and girls in the population is derived for this model. Simulation techniques are used to generate offspring. The results showed that the population size increases with the absolute differences, [B-G] for fixed N. They also suggested that son or daughter preference may be an important factor in fertility determinants, which may have important implications in population policies.     

Polymorphism of melanic ladybirds maintained by frequency-dependent sexual selection

Sexual selection, whether by female preference or male competition, is almost inevitably frequency-dependent. Female preference gives rise to a 'rare male effect', by which the rarer male phenotypes gain a relatively greater selective advantage. In addition to this effect, the proportion of females expressing a preference may also be frequency-dependent.
Frequency-dependent expression of mating preference can arise in at least two ways: (1) when females encounter a succession of courting males while searching for a male they prefer; (2) when females chose a male from within a lek. Models of mating behaviour reveal a clear distinction between the frequency dependence in the expression of female preference and the frequency dependence in the consequent selection of the males. When expression of preference is highly dependent on frequency, the selection of males is constant or only slightly frequency-dependent: constant expression of preference produces high frequency dependence of selection. Analysis of general models shows that genetic polymorphisms can be maintained under a wide range of conditions.
The ladybird, Adalia bipunctata , is polymorphic for several melanic and non-melanic phenotypes. Females have a genetically determined preference for melanic males. Non-melanic phenotypes mate assortatively. By estimating the parameters of a detailed model of natural selection, sexual selection and assortative mating, it has been shown that the Adalia bipunctata polymorphism will be maintained at frequencies observed in the wild.     

Control of the rate of photosynthetic carbon dioxide fixation

A simplified model of the reductive pentose phosphate pathway of photosynthesis is analysed in order to quantify the degree to which each of the constituent reactions controls the rate of CO₂ fixation (given by the control coefficient). The analysis focuses on the four largely irreversible reactions of the cycle together with the first irreversible reaction in the sucrose and starch synthetic pathways. The model assumes that the other reactions are at equilibrium. The photorespiratory and electron transport systems are not included in the model. The analysis demonstrates that: (1) an analytical approach can be used to investigate the distribution of flux control in autocatalytic and moiety-conserved cycles; (2) measurements of enzyme kinetic parameters and certain fluxes and substrate concentrations can be used to solve the equations defining the enzyme control coefficients; (3) the conservation of total stromal phosphate and the intricate regulatory mechanisms of the photosynthetic system result in a relationship between the control coefficients that is complex and may defy any intuitive assessment of ‘rate limitation’; (4) ribulose-1,5-bisphosphate carboxylase / oxygenase may, under certain conditions, be a major controller of the rate of CO₂ fixation and, by regulating the concentration of ribulose 1,5-bisphosphate, may be important in governing the ratio of organic to inorganic phosphate in the stroma; (5) the other enzymes may also serve an important role in determining the distribution of phosphate between organic and inorganic species because they catalyze reactions at the branch points between starch and sucrose synthesis and ribulose 1,5-bisphosphate regeneration; (6) these enzymes that catalyze ‘branch-pint’ reactions may have negative control coefficients because of their ability to reduce the total concentration of cycle intermediates; (7) an approach combining the use of the equations presented in this paper and flux and substrate concentration measurements may be adequate for determining the control coefficients of several enzymes of the reductive pentose phosphate pathway.     

A Stochastic Model for Estimating Adolescent Sterility

In this article, a simple stochastic model for the time to first conception of a cohort of married women is developed, by identifying three states, ‘adolescent sterile’, ‘ovulating’ and ‘conceived‘, into which they can be placed. It is demonstrated that the model provides a close fit to observed data. The estimates of the parameters in the model, can be used to calculate the number of women in each state at different points of time and also to obtain estimates of the probabilities of conception for the two categories of women, adolescent sterile and biologically mature.     

Logical Probability and Risk Assessment

A risk assessment is intended to provide a statement of current knowledge which is intended to inform a decision-maker of the current state of knowledge in response to a particular concern. Because answering the concerns of decision-makers often requires inferences to be drawn, doubt often arises over how the inference is to be drawn. In quantitative risk assessment, where a mathematical equation or model is used to draw the inference, this uncertainty is referred to as model uncertainty. A two-step process, which is referred to as logical probability, is proposed as a technique for representing model uncertainty in a risk assessment. The first step involves assigning model weights in which the degree of evidential support for each of the alternative models is considered. The second step involves assigning a unique interval in the range of 0 to 1 for each model which reflects the models' weight, to form a probability distribution. While the second step is straightforward, the first step is not. Assigning model weights requires consideration of any line of evidence that may reasonably impact the validity of the assertion of a model. While the development of a procedure for doing so may be expected to be a process which reflects the subjective preferences of whomever is involved in creating it, there are some historical precedents on which to build. Foremost among these are (1) the use of a correlation coefficient or other goodness-of-fit criteria to measure the degree of correspondence between a given model and a set of observations which are used as evidence to support it, and (2) preference given to models which are simpler, which may be ascertained as the number of adjustable parameters the model contains. Additional principles, which have little or no tradition to stand on, must be used to reflect the impact of other empirically supported beliefs on model preference. The procedure proposed is comparable to the procedure known as decision analysis, in which probabilities are assigned to alternative models based on expert or subjective input. The principal difference in the present case is that it is suggested that principles which transcend the decision at hand should be sought and articulated in order to generate a consistent measure of uncertainty arising from interpretation.