Bayesian Modeling of the Level and Duration of Fertility in the Menstrual Cycle

Time to pregnancy studies that identify ovulation days and collect daily intercourse data can be used to estimate the day-specific probabilities of conception given intercourse on a single day relative to ovulation. In this article, a Bayesian semiparametric model is described for flexibly characterizing covariate effects and heterogeneity among couples in daily fecundability. The proposed model is characterized by the timing of the most fertile day of the cycle relative to ovulation, by the probability of conception due to intercourse on the most fertile day, and by the ratios of the daily conception probabilities for other days of the cycle relative to this peak probability. The ratios are assumed to be increasing in time to the peak and decreasing thereafter. Generalized linear mixed models are used to incorporate covariate and couple-specific effects on the peak probability and on the day-specific ratios. A Markov chain Monte Carlo algorithm is described for posterior estimation, and the methods are illustrated through application to caffeine data from a North Carolina pregnancy study.     

A survival analysis approach to modeling human fecundity

Understanding conception probabilities is important not only for helping couples to achieve pregnancy but also in identifying acute or chronic reproductive toxicants that affect the highly timed and interrelated processes underlying hormonal profiles, ovulation, libido, and conception during menstrual cycles. Currently, 2 statistical approaches are available for estimating conception probabilities depending upon the research question and extent of data collection during the menstrual cycle: a survival approach when interested in modeling time-to-pregnancy (TTP) in relation to women or couples' purported exposure(s), or a hierarchical Bayesian approach when one is interested in modeling day-specific conception probabilities during the estimated fertile window. We propose a biologically valid discrete survival model that unifies the above 2 approaches while relaxing some assumptions that may not be consistent with human reproduction or behavior. This approach combines both the survival and the hierarchical models allowing investigators to obtain the distribution of TTP and day-specific probabilities during the fertile window in a single model. Our model allows for the consideration of covariate effects at both the cycle and the daily level while accounting for daily variation in conception. We conduct extensive simulations and utilize the New York State Angler Prospective Pregnancy Cohort Study to illustrate our approach. We also provide the code to implement the model in R software in the supplemental section of the supplementary material available at Biostatistics online.     

Modeling human fertility in the presence of measurement error

The probability of conception in a given menstrual cycle is closely related to the timing of intercourse relative to ovulation. Although commonly used markers of time of ovulation are known to be error prone, most fertility models assume the day of ovulation is measured without error. We develop a mixture model that allows the day to be misspecified. We assume that the measurement errors are i.i.d. across menstrual cycles. Heterogeneity among couples in the per cycle likelihood of conception is accounted for using a beta mixture model. Bayesian estimation is straightforward using Markov chain Monte Carlo techniques. The methods are applied to a prospective study of couples at risk of pregnancy. In the absence of validation data or multiple independent markers of ovulation, the identifiability of the measurement error distribution depends on the assumed model. Thus, the results of studies relating the timing of intercourse to the probability of conception should be interpreted cautiously.     

A new approach to modeling daily probabilities of conception

Standard conception probabilities models assume that different acts of intercourse make independent contributions to the probability of conception in viable cycles. We propose an alternative, approximate model based on the assumption that the act of intercourse closest to the time of maximum fertility is the one most likely to have caused conception. We describe an adaptive algorithm [the most fertile intercourse day (MFID) algorithm] that estimates the most fertile intercourse day in each cycle. The approach is easily extended to include covariates and random between-couple differences in fecundability that affect the probability of conception in a given cycle. Reanalyses of two data sets reported in the literature are presented. Estimates of the probability of conception during the most fertile period of the cycle and of the effects of covariates are similar to estimates found using standard models.     

Hierarchical models for the probabilities of conception

In the last thirty years, there has been considerable interest in finding better models to fit data for probabilities of conception. An important early model was proposed by Barrett and Marshall (1969) and extended by Schwartz, MacDonald and Heuchel (1980). Recently, researchers have further extended these models by adding covariates. However, the increasingly complicated models are challenging to analyze with frequentist methods such as the EM algorithm. Bayesian models are more feasible, and the computation can be done via Markov chain Monte Carlo (MCMC). We consider a Bayesian model with an effect for protected intercourse to analyze data from the California Women's Reproductive Health Study and assess the effects of water contaminants and hormones. There are two main contributions in the paper. (1) For protected intercourse, we propose modeling the ratios of daily conception probabilities with protected intercourse to corresponding daily conception probabilities with unprotected intercourse. Due to the small sample size of our data set, we assume the ratios are the same for each day but unknown. (2) We consider Bayesian analysis under a unimodality assumption where the probabilities of conception increase before ovulation and decrease after ovulation. Gibbs sampling is used for finding the Bayesian estimates. There is some evidence that the two covariates affect fecundability.     

Menstrual Cycle Phase Does Not Predict Political Conservatism

Recent authors have reported a relationship between women''s fertility status, as indexed by menstrual cycle phase, and conservatism in moral, social and political values. We conducted a survey to test for the existence of a relationship between menstrual cycle day and conservatism.2213 women reporting regular menstrual cycles provided data about their political views. Of these women, 2208 provided information about their cycle date, 1260 provided additional evidence of reliability in self-reported cycle date, and of these, 750 also indicated an absence of hormonal disruptors such as recent hormonal contraception use, breastfeeding or pregnancy. Cycle day was used to estimate day-specific fertility rate (probability of conception); political conservatism was measured via direct self-report and via responses to the "Moral Foundations” questionnaire. We also recorded relationship status, which has been reported to interact with menstrual cycle phase in determining political preferences.We found no evidence of a relationship between estimated cyclical fertility changes and conservatism, and no evidence of an interaction between relationship status and cyclical fertility in determining political attitudes. Our findings were robust to multiple inclusion/exclusion criteria and to different methods of estimating fertility and measuring conservatism. In summary, the relationship between cycle-linked reproductive parameters and conservatism may be weaker or less reliable than previously thought.     

Patterns of First Intercourse: A Survey Among Israeli Women

Seasonality among animals has an important function in breeding cycles. Research focusing on conception and coital activity in humans has called attention to the possible existence of circannual rhythms in human sexual activity. We studied the diurnal variations, seasonality and subjective sexual preference among young women in Israel-focusing on their first intercourse. The survey included 135 college students who completed a self-report questionnaire. Half of the subjects (50%) reported loss of virginity during the summer with an almost equal distribution among the other seasons. Over 85% of the subjects had their first intercourse either in the evening or at night. Subjective sexual – seasonal – preference was reported by 44.4% of the subjects but no correlation was found between the preferred season and actual loss of virginity; except for the autumn “preferers”. Our study supports previous research generalizing the sociological explanation for seasonality of the first intercourse.     

Patterns of First Intercourse: A Survey Among Israeli Women

Seasonality among animals has an important function in breeding cycles. Research focusing on conception and coital activity in humans has called attention to the possible existence of circannual rhythms in human sexual activity. We studied the diurnal variations, seasonality and subjective sexual preference among young women in Israel-focusing on their first intercourse. The survey included 135 college students who completed a self-report questionnaire. Half of the subjects (50%) reported loss of virginity during the summer with an almost equal distribution among the other seasons. Over 85% of the subjects had their first intercourse either in the evening or at night. Subjective sexual - seasonal - preference was reported by 44.4% of the subjects but no correlation was found between the preferred season and actual loss of virginity; except for the autumn “preferers”. Our study supports previous research generalizing the sociological explanation for seasonality of the first intercourse.     

Fertility of couples following cessation of contraception

A retrospective study of the delay in conception after stopping different methods of contraception is undertaken among 5108 pregnant women residing in a departement in the Paris region. Particular attention is paid to the delay that sometimes separates the cessation of contraception and the beginning of intercourse without precautions. The counfounding factors considered are: the observation of a waiting period following cessation of contraception; the existence of a strategy, i.e., a coital pattern adopted by certain couples who have intercourse more often during the middle of the cycle, hoping to conceive more rapidly; the regularity of the menstrual cycle and the woman's age at the beginning of unprotected intercourse. The statistical techniques include chi-square analysis, analysis of variance correlation, partial multiple regression and actuarial rates of infecundity. The mean age of women at their last period was 26.3 years; 37.6% were primagravida; 87.3% worked outside the home and 36.6% were smokers. 72% of the women used oral contraceptives; 14% the IUD, and 14% other methods. Among all factors studied, 4 are strongly related both to fertility and to mode of contraception: the waiting period, coital strategy, regularity of the menstrual cycle and the woman's age. Shorter average time required to conceive (TRC) and greater fecundability are associated with the observation of a waiting period, no coital strategy, and regular cycles. The woman's age at the time of unprotected intercourse shows no correlation with TRC; it does, however, show a correlation with fecundability, which increases with age to reach a maximum in the age group 25-29 and then decreases progressively. These 4 factors are also related to the mode of contraception. Those using hormonal contraception were more likely to observe a waiting period. However, they report a coital strategy less frequently, their usual cycle is more often irregular and they are younger than IUD or other method users. Results demonstrate a decreased fertility for couples after the pill and, to a lesser degree, after IUD use, but only among those who did not take other precautions (waiting period). For couples observing a waiting period, there is no decrease of fertility following the initial exposure to the risk of conception. This comparison of return of fertility following cessation of different contraceptive methods shows that raw data must be interpreted with caution. It is essential to take into account the possibility of delay in becoming exposed to the risk of conception while various confounding factors must also be considerd.     

Bayesian adaptive regression splines for hierarchical data

This article considers methodology for hierarchical functional data analysis, motivated by studies of reproductive hormone profiles in the menstrual cycle. Current methods standardize the cycle lengths and ignore the timing of ovulation within the cycle, both of which are biologically informative. Methods are needed that avoid standardization, while flexibly incorporating information on covariates and the timing of reference events, such as ovulation and onset of menses. In addition, it is necessary to account for within-woman dependency when data are collected for multiple cycles. We propose an approach based on a hierarchical generalization of Bayesian multivariate adaptive regression splines. Our formulation allows for an unknown set of basis functions characterizing the population-averaged and woman-specific trajectories in relation to covariates. A reversible jump Markov chain Monte Carlo algorithm is developed for posterior computation. Applying the methods to data from the North Carolina Early Pregnancy Study, we investigate differences in urinary progesterone profiles between conception and nonconception cycles.     

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.     

Semiparametric bayesian inference for multilevel repeated measurement data

We discuss inference for data with repeated measurements at multiple levels. The motivating example is data with blood counts from cancer patients undergoing multiple cycles of chemotherapy, with days nested within cycles. Some inference questions relate to repeated measurements over days within cycle, while other questions are concerned with the dependence across cycles. When the desired inference relates to both levels of repetition, it becomes important to reflect the data structure in the model. We develop a semiparametric Bayesian modeling approach, restricting attention to two levels of repeated measurements. For the top-level longitudinal sampling model we use random effects to introduce the desired dependence across repeated measurements. We use a nonparametric prior for the random effects distribution. Inference about dependence across second-level repetition is implemented by the clustering implied in the nonparametric random effects model. Practical use of the model requires that the posterior distribution on the latent random effects be reasonably precise.     

Natural Conception May Be an Acceptable Option in HIV-Serodiscordant Couples in Resource Limited Settings

Many HIV serodiscordant couples have a strong desire to have their own biological children. Natural conception may be the only choice in some resource limited settings but data about natural conception is limited. Here, we reported our findings of natural conception in HIV serodiscordant couples. Between January 2008 and June 2014, we retrospectively collected data on 91 HIV serodiscordant couples presenting to Beijing Youan Hospital with childbearing desires. HIV counseling, effective ART on HIV infected partners, pre-exposure prophylaxis (PrEP) and post-exposure prophylaxis (PEP) in negative female partners and timed intercourse were used to maximally reduce the risk of HIV transmission. Of the 91 HIV serodiscordant couples, 43 were positive in male partners and 48 were positive in female partners. There were 196 unprotected vaginal intercourses, 100 natural conception and 97 newborns. There were no cases of HIV seroconversion in uninfected sexual partners. Natural conception may be an acceptable option in HIV-serodiscordant couples in resource limited settings if HIV-positive individuals have undetectable viremia on HAART, combined with HIV counseling, PrEP, PEP and timed intercourse.     

Phytoplankton stoichiometry

Because phytoplankton live at the interface between the abiotic and the biotic compartments of ecosystems, they play an important role in coupling multiple nutrient cycles. The quantitative details of how these multiple nutrient cycles intersect is determined by phytoplankton stoichiometry. Here we review some classic work and recent advances on the determinants of phytoplankton stoichiometry and their role in determining ecosystem stoichiometry. First, we use a model of growth with flexible stoichiometry to reexamine Rhee and Goldman’s classic chemostat data. We also discuss a recent data compilation by Hall and colleagues that illustrates some limits to phytoplankton flexibility, and a model of physiological adaptation that can account for these results. Second, we use a model of resource allocation to determine the how the optimal nitrogen-to-phosphorus stoichiometry depends on the ecological conditions under which species grow and compete. Third, we discuss Redfield’s mechanism for the homeostasis of the oceans’ nitrogen-to-phosphorus stoichiometry and show its robustness to additional factors such as iron-limitation and temporal fluctuations. Finally, we suggest areas for future research.     

A direct optimization approach to hidden Markov modeling for single channel kinetics

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.     

Mixture models with multiple levels, with application to the analysis of multifactor gene expression data

Model-based clustering is a popular tool for summarizing high-dimensional data. With the number of high-throughput large-scale gene expression studies still on the rise, the need for effective data- summarizing tools has never been greater. By grouping genes according to a common experimental expression profile, we may gain new insight into the biological pathways that steer biological processes of interest. Clustering of gene profiles can also assist in assigning functions to genes that have not yet been functionally annotated. In this paper, we propose 2 model selection procedures for model-based clustering. Model selection in model-based clustering has to date focused on the identification of data dimensions that are relevant for clustering. However, in more complex data structures, with multiple experimental factors, such an approach does not provide easily interpreted clustering outcomes. We propose a mixture model with multiple levels, , that provides sparse representations both "within" and "between" cluster profiles. We explore various flexible "within-cluster" parameterizations and discuss how efficient parameterizations can greatly enhance the objective interpretability of the generated clusters. Moreover, we allow for a sparse "between-cluster" representation with a different number of clusters at different levels of an experimental factor of interest. This enhances interpretability of clusters generated in multiple-factor contexts. Interpretable cluster profiles can assist in detecting biologically relevant groups of genes that may be missed with less efficient parameterizations. We use our multilevel mixture model to mine a proliferating cell line expression data set for annotational context and regulatory motifs. We also investigate the performance of the multilevel clustering approach on several simulated data sets.     

Age and fertility: can women wait until their early thirties to try for a first birth?

Postponing the start of childbearing raises the question of fertility postponed versus fertility foregone. One of the limitations of previous studies of 'How late can you wait?' is that any observed decline in the probability of conception with age could be due to a decline in fecundability with age or due to a decline in coital frequency with age or due to both factors. Using data from a multinational longitudinal study conducted to determine the daily probability of conception among healthy subjects, a discrete-time event history model with long-term survivors (sterile population) is used to study the relationship between age and fecundability for childless women, while controlling for the pattern of intercourse within a menstrual cycle. The findings suggest that women can wait until their early thirties to try for a first birth, providing that they are not already sterile, as the magnitude of the decline in fecundability is very modest and of little practical importance.     

Synchronization of ovulation in beef herds: improved conception rate after and interrupted course of progesterone administration

Administration of 50 mg progesterone in 6 im injections every 48 hours was used to synchronize the estrous cycles of 6 herds of cows and/or heifers. Because of a previous finding that cows in the last 3rd of their estrous cycles when progesterone treatment is begun exhibit lower fertility, such cows were excluded by presynchronization of the herd 8 days prior to the start of treatment by 4 48-hourly injections of 50 mg progesterone. 69%-86% of the animals became pregnant in 4 of the 6 herds during the synchronized estrus, which compares favorably with the average 1st insemination conception figure without synchronization (approximately 70%). Previous conception rates observed averaged approximately 47%.     

A simple algorithm for averaging multiple cycles of waveforms having varying cycle periods

We present a simple method for averaging multiple cycles ofa waveform having cycles with varying periods. This averagingprocess preserves the morphology of the waveform by convertingfixed time per point data into fixed fraction of cycle data.The algorithms are implemented using the ASYST system, whichprovides efficient code for performing the computations andpermits straightforward integration of the data acquisitionand analysis tasks. An example is given of the results of theprogram in processing cyclical waveforms obtained in respiratoryresearch. Received on December 13, 1990; accepted on March 21, 1991     

Sexuality and reproduction

The relationship between coital rates, a quantitative index of sexuality, and the quality and quantity of reproduction is discussed in terms of sex ratio, delayed fertilization, frequency of intercourse, intercourse distribution within the menstrual cycle, acceleration of ovulation, and the quality of intercourse. Frequency of intercourse affects chances of conception, but its effect on twinning and sex ratio are unclear. Current research on reproduction offers little evidence of the effect of intercourse quality on reproduction in fully fecund individuals; research into the possible effect on reduced fecundity would be desirable.