TTP is defined as the number of menstrual cycles it takes for a couple, having regular sexual intercourse without the use of contraception, to conceive. It has been used as a measure to study the effect of various exposures on fecundity, see, for instance, Baird and others (1986), Law and others (2005), and Buck Louis and others (2009). Weinberg and Gladen (1986) first proposed a beta-geometric model for the probability distribution of the TTP. Subsequently, Scheike and Jensen (1997) proposed a discrete survival model for the hazard of conception in a cycle j given cycle level covariates , aswhere denotes a subject-specific random effect. A particular feature of this model is that the hazard is related linearly to the covariates when transformed by a complementary log–log function. This is precisely the proportional hazards model with random effects for grouped data. The Scheike and Jensen (1997) model allows for the inclusion of cycle-varying covariates but cannot incorporate the effects of the day-level covariates, typically collected in prospective pregnancy studies. Due to the inherent biases and recall errors in retrospectively ascertained TTPs [see, for instance, Weinberg and others, 1994a; Cooney and others, 2009), the current emphasis is on prospectively designed studies where extensive information is collected from a couple on various time scales including the woman level (e.g., previous reproductive history), cycle level (e.g., biomarkers for stress), and daily level (e.g., intercourse behavior, various reproductive hormonal levels, smoking, caffeine and alcohol consumption, and other lifestyle factors). Another limitation of using (1.1) is that it does not account for “immaculate conception,” that is, the hazard for conception in a cycle should be zero if no intercourse occurs.

A question of considerable interest, in the presence of detailed daily level information, is the probability of conception due to a single act of intercourse on a given day relative to ovulation. This is known as the day-specific conception probability. This enables one to determine the “fertile window,” a quantity of considerable interest to couples planning or trying to avoid pregnancy. The days outside of the “fertile window” have a small chance of pregnancy. The estimation of day-specific conception probabilities is complicated by the fact that it is unusual for a sexually active couple to have a single act of intercourse during the fertile window. Additionally, only a proxy for ovulation is available as the gold standard requires direct visualization of the ovaries via laparoscopic or ultrasonographic techniques that are available only for women seeking medical or infertility treatment such as in vitro fertilization or intracytoplasmic sperm injection. Consequently, differing precision exists in such identification.

The original seminal work of Barrett and Marshall (1969) proposedwhere denotes the indicator vector of intercourse in the fertile window and is the day-specific probability of conception. The 2 main assumptions underlying this model are as follows.

Since then, the literature has responded to the second assumption (A2) to a great extent starting with the work of Schwartz and others (1980) who noted that conception is not only dependent on the timing of the intercourse but also on several biological factors such as the penetrability of the mucus, the capacity of the ovum to be fertilized and the receptivity of the uterine lining for implantation. This led to the modelwhere ω is the probability of a cycle being viable. Modifications of this model include the Weinberg and others (1994c) model that accommodates cycle-specific covariates; the Zhou and Weinberg (1996) model that incorporates day-specific covariates; the Zhou and others (1996) model that accounts for the within-woman dependency; the Dunson and Zhou (2000) model that incorporates both within-woman dependency and a sterile fraction. Zhou and Weinberg (1996) proposed an expectation–maximization (EM) algorithm for the maximum likelihood estimation of parameters and used a sandwich estimator of the variance to adjust for within-woman dependency. However, their estimates of day-specific probabilities are biased downward, and the sandwich estimator is not valid because less fertile women contribute more cycles to the data. This class of models culminated in the work of Dunson and Stanford (2005), who note the weak identifiability of the cycle viability term ω. Consequently, they proposed a hierarchical modelwhere denotes a couple-specific random effect. Note that in this model, the effects of covariates (possibly day-level) on the probability of conception in a menstrual cycle is mediated only through the daily level probabilities of conception.

An assumption that is common to all day-specific models is (A1), namely that the ejaculates of sperm introduced by the intercourse acts on different days compete with each other independently in an attempt to fertilize the ovum. In particular, the independence assumption necessitates the relationwhere denotes the event that a sperm from kth intercourse act fertilizes the ovum, and denotes the indicator of intercourse occurring on the kth day of the cycle. This may not be a reasonable biological assumption. Each intercourse act in the fertile window introduces a fresh ejaculate of sperm in the reproductive tract that can potentially fertilize the ovum. It is well known in the clinical literature (van Duijn and Freund, 1971; Tyler and others, 1985; Levin and others, 1986; Carlsen and others, 2004) that tremendous inter- and intravariability exists in sperm quality ranging from azoospermic (absence of sperm) to high-quality sperm. In fact, studies have shown that there is a 29.2% reduction in sperm concentration when ejaculatory frequency went from one to two episodes during a 7-day period prior to semen collection following 2 days of abstinence. For 3 or more ejaculations, the level was reduced by about 41% as compared to the one ejaculate group (Carlsen and others, 2004). Consequently, this implies that the number of sperm available for fertilization does not just vary from day to day (assuming one ejaculate per day) but also depends on how many intercourse acts occurred previously (that is, on the previous 's). Sperm concentration and number per ejaculate may affect the probability of conception. These data suggest that (1.3) may not be reasonable. Although we focused on sperm concentration for the purpose of illustration, a host of other semen characteristics, such as motility and morphology also impact conception probability and are dependent upon previous intercourse pattern (Carlsen and others, 2004), supporting the need to relax the independence assumption.

Royston and Ferreira (1999) proposed an alternative approach that does not require the independence assumption in that they assume that in cycles with multiple intercourse acts only the most fertile one contributes to the probability of conception. Although this may be a reasonable approximation in some cases, sperm introduced on less optimal days in the fertile window can also compete to fertilize the ovum and should not be ignored, as normal appearing sperm may survive up to 5 or 6 days in the female reproductive tract.

Let be the event that the ovum is fertilized in the jth cycle, and be the event that a sperm from the kth intercourse fertilizes the ovum. Note that the fertilization of an ovum normally requires a sperm originating from one of the potential intercourse acts that the couple may have had in the fertile window. Then (disjoint union of events). So,Under the independence assumption (A1) of the events ,To avoid requiring the independence assumption (A1), we mimic the mixing of ejaculates of sperm from different intercourse acts in the reproductive tract of a woman by using an arbitrary linear combination of intercourse acts in the fertile window. In other words, we weigh separately the intercourse acts on different days so as to discriminate between an intercourse act occurring on day k with that occurring on day in the fertile window. These weights are estimated based on the observed sample. Furthermore, we propose to directly model conception in cycle j, given that conception has not occurred so far, byObserve that is the hazard for conception in cycle j. In other words, we propose the following discrete survival model for TTP:We assume that the random effects, , follow a Gamma distribution with mean 1 and variance η. Observe that the proposed model corrects for “immaculate conception,” that is, the hazard for conception in a cycle is zero if the couple does not have any intercourse in the fertile window of that cycle. The regression coefficients capture the baseline kth day effect of intercourse on the probability of conception in cycle j. The cycle-varying parameter denotes the cycle-specific baseline, a quantity of considerable interest (Weinberg and others, 1994b). The regression coefficients β capture the effect of the covariates . Observe that if a couple had intercourse only on day k, that is, , then, under the proposed model, the probability of conception in cycle j is given byThis is the probability of conception in cycle j if the couple had intercourse only on a specific day in the fertile window of cycle j. This is analogous to the day-specific probabilities of conception in the day-specific models for conception. Consequently, we refer to it as the kth day-specific conditional hazard of conception. Also, note that the effect of covariates on can be viewed as additive effects on complementary log–log scale.

Furthermore, we can also estimate the effect of covariates directly on the probability for conception in cycle j as follows:Consequently, the probability mass function can be expressed asThis yields the survival function for as follows:Consequently, using the proposed model (2.1), one can model the day-level hazard for conception via (2.2) as well as model the effects of covariates on the survival function (2.3) in the same model. Also interesting to note is the constant ratio of log survival functions for the time-independent covariates (assuming equal time-dependent covariates).

Similar to Scheike and Jensen (1997), the marginal forms (with respect to random effect ) for the probability of conception in cycle j and the hazard for conception in cycle j can be expressed asUnder the assumption of a Gamma distribution for with mean 1 and variance η, the hazard rate is given byThus, the marginal kth day-specific conditional probability of conception in cycle j is given by

Most prospective pregnancy studies design data collection to include the use of daily diaries to ascertain daily level covariates such as menstruation and sexual intercourse and, possibly, factors purported to impact couple fecundity (e.g., cigarette smoking, alcohol, and caffeine consumption). One can also estimate the effect of such covariates in the model by viewing them as a vector , the number of days L of interest need not be the same as the fertile window. One can incorporate the daily-level covariate into the model as follows:The observed likelihood for the discrete survival model (2.1) is similar to that of binary data with probability of success However, this is different from the Dunson and Stanford (2005) approach, where they pose the problem in terms of binary outcomes and model the covariate effects only through day-specific probabilities (1.2) coupled with the independence assumption (A1). Additionally, one cannot incorporate the cycle-varying intercept in (1.2) due to lack of identifiability with day-varying baseline intercept, As mentioned previously, the effect may be important in these prospective pregnancy studies.

We generated the data as follows: for subject i, the frailty variable was generated from a gamma distribution with mean 1 and variance η, the day-level intercourse behavior were generated such that with probability of one as for all , , and the covariates were generated such that variable with probability of one as each and . Subjects who had not experienced an event at were censored. Note that in this setup, we have incorporated approximately 8% zero-risk sets which is close to the percentage encountered in the real data analysis in the next section.

Tables 1 and ​and22 present the performance of the likelihood-based estimators accounting for the zero-risk sets as introduced in (2.1) or ignoring its effect. We made the comparison at various sample sizes ranging from to even motivated by some recently completed prospective pregnancy studies, for example, the Life Study, the Oxford Conception Study. The censoring percentage was also varied from 10% to 30%. The results presented are based on 1000 replicates. The column bias refers to the average of the difference between the estimated value of the parameter and the true value, the Avg(SE) refers to the average of the asymptotic standard deviation (calculated using the estimated observed fisher information), SE(est) refers to the standard deviation of the estimates and CP refers to the coverage probability for a 95% confidence interval.

Observe that the bias of the estimates of the regression coefficients for the zero-risk set corrected model are reasonably small even for small sample size, and their performance improves considerably as n increases. Furthermore, the estimated standard deviation gets closer to the sampling standard deviation with the sample size. The coverage probability for the estimates are close to the nominal 95% and become closer as the sample size increases. However, the considerable bias in estimates of the 's when the zero-risk sets are ignored causes the coverage probabilities to be much lower than the nominal value. In addition, the estimates of β and η have higher bias and standard error when zero-risk sets are ignored for both settings and all n. Overall, the estimates for the corrected model perform very well even for small sample size. We further observe that the coverage probability for the 95% confidence interval for the variance of the random effect was lower than that for other parameters for However, it becomes much closer to the nominal level of 95% as the sample size increases from to and

In Figure 1, we present the conditional day-specific probabilities of conception in cycle j for a non-smoking woman aged 30 years (the mean), and with parity = 1 versus 0. Note that the Figure 1(a) indicates the probability of conception in the first cycle given that the couple had intercourse on a specific day () in the fertile window. Figure 1(b) indicates the probability of conception in the second cycle given that the couple had intercourse only on a specific day in the fertile window and had not conceived in the first cycle, (c) and (d) denote these probabilities given that the couple had not conceived by the second and third cycles, respectively. These plots do indicate that parity significantly increases the conception probabilities at the daily-level. Also, the graphs do indicate considerable measurement error associated with identifying the day of ovulation by the Ogino-Knauss method, which is consistent with findings of using other proxies of ovulation with similar measurement errors. This is acknowledged in the reproductive literature, see Lynch and others (2006) for a summary.

Figure 2 displays the effect of parity on the TTP survival distribution for a nonsmoking woman aged 30 years with parity = 1 versus 0 and having intercourse on days −4 through 2. Here, it is clear that the parity significantly reduces the TTP, where the estimated median TTP reduces from 11 to 3 cycles.