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.

The beta distribution has two parameters, α and β. The probability density function is given by ) , ( . Note that Forbes et al. refer to the distribution parameters as ν and ω, rather than α and β; we have chosen (α, β) because this appears to be the more common notation.
For examples 1 to 3 in the main text, the distribution of intrinsic conception rates in the non-sterile population is assumed to be a beta distribution. For example 4 in the main text, the distribution of intrinsic conception rates in the non-sterile population is assumed to be a mixture of two beta distributions.
For additional examples S1.4, S1.5 and S1.6, described in Appendix B, the non-sterile population is drawn from a compressed beta distribution. This is defined as follows: the intrinsic conception rate y is given by y = kz, where z has a beta distribution and k is a positive constant, with k < 1. This means that the maximum possible value of the intrinsic conception rate within the population is k. For additional examples S1.4, S1.5 and S1.6 we take k = 0.6.

Appendix B: additional examples and further discussion
The question of how sensitive results are to the assumed form of the prior distribution of intrinsic conception rates can be explored by examining how the results change when other forms of distribution are considered. Examples S1.1 to S1.3 are qualitatively similar to examples 1 to 3 in the main text, except that the non-sterile population is described by a triangular distribution (see for example Potter [36] (reference 36 in the main text)) rather than a beta distribution. Examples S1.4 to S1.6 are again qualitatively similar to examples 1 to 3 in the main text, except that the nonsterile population is now described by a compressed beta distribution in which the maximum possible intrinsic conception rate is 0.6. Formally, we can say that the intrinsic conception rate is given by y = 0.6z, where the probability density function for z has a beta distribution. A simple rationale for using a compressed beta distribution rather than the non-compressed form used in the examples in the main text is that it may be plausible to assume that for even the most fertile couples there is an upper limit to their probability of conception per cycle which is considerably less than one. A compressed beta distribution would arise if, for a couple to conceive on a given cycle, two independent processes must both have a positive outcome, where the success rate among couples of the first process has a beta distribution, and the second process has fixed probability k of success. For examples S1.4 to S1.6 we assume that k = 0.6.

The additional examples: specification/parameters
Example S1.1: 5% of couples are assumed to be sterile, i.e. have a zero probability of conceiving without medical assistance; their intrinsic conception rate is zero. The remaining 95% are assumed to have intrinsic conception rates described by a triangular distribution, with intrinsic conception rates ranging from 0 to 0.8 and peak probability density at an intrinsic conception rate of 0.4. Figure S1.1(a) shows the probability density function for the intrinsic conception rate for the initial population, together with the posterior distributions conditional on non-conception after 6, 12, 18 and 24 cycles. Example S1.2: 5% of couples are assumed to have a zero probability of conceiving without medical assistance. The remaining 95% again are assumed to have intrinsic conception rates described by a triangular distribution, but with intrinsic conception rates now ranging from 0 to 0.5 and peak probability density at an intrinsic conception rate of 0.25. Just as example 2 in the main text described a less fertile population than that in example 1 in the main text, so example S1.2 describes a less fertile population than example S1.1: whilst 5% of the population are identically sterile in the two examples, for the upper 95% of the distribution all percentiles of the prior distribution of the intrinsic conception rate are lower for population represented by example S1.2 than for population represented by example S1.1. Figure S1.1(b) shows the probability density function for the intrinsic conception rate for the initial population, together with the posterior distributions conditional on non-conception after 6, 12, 18 and 24 cycles.
For examples S1.1 to S1.6, figure S1.1 shows the distribution of the intrinsic conception rate conditional on different numbers of cycles of attempted conception without success. Tables S1.1 to S1.6 show, for these examples, fertility metrics after 0, 1, 3,6,9,12,18,24 and 36 elapsed cycles without conception. These are the same fertility metrics given in the main text, for examples 1 to 4, in Tables 2 to 5. As in the main text, the last column of each table shows the cumulative probability of conception. Fuller versions of these tables are given in Appendix D: these full tables show the same metrics, together with the probability of conceiving in the following 24 cycles, for all values of number of cycles elapsed from 0 to 36.
The effect of changing the general form of the prior distribution, whilst choosing new parameters to maintain a reasonable fit to the relevant data, can be seen by comparing the relevant figures and tables corresponding to the same data. Specifically, Figures S1.1(a) and (d), and Tables S1.1 and S1.4, should be compared to Figure 2(a) and Table 2 in the original text; Figure S1.1(b) and (e), and Tables S1.2 and S1.5, should be compared to Figure 2(b) and Table 3 in the main text; and Figures S1.1(c) and (f) and Tables S1.3 and S1.6 should be compared to Figure 2(c) and Table 4 in the main text. The general picture from the figures is that, whilst the prior distribution clearly looks different for different forms of prior distribution fitted to the same data, the posterior distributions conditional on six or more cycles of non-conception look more similar.
We now consider more detailed comparisons between examples based on different forms of prior distribution but fitted to the same data.
-Comparison 1, for examples 1, S1.1 and S1.4, fitted to the data given in Gnoth et al. [1] (reference 1 in the main text): Figure S1.2 shows (a) how the probability of conception on the next cycle depends on the number of elapsed cycles, and (b) corresponding cumulative conception probabilities; and Table S1.7 shows how the number of cycles of nonconception which must elapse for the same metrics of infertility or subfertility considered in the main text to be reached (shown in Table 6 for the four examples considered in the main text). -Comparison 2, for examples 2, S1.2 and S1.5, fitted to the cumulative conception probabilities quoted by the UK's National Institute for Health and Clinical Excellence [33] (reference 25 in the main text) after 1, 2 and 3 years respectively: Figure S1.3 shows (a) how the probability of conception on the next cycle depends on the number of elapsed cycles, and (b) the corresponding cumulative conception probabilities; and Table S1.8 shows how the number of cycles of non-conception which must elapse for metrics of infertility or subfertility to be reached. -Comparison 3, for examples 2, S1.2 and S1.5, fitted to the conception rates reported by Wang et al. [2] (reference 2 in the main text): Figure S1.4 shows (a) how the probability of conception on the next cycle depends on the number of elapsed cycles, and (b) the corresponding cumulative conception probabilities; and Table S1.9 shows how the number of cycles of non-conception which must elapse for metrics of infertility or subfertility to be reached.  Figure S1.1. Probability distributions for the intrinsic conception rate for: (A) example S1.1, (B) example S1.2, (C) example S1.3, (D) example S1.4, (E) example S1.5, (F) example S1.6. In each case the plot labelled I shows the initial (prior) distribution. The other plots show the distribution conditional on non-conception after 6, 12, 18 and 24 cycles. A thick line represents the finite proportion of the population with an intrinsic conception rate of zero. The total area underneath each curve corresponds to the proportion of the remaining population which is not sterile: this decreases with the number of cycles as those becoming pregnant are lost from the population; consequently the proportion of the remaining population who are sterile increases (see first column of tables S1.1 to S1.6)   The general pattern from these comparisons is that different forms of prior distribution fitted to the same dataset tend to give similar patterns of conception over time, i.e. the choice of the parametric form of the prior distribution does not appear to greatly influence the predicted pattern of conception over time. However, there is important variation between these comparisons, with the variation in the patterns of how the probability of conception on the next cycle depends on the number of elapsed cycles being greatest for comparison 2 (contrast Figure S1.3(a) with Figure  S1.2(a) and Figure S1.4(a)). Comparison 2 also exhibits variation of up the three cycles in the number of cycles that must elapse for a given metric of subfertility to be reached (Table S1.8), whereas comparisons 1 and 3 exhibit variation of no more than one cycle (Tables S1.7 and S1.9). The difference may be because for comparison 2, comprising examples 2, S1.2 and S1.5, the first data point specifying the proportion of couples who have conceived occurs after 12 cycles. In contrast for comparison 1, comprising examples 1, S1.1 and S1.4, the Gnoth et al. data [1] ( Table  1) on which these examples are based gives four data points for the cumulative conception probability up to 12 cycles; and for comparison 3, comprising examples 3, S1.3 and S1.6, the Wang et al.
[2] data ( Table 2) on which these examples are based gives nine data points for the clinical pregnancy rate up to 12 cycles. Thus, for comparisons 1 and 3 there is more early data which enables the distribution to be "pinned down" more strongly, whereas for comparison 2 the relative lack of early data means that the fitted distribution is less constrained.   Table S1.9: Comparison of examples 3, S1.3 and S1.6: number of cycles of non-conception which must elapse for various metrics of infertility or subfertility to be reached.

Derivation of the formula for updating the distribution of the intrinsic conception rate
As in the main text, let f(y) be the initial (prior) distribution of the intrinsic conception rate for a couple who are trying to conceive. This applies to some population, or to a couple with a given set of objective characteristics. Let f n (y) be the distribution conditional on n cycles of non-conception. So, for example, f 3 (y) is the distribution conditional on three cycles of non-conception. Note that we can write f(y) = f 0 (y), i.e. the prior distribution is the distribution conditional on zero cycles of nonconception.
Let X n denote the event that the couple have not conceived within n cycles. Applying Bayes' theorem for a continuous distribution as given in Lindley [30], the distribution of y conditional on X n is given by In this expression, p(X n |y) denotes the probability of non-conception within n cycles for a given value of y. This is given by 1 -P(y,n), where P(y,n) is given in equation (2) of the main text. Substituting from equation (2) Substituting from equation (3) where, as stated in the main text, s(n) is the probability that a couple whose intrinsic conception rate is drawn at random from the distribution f(y) will not conceive within n cycles. We give this result in the main text as equation (4).
It is also useful to have an expression for the probability that a couple drawn from some distribution of the intrinsic conception rate (which is not necessarily the prior distribution) will conceive within m months. Suppose the couple have had n cycles of non-conception where n ≥ 0. Then the conditional distribution for the couple's intrinsic conception rate is f n (y). (Note that if n = 0 this reduces to f 0 (y), which is the same as the prior distribution f(y).) The probability that such a couple will conceive within the next m cycles is given by

Computational method
A program has been written in C under the unix operating system. Given any prior distribution for the intrinsic conception rate, it will numerically compute the following: the probability that a couple drawn at random from the prior distribution will have failed to conceive within any specified number of cycles. This is s(n); -the posterior distribution of the intrinsic conception rate conditional on any specified number of cycles of non-conception. This is f n (y); -any specified percentiles of the distribution f n (y); -the probability that a couple drawn at random from the distribution f n (y) will conceive within a specified number m of cycles. This is therefore the probability that a couple who have failed to conceive within n cycles will conceive within m additional cycles.
The probability distribution of the intrinsic conception rate is stored in the following form: a number p 0 specifies the proportion of the distribution which corresponds to sterility (y = 0), and the remainder of the distribution, i.e. the continuous part, is stored as an array of 10001 numbers. These numbers represent values of y from 0.0 to 1.0 inclusive, in steps of 0.0001.
The program begins with the following inputs: the proportion of the prior distribution which is initially sterile. We can label this p 0 (0). Computation of a posterior distribution. The prior distribution of the intrinsic conception rate is stored as an array f 0 [i], where 0 ≤ i ≤ 10000, and an initial proportion that are sterile p 0 (0). From these inputs the program computes the posterior distribution of the intrinsic conception rate conditional on n cycles of non-conception. This involves computing a new proportion p 0 (n) corresponding to sterility and a new array f n [i] representing the continuous part of the distribution, by numerically applying expression (4) in the main text. This function also returns the probability of not conceiving within n cycles.
Probability of conception within m cycles. This is computed for a couple drawn from some general distribution of the intrinsic conception rate. The distribution used as an input to this function is the distribution conditional on n cycles of non-conception where n ≥ 0 (and n = 0 corresponds to the prior distribution). The specific inputs are the proportion p 0 (n) of the distribution corresponding to sterility and the array f n [i], where 0 ≤ i ≤ 10000, corresponding to the continuous part of the distribution. This function numerically evaluates expression (C3).

Methods for example 5
Example 5 differs from the other examples in that couples are subject to a declining intrinsic conception rate during the period of attempted conception.
The model is adapted from Leridon [24]. Considering first a single couple, the couple's intrinsic conception rate is assumed to decline linearly over 12.5 years until sterility occurs; before this decline begins, it is at a constant (peak) value. There is variation between couples: they vary in (i) their peak intrinsic conception rate, and (ii) the age at which sterility occurs. The distribution of peak intrinsic conception rate among couples is assumed to be a beta distribution with α = 3 and β = 10 (see Leridon & Slama [34]). The distribution among couples of the female age at which sterility occurs in given in Leridon [24], Table 2. Peak intrinsic conception rate and age of sterility are assumed to be independent.
The population is divided into 409 cohorts, according to the female age in months at which sterility occurs. The first cohort is already sterile at a female age of 25 years. The last cohort becomes sterile at a female age of 59 years. More generally, the nth cohort becomes sterile at a female age of 25 years + (n -1) months. (So, for example, the 100 th cohort becomes sterile at a female age of 25 years + 99 months, i.e. 33 years and 3 months.) Table 2 in Leridon [24] gives the proportion of the population that are sterile as a function of completed female years. The proportions that become sterile on a monthly basis are estimated using linear interpolation within each year. For example, Leridon [24], Table 2, gives the proportion of couples who are sterile at ages 25 and 26 at 10 and 11 per 1000 respectively. The interpolation we apply means that we assume that 10 + 1/12 per 1000 are sterile at 25 years and one month, 10 + 2/12 per 1000 at 25 years and two months, and so on.
A separate computation is carried out for each cohort, with the distribution of the intrinsic conception rate tracked numerically over successive cycles of attempted conception. The distribution is stored as an array of 10001 numbers, exactly as for examples 1 to 4 (although in those examples the distribution represents the whole population). After each cycle, the distribution is first updated, applying expression [4] in the main text: this calculates the proportion at each value of the intrinsic conception rate that leaves the population because a pregnancy has been achieved. Then, if the cohort is within 12.5 years of the age of sterility, intrinsic conception rates for the cohort as a whole are reduced by the appropriate proportion: this is 1/150 if the cohort goes from 150 to 149 months to sterility (i.e. the completed month takes the cohort to 149 months before sterility), 1/149 if the cohort goes from 149 to 148 months before sterility, and so on. This involves computationally compressing the distribution. For those cohorts which are below their peak fertility but not yet sterile at the start of the attempted conception process, intrinsic conception rates are appropriately reduced before the start of the computational process tracking a cohort over the duration of attempted conception. Some of these latter cohorts reach their age of sterility and therefore become sterile during the period that they are tracked.
A population with a given female age comprises a mixture of all the cohorts. Conception characteristics for such a population are obtained by appropriately combining the results from all cohorts. The population distribution of the intrinsic conception rate for a given number of cycles of non-conception is given by a superposition (i.e. a weighted sum) of the distributions for all the cohorts. The probability of conception in the next month, or more generally the next m months, is given by a weighted mean of the values of this probability over all cohorts. The weight applied to each cohort is given by the proportion of the population that constitutes that cohort.
We have also implemented a modified version of the program, in which reproductive ageing during the period that couples are attempting to conceive is switched off. The purpose was to explore the difference that reproductive ageing during the conception period makes to the calculated number of cycles of attempted conception before different metrics of subfertility are reached (see Table 12 in the main text).

Appendix D: Full tables of fertility metrics
These are fuller versions of tables giving fertility metrics. They give metrics for all values of number of months elapsed from 0 to 24, and include as an additional metric the probability of conception in the next 24 cycles. The tables concerned are: Tables 2 to 5 and 7 to 10 in the main text, for  examples 1 to 5, and Tables S1.1 to Table S1.6 for the additional examples S1.1 to S1.6.