The biology of human reproduction is complex and varies considerably between individuals. Attempts at estimating a set of simplified components date back to the 1950s–1960s with the work of Louis Henry (Henry, 1953, 1961, 1964), among others. These methods and estimates were later refined by scholars like Henri Leridon and John Bongaarts (Bongaarts, 1978; Leridon, 1977), and have since been implemented in various microsimulation models of human reproduction (Bongaarts, 1977; Ciganda and Todd, 2022; Granholm et al., 2026; Habbema et al., 2015, 2015; Leridon, 2004). For the purposes of our analysis, we model the following components: fecundability, age at sterility, foetal survival and the non-susceptible period.

In short, the procedure is as follows: we generate a distribution of the ages at which agents first attempt to conceive, run the simulation model and then verify how well the simulated distribution of the ages at first birth matches the actual distribution of the ages at first birth from the administrative data. Based on the similarity scores between the simulations and the actual distribution, a posterior distribution is approximated. We do not know the shape of the distribution of the age at which women first attempt to conceive and test the fit of five parametric distributions: the gamma distribution, the lognormal distribution, the Gumbel distribution, the logistic distribution and the normal distribution. All these distributions can be parametrised by a value for the mean and the variance, and we therefore specify prior distributions for the mean and the variance.

The implementation of the sequential ABC method that we use requires uniform prior distributions, as the first step applies Latin hybercube sampling, meaning that the full range of combinations of prior values are considered with better coverage of the entire parameter space. For the prior for the mean, we have chosen a minimum that is five years below the average age at first birth for that cohort. Prior research suggests that about half of couples conceive a child within six months of starting to attempt to conceive (Juul et al., 2000; Slama et al., 2012; Wesselink et al., 2017). Assuming nine months of gestation for live births, this means that half of couples take about 15 months or fewer to have a child. Our minimum for the prior is well below this duration. As an upper boundary, we have chosen a maximum that is two years above the true average age at birth. While highly unlikely, it is not impossible that the average age at first attempting to conceive is higher than the average age at first birth when those trying to conceive a child at high ages do not succeed and those trying to conceive a child at younger ages succeed at a rapid pace (see also Figure 5). For the prior for the variance, we have selected minimum and maximum values that are nine points below and above the true variance in the age at first birth for each cohort. This range is chosen because it lies well beyond the observed minimum and maximum ages at first birth, and because the variance in the age at first attempt to conceive is unlikely to differ substantially from that of the distribution of the age at first birth. Sequential ABC occurs in multiple steps through which the evolution of the posterior distribution can be traced. The outer boundaries of the posterior distributions are not close to the minimum or maximum of the limits set on our prior distributions in any of the models, which suggests that our chosen limits are not too narrow. Selected repetitions of the models with an even wider prior distribution further show that the limits we set are appropriate, while making the range narrower leads to similar posterior distributions (see Section S6.3, file “1_supplement.html”).

Here, we will walk through an example in which the number of prior values drawn is 2000 (e.g. 2000 sets of prior values for the mean and the variance) and the number of agents in the model is 5000. In the first step of sequential ABC, the randomly selected pairs of 2000 prior values are used to run 2000 simulation models with 5000 agents. These 2000 simulations lead to 2000 distance scores between the output of the simulation model (distribution of the ages at first birth) and the true distribution of the age at first birth. As a distance score, we have chosen the Chi-square distance (see also Lenormand et al., 2013). Distances are normalised by first dividing them by the standard deviation in distance scores from the original 2000 samples (at the first step), and then by calculating the Euclidean distance of these distance measures to zero (zero distance means that there is a perfect match between the simulated and the observed age distribution). The parameter alpha then determines how many of these 2000 prior values are kept. As recommended by Lenormand et al (2013), we have chosen alpha to equal 0.5, which means that 1000 (combinations of mean and variance) prior values are kept. These prior values are assigned weights, depending on their distance scores, generating a proposal distribution. In the second step, another 1000 prior values are randomly drawn, but these values are now drawn from the proposal distribution. To arrive at novel values for the prior, randomness is added corresponding to twice the variance of the intermediate posterior distribution in the previous step (i.e. a Gaussian kernel). For these new samples, 1000 distance scores are calculated. New samples are “accepted” if they have lower distance scores than the worst distance score in the previous round. These new samples are added to the previously accepted 1000 samples, and again, the best 1000 samples are kept for the next round and new weights are calculated.

The algorithm stops when only a small proportion of new samples have lower distance scores than the worst score from the last proposal distribution. As recommended (Lenormand et al., 2013), we set this proportion to 5%, which means that the model stops if less than 5% of the new samples fall below this worst score. Because only distance scores that are lower than the previously worst score are added, and then only the best 1000 samples are selected, the worst distance score gets progressively lower with each step, and it becomes increasingly difficult to find new samples with distance scores that fall below this threshold. An alternative way of stating this is that the algorithm stops when including the very few new samples that fall below the threshold would hardly change the posterior distribution. The final step retains 1000 sets of priors with weights, which is our final posterior distribution (see Sections S6.5 and S7.2, file “1_supplement.html”, in which we run through an example).

Given that we have to repeat these analyses for all 21 cohorts, and that each analysis is computationally expensive, smaller sample sizes (of either agents or priors drawn) that are large enough to avoid compromising the quality of the posterior distribution are preferred. Two adjustments can be made that co-depend on one another: changing the number of agents in the simulation model (a lower number of agents will lead to more variance in the simulated age distributions, and thus to more variation in the distance scores) and changing the number of priors sampled. Running a model for one of the birth cohorts with different sets of numbers of agents and priors drawn reveals that the number of agents has a larger impact on the width of the posterior distribution than the number of priors drawn. A combination of 50,000 agents per simulation and 1000 initially drawn priors leads to satisfactory results (see Sections S6.1 and S6.2, file “1_supplement.html”), and these numbers are chosen for all the models we present here.

To examine to what extent the choice of priors has an impact on the posterior distributions, we also vary the range of our priors and show that wider or narrower prior ranges lead to similar results (see Section S6.3, file “1_supplement.html”). We further examine whether sequential ABC is able to retrieve the correct means and variances from known (simulated) distributions, and whether it is able to find the correct parametric distribution from which the data are generated. Using simulated distributions of the age at first attempt to conceive based on different parametric distributions, and using these to simulate the ages at first birth, we show that sequential ABC based on the age at first birth retrieves the correct means and variances of the age at first attempt distribution, and finds the correct parametric distribution from which the data are generated (see Section S6.6, file “1_supplement.html”).

To evaluate the consequences of further postponement, we run simulation models in which we vary the average age at first attempt to conceive from 20 to 35, simulating the age at first birth outcomes for 50,000 agents for each age. We have to make assumptions about the variance associated with those ages that we base on the association between the estimated average age at first attempt to conceive and the estimated variance across birth cohorts (see Section S8.3.1, file “1_supplement.html”). For the average ages at first attempt to conceive of 27 and 28 that were observed for the 1962 and 1967 cohorts, respectively (see Figure 3), we use the estimates of variance for those cohorts (22.4 and 22.2; see Section S8.3.1, file “1_supplement.html”). For the average ages at first attempt to conceive below 27, we use the variance that is estimated for 1960 (22.0), when the lowest average age at first attempt is observed. For the starting ages above 28, we use the estimated variance from 1974 (20.3), as it is in this cohort that the highest average age at first attempt is observed (Figure 3). This latter estimate for the variance is likely an overestimate for situations with higher average ages at first attempt to conceive, because a higher average age will “compress” the age range, leading to lower variance due to limits on the reproductive lifespan. Because there is no way of establishing the fit to the observed distributions in our hypothetical examples, we have chosen the gamma distribution to represent the age at first attempt to conceive. Results using different distributions are remarkably similar (see Section S8.3.4.1.1, file “1_supplement.html”).

Given the importance of fecundability in our model and the difficulty of assessing this quantity, we compare the durations of attempts to conceive a child through simulations based on our biological parameters to the findings of a series of empirical studies. In these studies, such durations are presented for people trying to have their first child, and for whom (aggregate) information is available on their ages at first attempt to conceive (Bonde, Ernst, et al., 1998; Bonde, Hjollund, et al., 1998; Eisenberg et al., 2021; Karmaus et al., 1999; Schwartz and Mayaux, 1982; Slama et al., 2012). This comparison shows that our simulation model captures well the age-related decline in pregnancy rates observed in these studies, although our agents have higher chances of conceiving (see Section S6.8, file “1_supplement.html”). Differences across these studies and differences between these studies and our study may be attributed to variation in methodological approaches, e.g. lower chances of fertilisation with artificial insemination than with natural conception (in the case of Schwartz and Mayaux (1982); see also Bongaarts (1982)), inappropriate censoring, differences in the ability of couples to detect a pregnancy in the first month or less consistency in attempts to conceive (actual couples, as opposed to agents, may not have had consistent opportunities for intercourse for months on end). Nonetheless, given that our agents have slightly increased chances of conception facilitated by uninterrupted attempts to conceive, which may be harder to maintain in real life, our estimates of the ages at first attempt to conceive a child may be on the high side (i.e. actual couples may need longer to conceive and might therefore be younger at their first attempt to conceive; see Section S6.8.8, file “1_supplement.html” for further discussion). To further examine the effect of fecundability, a key parameter in our model, we run sensitivity analyses in which we either increase or decrease fecundability by 20%. These analyses show that 20% higher fecundability would, on average across cohorts, increase the estimated age at first attempt to conceive by 1.3 months, whereas 20% lower fecundability would decrease it by 2.5 months (see Section S6.7, file “1_supplement.html”).

All models have been programmed in R (R Core Team, 2018), with the most important packages used being tidyverse (Wickham, 2017) and EasyABC (Jabot et al., 2013). See https://doi.org/10.34894/HIPZHY for all code, walkthroughs and supplementary materials.

The age at first birth increases steadily between the 1960 and 1970 cohorts, but levels off after 1970 (Figures 2 and 3). Women born in 1970 have their first child on average at around 27.7 years, and this age increases to about 29.6 years for women born in 1975, after which it decreases slightly. Some of the levelling off is due to the fact that women in the later cohorts may still have a child at higher ages, which would raise the age at first birth (2022 is the last observed year, when women born in 1980 were 42 years old). The variance in the age at first birth decreases in the younger cohorts, and the age distributions go from relatively broad, right-skewed distributions to narrower, symmetrical distributions in the younger cohorts (Figure 2).

The estimated average age at first attempt to conceive closely follows the trend of the age at first birth, shifted by about 11.7 months (Figure 3), and increasing from about 26.7 in 1960 to 28.4 in 1980. The models are able to reproduce the observed age at first birth, as the estimated average age at first birth is on average about 0.3 months below or above the actual average age at first birth. Figure 4 shows more detailed outcomes from our models for three birth cohorts with respect to the posterior distribution for the mean (panels a) and the variance (panels b), the resulting distribution of the age at first attempt to conceive (panels c) and the fit between the observed and the simulated distributions of the age at first birth (panels d). The distribution of the age at first birth arising from the simulations fits the true age distributions reasonably well (see Figure 4d and Section S8.1.3.4, file “1_supplement.html” for all years of birth). The estimated average age at first attempt to conceive includes both women who will have a child (which results in the distribution of the age at first birth) and women who will not have a child (which results in the rates of childlessness). Across the birth cohorts, agents who end up having children start attempting to conceive slightly earlier than the average, by about 2.6 months (Figure 3). This means that it takes about 14.2 months on average to conceive a child.

Similar to the trend in the age at first attempt to conceive, the rates of estimated childlessness among agents also increase, and then level off, although the differences between cohorts are minimal (Figure 3). Across the cohorts, the average rate of estimated childlessness is about 4%, which is much lower than the observed population rates of about 20%, reflecting both voluntary and involuntary childlessness (Figure 2).

To better understand the effects of the age at first attempt to conceive on the age at first birth and childlessness, we run models in which we vary the mean age at first attempt from 20 to 35 (see Figure 5). A striking pattern arises whereby the mean age at starting to attempt to conceive gets closer to the average age at first birth with increasing ages at first attempt to conceive (Figure 5a). At an average age at first attempt of 35, the average age at first birth is only six months later. This pattern can be explained by shifts in the age composition of agents and the marked increase in childlessness (Figure 5a; grey boxes). As the average age at first attempt to conceive rises, a growing share of agents begin trying to conceive at relatively advanced ages, when the probability of remaining childless is high. For instance, at an average age at first attempt of 35, over 15% of the agents start attempting to conceive at age 40 or higher, and these agents have childlessness rates close to 50% (see Section S8.3.5, file “1_supplement.html”). These childless agents who start late therefore raise the average age at first attempt to conceive, but do not have an impact on the average age at first birth. In this scenario, agents who begin attempting to conceive at younger ages are much more likely to have children and thus contribute more to the average age at first birth.

Examining only agents who have children, we observe that the difference between the average age at first attempt to conceive and the age at first birth is on average 15 months and varies little regardless of the average age at first attempt to conceive (from 14 months at age 20 to 16 months at age 35; Figure 5b). Given the decline in fecundability and foetal survival with age, we would not expect that the population of agents who first attempt to conceive at older reproductive ages (on average) need a similar amount of time to conceive a child as the population of agents who start at much younger ages (on average). In addition to the changes in the age composition and the rates of childlessness, another reason for this finding is a selection effect on fecundability. An agent with low fecundability at age 15, who has a low monthly probability of getting pregnant, will nonetheless have a high probability of giving birth to a child if the agent starts trying to conceive at a young age (Figure 5c). Although the monthly probability is low, the chance of eventually conceiving a child is high given many attempts. The fecundability of all agents declines with age, which means that agents with low fecundability at younger ages may not be able to conceive if they postpone their first attempt to conceive to later reproductive years. Agents with high fecundability have high probabilities of conceiving a child, even with increasing age (Figure 5c). This means that if the ages at first attempt to conceive are higher, there will be a stronger bias towards the agents with higher fecundability being the ones who go on to have a child than would be the case if the ages at first attempt to conceive were lower (see Sections S8.3.5 and S8.4.3, file “1_supplement.html”). The proposed mechanism is thus the following: when an entire birth cohort of agents has lower ages at first attempt to conceive, many agents with low fecundability will still have children, although it will take them longer on average than agents with high fecundability. These agents raise the average time to conception. In contrast, in populations in which the average age at first attempt to conceive is higher, agents with low fecundability will have low probabilities of having children, and their long durations of attempting to have children will not be included in the cohort average time to conception and age at first birth. Thus, the duration of the time to conception is shaped by two opposing forces as age increases. On the one hand, advanced reproductive age is associated with longer times to conception due to age-related declines in fecundability and foetal survival. On the other hand, durations may appear shorter at higher ages because agents who are still able to conceive at those ages are a selected group with relatively high fecundity.

In the above model, only the average age at first attempt to conceive and the variation in that age is fixed, meaning that in the population of agents there is substantial variation in the age at starting to attempt to conceive. To understand the effect of age on the probability of conceiving a child and the duration of attempting to conceive, we also run simulations in which we fix the ages at first attempt to conceive to the same value for all agents. Thus, we generate a sample of 50,000 agents with randomly assigned biological features and then fix the ages at first attempt to conceive for that same group of agents in one-year increments from age 20 to age 40 (see Section S8.4, file “1_supplement.html”). Here, we present results for the starting ages 20, 25, 30, 35 and 40. Between the ages of 20 and 30 there are some minor differences in outcomes (Figure 6): compared to 30-year-old agents, 20-year-old agents have a slightly higher chance of having a successful conception that leads to a birth in the first month of trying and take on average one month less to conceive, with 1% remaining childless at age 50, instead of 4%. If we compare 30-year-olds to older agents, we find that the percentage who remain childless at age 50 increases from 4% in 30-year-old agents to 9% in 35-year-old agents and then to 30% in 40-year-old agents. The increase in the average time until conception is less pronounced, with 40-year-olds taking on average four months longer to conceive than 30-year-olds. The differences across the ages at first attempt to conceive become much more pronounced when examining the Kaplan–Meier median survival – the point at which half of the agents have had a child. This measure takes into account the conception attempts of agents who ultimately do not have children. The median time to conception is three months for ages at first attempt to conceive up to age 30 and increases to five months at age 35 and to 13 months at age 40.