Early embryo mortality in natural human reproduction: What the data say

1. Where have all the conceptions gone?

In 1975, a short hypothesis published in The Lancet entitled “Where Have All The Conceptions Gone?” concluded that 78% of all conceptions were lost before birth³⁴. It has been widely cited by both scientists^{9,25,27,28,41} and non-scientists^42,43 alike. Conceptions among married women aged 20–29 in England and Wales in 1971 were estimated and compared to infants born in the same period. In this analysis (Table 1) there are reliable values, e.g., census data, and simple arithmetical calculations. However, speculative values are necessary to perform the calculations. Three are biological: (1) fertilisation rate following unprotected coitus during the fertile period was estimated as 50% and supported by reference to Hertig⁴⁴ (although his estimate was 84%⁵); (2) the length of a menstrual cycle (28 days); and (3) the duration of the fertile period (2 days). These latter values are plausible, but also variable (Figure 1). No justification is provided for three behavioural variables: (1) coital frequency estimated at twice per week; (2) proportion of unprotected coital acts estimated at 25%; and (3) either a random or regular distribution of coital acts during menstrual cycles such that 1/14 of all coital acts fall within a fertile period.

The validity of Roberts & Lowe’s conclusion depends largely on the accuracy and precision of these speculative values. The following two simple analyses illustrate the sensitivity of their conclusion on the speculative values.

1. When four of the speculative values are reduced by 25% (e.g., coital frequency reduced to 1.5/week) and cycle length increased by 10% (from 28 days to 31 days³⁸), the estimate for embryo loss drops to 22%. The opposite operation (e.g., coital frequency increased to 2.5/week) results in an estimate of 92% (Table 1). Embryo loss of 22% is barely sufficient to account for observed clinical losses, and 92% indicates a maximum FEC_LB of 8%. Neither scenario is biologically plausible.

2. A non-zero variance was applied to each speculative value reflecting their uncertain nature. Using the random number generator in Microsoft^® Excel (Office 2010) simulated values were obtained by random sampling from normal distributions with means equal to Roberts & Lowe’s speculative values with coefficients of variation equal to 20%. For simplicity, it was assumed that there was no covariance between the different speculative values. Table 1 shows the expected range within which 95% of these simulated values fall (e.g., coital frequency is 1.2–2.8/week). For each simulated record, a new estimate of embryo loss was calculated, and from 10,000 of these, the mean, median and 2.5^th and 97.5^th percentiles of embryo loss were determined. This step was repeated 1,000 times: the mean value of the simulated means was 73.3% and of the simulated medians was 76.5%. The mean values of the 2.5^th and 97.5^th percentile boundaries for embryo loss were 37% and 90% (Table 1). Separately, the same simulation was performed using NONMEM 7.3.0^® (Icon PLC, Dublin, Eire) to generate 100,000 data records which are represented in Figure 2. The code and simulated data values are in Dataset 1.

Figure 2. Distribution of embryo loss estimates from fertilisation to birth derived using a modified version of the model of Roberts & Lowe³⁴.

Embryo loss values were calculated using alternative speculative values (see text and Table 1) obtained by randomly sampling from normal distributions with means equal to the Roberts & Lowe values and a coefficient of variation of 20%. 100,000 simulated embryo loss values were obtained. Frequencies within a bin size of 0.25% are shown. The 2.5^th and 97.5^th percentiles are indicated. The simulation was performed using NONMEM 7.3.0® (Icon PLC, Dublin, Eire). Simulated values are in Dataset 1.

The sole purpose of these simple sensitivity analyses is to illustrate that modest adjustments to Roberts & Lowe’s original speculative values can result in any biologically plausible estimate for embryo loss. Whilst their analysis is useful for highlighting factors that influence observed fecundity, the output from the calculation remains substantially dependent on the subjectively selected input. Consequently, their analysis has no practical quantitative value.

Other sources of bias in their model include the failure to account for intentionally terminated pregnancies and the reduced fecundability of already pregnant women and nursing mothers. Despite this, it was described as “persuasive”⁵⁰ and it has been claimed that “it is still difficult to better the original calculations of Roberts and Lowe (1975)”²⁷. By contrast, others have noted that “their calculations can be criticized”⁹ and are “tenuous”⁵¹. Considering its quantitative limitations, it has been cited surprisingly often^13,28,52,53.

2. Life tables of intrauterine mortality

Constructing a life table of intrauterine mortality is challenging since embryonic death may occur even before the presence of an embryo is recognised. Nevertheless, in 1977, the distinguished demographer Henri Leridon published an impressive critique and analysis of pregnancy loss data, and a complete life table of intrauterine mortality²⁶. Leridon highlighted the consequences of inappropriate analysis and the quantitative biases produced by alternative numerical methods. Overall, he discussed sixteen studies, and provided detailed commentary on six published between 1962 and 1970^47,54–58. These data are summarised in Figure 3 and suggest that 12–24% embryos alive at 4 weeks’ gestation (i.e., approx. 2 weeks’ post-fertilisation, see Figure 1) will perish before birth.

Figure 3. Graphical representation of the fate of 1,000 pregnancies in progress at 4 weeks’ gestation (2 weeks’ post-fertilisation).

The figure is generated using values in Table 4.3 of Leridon (1977)²⁶ and are derived from six different studies (see text). The Kauai Pregnancy Study data published by French & Bierman (1962)⁴⁷ are shown in thick black. Data from Shapiro (1970)⁵⁷ were analysed either with all pregnancies included (ALL) or with those pregnancies excluded that aborted within one week of study entry (EXCL.). The greater loss observed with ALL may be due to a correlation between study entry and abortion risk. Based on these data, the risk of losing a pregnancy ongoing at 4 weeks’ gestation ranges from 12.5% to 23.7% (excluding Shapiro (1970) ALL). Values are in Dataset 2.

Leridon described the Kauai Pregnancy Study published by French & Bierman (1962)⁴⁷ in particular detail. In this study, an attempt was made to identify every pregnancy on Kauai from 1953–56. Women were encouraged to enrol as soon as they missed a period. Pregnancy loss may therefore have been overestimated, since not all amenorrhoea is caused by conception, although other studies that relied upon medically-identified pregnancies probably underestimated pregnancy loss by not capturing all cases¹⁶. Whatever the truth, it is clear that, among the studies reviewed by Leridon, the Kauai Pregnancy Study revealed the highest levels of pregnancy loss (Figure 3).

All recorded pregnancies in the Kauai study were categorised by date of enrolment in four week intervals, beginning with 4–7 weeks’ gestation. This time-staggered approach enabled risk of miscarriage to be associated with stage of gestation. However, despite considerable efforts, only 19% of the 3,197 recorded Kauai pregnancies were enrolled between 4–7 weeks’ gestation, thereby reducing the precision of pregnancy loss estimates for this earliest of time intervals. Although pregnancies were grouped in four week periods, Leridon suggested that early mortality may change week by week, resulting in underestimation of pregnancy loss. He re-allocated the 592 study entries and 32 pregnancy losses for weeks 4–7 (Table 2) generating an overall probability of pregnancy loss during this period of 15.0%, higher than 10.8% originally reported⁴⁷. Leridon’s own description of this interpolation as “risky” can be illustrated by adjusting his re-allocation²⁶. Transferring just two of the pregnancy losses out of or into the first week results in estimates of the 4–7 week pregnancy loss of 10.9% and 19.1% respectively (Table 2). The validity of adjusting Leridon’s re-allocation may be questioned. However, pregnancy loss in week 4–5 of the Kauai Study would manifest as a menstrual period delayed by up to one week. This is far from being a robust pregnancy diagnosis and in a different study⁵⁷, exclusion of pregnancy losses reported within one week of study entry resulted in substantially different loss probabilities (Figure 3) suggesting a confounding correlation between entry and loss²⁶. Nevertheless, the re-allocation does reinforce a concern highlighted by Leridon, namely the uncertainty that affects the first probability. Clearly, these estimates of early loss should be treated with caution.

A more fundamental problem is that these data offer no insight into the fate of embryos prior to the earliest possible point of clinical pregnancy detection. Leridon completed his life table with values from Hertig’s 1967 analysis⁵. He concluded that among 100 ova exposed to the risk of fertilisation, 16 are not fertilised, 15 die in week one (between fertilisation and implantation), and 27 die in week two (between implantation and the time of the first missed period). After two weeks his life table follows the Kauai probabilities closely ending with 31 live births. Leridon’s table therefore indicates an embryo mortality of 50% (42/84) within the first two weeks after fertilisation and a total mortality of 63% (53/84) from fertilisation to birth.

Leridon’s account of intrauterine mortality has been widely cited. However, its accuracy depends entirely on the quality and interpretation of the data from Hertig⁵ and French & Bierman⁴⁷. French & Bierman’s approach probably resulted in an overestimate of total pregnancy loss and is certainly imprecise in its estimate of embryo loss in the four weeks following the first missed menstrual period. The reliability of Hertig’s estimates of embryo loss in the two weeks following fertilisation is considered below.

3. Biochemical detection of pregnancy using hCG

Quantification of pregnancy loss requires pregnancy diagnosis. The earliest outward sign of pregnancy is a missed menstrual period, approximately 2 weeks after fertilisation, although amenorrhoea in women of reproductive age is not exclusively associated with fertilisation^59,60. Several potentially diagnostic pregnancy-associated proteins have been identified⁶¹ of which only one, Early Pregnancy Factor (EPF)⁶², has been claimed to be produced by embryos within one day of fertilisation. However, there is doubt about the utility of EPF for diagnosing early pregnancy⁶³ and little has been published on it in the past five years.

Modern pregnancy tests detect human chorionic gonadotrophin (hCG), a highly glycosylated 37 kDa protein hormone produced by embryonic trophoblast cells⁶⁴. Elevation of hCG around 6–7 days after ovulation is associated with embryo implantation^27,28,65 (Figure 1). Early assays for the detection of hCG were probably confounded by antibody cross-reactivity with luteinizing hormone⁶⁶ but modern tests are more specific and a positive result is a reliable indicator of early pregnancy. Highly sensitive assays have revealed low levels of hCG in non-pregnant women and healthy men⁶⁷; hence, quantitative criteria and appropriate design are required to distinguish between non-pregnant women and those harbouring early embryos^65,68,69.

Figure 4 and Table 3 summarise findings from thirteen studies that used hCG to identify so-called early, occult or biochemical pregnancy loss, i.e., pregnancy loss between the initiation of implantation and clinical recognition^{46,48,49,70–79}. (Ellish et al. (1996)⁶⁹ is not included since the hCG assay was positive for only 72.5% of clinical pregnancies. By contrast, among the thirteen studies in Table 3 only one clinically-recognised pregnancy was reported undetected by hCG testing⁷⁰. Nevertheless, their estimates of early pregnancy loss (17.4%) and clinical loss (13.7%) are comparable to these other studies.) Each study measured urinary hCG levels except two, which measured hCG in serum^72,73. Notwithstanding design and subject differences, estimates for clinical pregnancy loss, ranging from 8.3% – 21.2% (Figure 4), are similar to previous estimates (Figure 3). Estimates for early/occult pregnancy loss ranged from 0% to 58.3% in studies^70–74 prior to Wilcox (1988)⁴⁹. This high variance was probably due to reduced specificity and sensitivity of the hCG assays and sub-optimal study design^{16,61,80–83}. Studies from Wilcox (1988)⁴⁹ onwards have produced more consistent data indicating early/occult pregnancy loss of approximately 20% (Figure 4). In the largest studies^48,79 pregnancies were clinically recognised only if they lasted ≥6 weeks after the onset of the last menstrual period; hence, early pregnancy losses in these studies included those lost up to approximately two weeks after a missed menstrual period. Definition of clinical pregnancies can influence comparison of study results^39,82. For example, Wilcox originally reported 43 (later 44⁸⁵) pre-clinical and 19 clinical losses from 198 detected pregnancies⁴⁹, giving 21.7% early/occult and 12.3% clinical loss rates. In later reports, 4 cases were re-allocated resulting in 48 early and 15 clinical losses (i.e., 24.1% early/occult and 9.9% clinical loss rates)^40,86–88. Variable definitions can generate confusion, although in this case the overall picture is not greatly affected. Based on the eight most recent studies, beginning with Wilcox (1988), pregnancy loss from first detection of hCG through to live birth is approximately one third (Table 3). This is consistent with another recent study which found that 98 out of 301 (32.6%) singleton pregnancies diagnosed by an early positive hCG test and followed-up to either birth or miscarriage were lost⁸⁹.

Figure 4. Summary of findings from thirteen studies that used hCG detection to diagnose early pregnancy.

Data are arranged by publication date and the first author of the study is shown. Three datasets are shown: (i) the percentage of at risk menstrual cycles that were hCG positive; (ii) the percentage of hCG positive cycles that did not manifest as clinical pregnancies = early pregnancy loss; and (iii) the percentage of clinical pregnancies lost prior to 12 or 28 weeks or live birth (definitions vary between studies). A clinical pregnancy may be manifest by a missed period although criteria vary between studies. Videla-Rivero et al.⁷³, Sasaki et al.⁷⁶, Cole⁷⁸ and Mumford et al.⁷⁹ do not report sufficient data to calculate all three values. Values are in Dataset 3.

The much cited Wilcox (1988) study⁴⁹ is the earliest of several large well-designed studies that made use of a specific and sensitive hCG assay and led to numerous further publications^{40,84–87,90–92}. Two other studies (Zinaman (1996)⁴⁶ and Wang (2003)⁴⁸) were similar in purpose, design and execution. These studies provide some of the best available data to calculate pregnancy loss between implantation and birth³⁹. In each study, women intending to become pregnant and with no known fertility problems were recruited and hCG levels monitored cycle by cycle in daily urine samples until they became pregnant. Most women were followed through to late pregnancy or birth. Although these studies provide evidence regarding the outcome of both clinical and hCG pregnancies, determining the fate of embryos prior to implantation is more difficult. To relate the study results to pre-implantation embryo loss, it is necessary to determine fecundability. In each study FEC_CLIN declined in successive cycles as the proportion of sub-fertile women increased. Hence, reported raw FEC_HCG values of 30%⁴⁶ and 40%⁴⁸, and FEC_CLIN values of 25%⁴⁹ and 30%⁴⁸ are biased underestimates of the fecundability of normal fertile women. A recent re-analysis of these data provides statistical evidence for discrete fertile and sub-fertile sub-cohorts within the study populations³⁹. The proportions of sub-fertile women (mean [95% CI]) were estimated as 28.1% [20.6, 36.9] (Wilcox); 22.8% [12.9, 37.2] (Zinaman); and 6.0% [2.8, 12.3] (Wang). For normally fertile women, FEC_HCG was, respectively: 43.2% [35.6, 51.1]; 38.1% [32.7, 43.7]; and 46.2% [42.8, 49.6]. FEC_CLIN was: 33.9% [29.4, 38.6]; 33.3% [27.6, 39.6]; and 34.9% [33.0, 36.8]. There was no apparent difference in π_CLIN between fertile and sub-fertile sub-cohorts, which was estimated as: 78.3% [69.2, 85.3]; 87.5% [76.0, 93.9]; and 75.4% [71.5, 79.0]³⁹.

Why do a proportion of menstrual cycles in women attempting to conceive fail to show any increase in hCG? Since FEC_HCG = π_SOC × π_FERT × π_HCG , there can be various causes for this failure including mistimed coitus, anovulation, failure of fertilisation or pre-implantation embryo death. Although FEC_HCG puts limits on the extent of pre-implantation embryo loss, uncertainty in the estimates of π_SOC, π_FERT and π_HCG translates into uncertainty in estimates of pre-implantation embryo mortality. In the Wang study, for normally fertile women, FEC_HCG = 46.2%; hence, the absolute maximum value for pre-implantation embryo loss must be 53.8%, although only if π_SOC = π_FERT = 1, conditions both extreme and unlikely³⁹. Studies of the relationship between coital frequency and conception indicate that fecundability is greater with daily compared to alternate day intercourse^39,93,94. Hence, when coital frequency is less than once per day a proportion of reproductive failure will be due to mistimed coitus, i.e., π_SOC < 1. In the Wilcox study, coitus occurred on only 40% of the six pre-ovulatory days^39,40, and in the Zinaman study participants were advised that alternate day intercourse was optimal⁴⁶. Based on the difference in fecundability between daily and alternate day intercourse as modelled by Schwartz⁹⁴, a value of π_SOC = 0.80 was used to calculate pre-implantation embryo mortality³⁹. However, this is a speculative estimate, and in reality the value may be higher, or lower.

A further critical missing piece of the equation is knowledge of the efficiencies of fertilisation and implantation under normal, natural, propitious circumstances. Assuming that either of these processes may be up to 90% efficient, and based on data from the three hCG studies^46,48,49, a plausible range for pre-implantation embryo loss in normally fertile women is 10–40% and for loss from fertilisation to birth, 40–60%³⁹. Even with these wide ranges of mathematically possible outcomes, it is clear that estimates for total embryonic loss of 90%³⁷, 85%³⁶, 83%², 80–85%^11,35, 78%³⁴, 76%^10,33 and 70%^27–31 are excessive.

In 1990, Charles Boklage concluded that “at least 73% of natural single conceptions have no real chance of surviving 6 weeks of gestation”^10,95. Live birth fecundability was estimated as “not over 15%”, substantially lower than Leridon’s 31%. Despite this discrepancy, Boklage’s conclusions were derived from a review of data including several hCG studies^{49,65,70–73} and Leridon’s analysis²⁶. He derived a model describing the survival probability of human embryos comprising the sum of two exponential functions:

P_t(pregnancy survival) = 0.73e^-0.155t + 0.27e^-0.00042t

in which t is the time in days post-fertilization. This is the source of the 73% in the conclusion.

There are, however, serious problems with this analysis. Firstly, data presented as embryo survival probabilities at different times post-fertilization^{49,65,70,71,73} are fecundabilities, i.e., successes per cycle, not per fertilised embryo. Secondly, for reasons that are unclear, data from Whittaker⁷² and Leridon²⁶ were excluded from the modelling analysis and the data from an earlier Wilcox report⁶⁵ were included twice since this preliminary data had been incorporated into the later report⁴⁹. Thirdly, the modelled data were normalised to a survival probability of 0.287 at 21 days post-fertilization. This value was derived from data published by Barrett & Marshall (1969) on the relationship between coital frequency and conception⁹³. Barrett & Marshall had concluded that coitus during a single day alone, 2 days before ovulation, resulted in a conception probability of 0.30. Boklage’s value of 0.287 is his calculated equivalent. However, conception in this study was “identified by the absence of menstruation, after ovulation”⁹³. Hence, 0.30 (and similarly, 0.287) is a clinical fecundability and not a measure of embryo survival. Furthermore, 0.30 is a non-maximal fecundability, since it was an estimate based on coitus on a single day (2 days before ovulation) within the cycle. Barrett & Marshall clearly report that as coital frequency increased so did the fecundability, up to a maximum of 0.68 associated with daily coitus⁹³.

Boklage’s analysis can only make biological sense if it is assumed that every cycle in the Barrett & Marshall study resulted in fertilisation. Under these circumstances, failure to detect conception in 71.3% (1 – 0.287) of cycles would be due entirely to embryo mortality. However, this is highly implausible and explicitly contradicted by the higher estimate of fecundability reported⁹³. Boklage’s implicit assumption also contradicts his further conclusion that “only 60–70% of all oocytes are successfully fertilized given optimum timing of natural insemination”¹⁰. The vertical normalisation of the hCG study data to a value of 0.287 at 21 days is the principal determinant of the parameters that define the two exponential model. Any change in this value would commensurately alter the balance between the two implied sub-populations of embryos. Since it is evident that the value of 0.287 is neither an embryo survival rate nor even a maximal fecundability, it follows that quantitative conclusions from this analysis in relation to the survival of naturally conceived human embryos are of doubtful validity.

However, Boklage was right about two things. Firstly, the difficulty of calculating pre-clinical losses: as he put it, “In the place of the necessary numbers for the first few weeks of pregnancy we find editorially acceptable estimates which, while perhaps not far wrong, are difficult to defend with any precision”. Secondly, the source of some of the only directly relevant data (even though he excluded it from his modelling analysis), namely, “Hertig’s sample is, and will probably remain, unique”.

4. The anatomical studies of Dr Arthur Hertig

At the start of the 1930s, no-one had ever seen a newly fertilised human embryo. It was barely 60 years since Oscar Hertwig had first observed fertilisation in sea urchins⁹⁶, and just 40 years before the birth in 1978 of Louise Brown, the first test tube baby^97,98. In Boston, Dr Arthur Hertig and Dr John Rock’s search for early human embryos generated an irreplaceable collection, and set an influential benchmark for the scale of early human embryo mortality.

The so-called “Boston Egg Hunt” began in 1938⁹⁹. Hertig and Rock recruited 210 married women of proven fertility who presented for gynaecological surgery⁴⁴. (In most of their publications, the number is given as 210^5,100,101 although 211 subjects are mentioned elsewhere⁴⁴.) Of these, 107 were considered optimal for finding an embryo because they apparently: (i) demonstrated ovulation; (ii) had at least one recorded coital date within 24 hours before or after the estimated time of ovulation; (iii) lacked pathologic conditions that would interfere with conception. Hertig examined the excised uteri and fallopian tubes, and over fifteen years found 34 human embryos aged up to 17 days^{5,44,100–107}. Of these, 24 were normal and 10 abnormal^5,100. (There is some confusion over this: in three publications^44,101,107, 21 embryos are described as normal and 13 as abnormal. It appears that the three alternatively described embryos (C-8299; C-8000; C-8290) were originally defined as abnormal based on their position or depth of implantation⁴⁴.) Table 4 provides information about the 34 embryos found in these 107 women. Although the study was primarily intended to find and describe early human embryos, Hertig subsequently used the data to derive estimates of reproductive efficiency including early embryo wastage^5,100.

Hertig’s analysis^5,100 relies heavily on the 15 normal and 6 abnormal implanted embryos found in 36 women from cycle day 25 onwards. He assumed the 6 abnormal embryos would perish around the time of the first period concluding that fertility (% pregnant) at this stage = 42% (15/36). Of the 8 pre-implantation embryos identified (7 in the uterus and 1 in the fallopian tubes), 4 were abnormal. Hertig assumed the 4 normal embryos would implant successfully but that some of the abnormal ones would not, such that the proportion of normal embryos would increase from 50% (4/8) before implantation to 71% (15/21) after implantation as observed. Hence, among the 36 post-cycle day 25 cases, in addition to the 15 normal embryos, there must have been 15 abnormal pre-implantation embryos of which 60% (9/15) failed to implant and were not observed, and 40% (6/15) did implant and were observed, although these 6 would have perished shortly afterwards. This left 6/36 eggs that must have been unfertilised. The ratio of ‘unfertilised’: ‘fertilised abnormal’: ‘fertilised normal’ was therefore 6:15:15, matching the 16% infertility (no fertilisation), 42% sterility (post-fertilisation death) and 42% fertility (reproductive success) reported in Figure 9 of Hertig’s 1967 article, “The Overall Problem in Man”⁵. This is the source of Hertig’s 84% fertilisation rate and 50% embryo loss before and during implantation, and is reproduced in Leridon’s life table²⁶ as 84/100 eggs surviving at time zero (ovulation and fertilisation) and 42 surviving to 2 weeks (time of first missed period).

Hertig provides almost the entire body of evidence used to quantify natural human embryo loss in the first week post-fertilisation. Most claims regarding early human embryo mortality find their source here. Before considering how reliable the figures are, it is worth repeating Hertig’s own caveat, namely, the lack of data on the efficiency of natural fertilisation⁵. All estimates of embryo mortality from fertilisation onwards are subject to commensurate inaccuracy in the absence of reliable fertilisation probabilities (i.e., π_FERT), which are “surprisingly difficult to estimate”²¹.

There are several problems with Hertig’s analysis. As noted by others, the observations are cross-sectional, but the inferences are longitudinal¹⁰⁸. Hertig detected 21 embryos from 36 cases (58.3%) from cycle day 25 onwards. If this detection rate were representative, then on average, prior to day 25, the detection rate should either be the same or higher; however, they are all lower, and substantially so (Table 4). Hertig suggested that this was due to the technical difficulty of finding newly fertilised embryos. However, the detection rate for cycle days 18–19 was good (46.7%) and embryos one or two days younger would not have been much smaller, at which stage the detection rate was poor (11.1%). An alternative explanation for this discrepancy might simply be random variation. Furthermore, from cycle day 25 onwards, embryos would probably have produced hCG and therefore FEC_HCG would have been at least 58%. This is approximately double the equivalent values observed in more recent and robust hCG studies (Table 3) further suggesting that this subset of the data is not representative.

Despite having proven fertility, these women presented for gynaecological surgery which, according to Hertig, was “medically essential”⁹⁹. This suggests that the women may have had sub-optimal reproductive function, although the effect of this on the quantitative outcome of the study is difficult to gauge. Furthermore, Hertig’s reproductively ‘optimal’ coital pattern does not include 2 days pre-ovulation and does include one day post-ovulation, conditions which are known not to maximise fertilisation^{39,40,93,94,109}. Hence, detection rates before cycle day 25 may be more representative than those after. Given the numerical discrepancies, they cannot both be.

Hertig does not provide error estimates with his conclusions. In order to estimate the precision of his derived proportions, a bootstrap analysis was performed as follows: Hertig’s 107 optimal cases were categorised according to stage of cycle (Category 1 = cycle days 16–19 (n=24); Category 2 = cycle days 20–24 (n=47); Category 3 = cycle days ≥25 (n=36)), and presence and type of embryos (Category 0 = no embryo (n=73); Category 1 = normal embryo (n=24); Category 3 = abnormal embryo (n=10)). Five hundred pseudo-datasets each containing 107 cases were generated using a balanced random re-sampling method using Microsoft Excel^®. The original and pseudo datasets are in Dataset 4.

Hertig’s numerical calculations, as detailed above, were repeated for each pseudo-dataset thereby generating 500 estimates for each parameter, from which were derived median values and [95% CIs] using the percentile method110: fertility = 42% [26%, 59%]; sterility = 42% [5%, 182%]; infertility = 16% [-127%, 61%]; pre-implantation embryo survival probability = 69% [27%, 128%]; post-implantation to week two survival probability = 71% [50%, 91%]; detection rate for cycle day 25 onwards = 58% [41%, 74%]. Median values matched estimates calculated from the original dataset. Bootstrap 95% CIs for the day 25 detection rate (58%) matched those calculated using the “exact” method of Clopper & Pearson¹¹¹, [41%, 74%], which are a little wider than those calculated using the “more exact” method of Agresti & Coull¹¹², [42%, 73%]. (These analyses were performed using an online GraphPad^® calculator accessed on 18^th April 2017: http://www.graphpad.com/quickcalcs/ConfInterval1.cfm.) The congruence between these confidence intervals and the point estimates provides some reassurance that that the bootstrap procedure worked effectively. Estimates of parameters other than the day 25 detection rate (58%) are derived from more complex proportional relationships, and are therefore less precise. Table 5 reproduces a life table in the style of Leridon²⁶ and includes probabilities for each reproductive step with confidence intervals. These intervals (and some noted above) are impossibly wide highlighting further problems with Hertig’s analysis.

Hertig’s analysis omits 47 cases from cycle days 20–24, comprising 44% of his data. It is clear why he cannot use it, since all five embryos were normal and, given his mathematical and biological assumptions, five normal implanting embryos could not become 29% (6/21) abnormal post-implantation. Others have also noted that these “missing data are sufficient to engender an entirely different result”¹⁶. Furthermore, the data that define the 50% proportion of abnormal pre-implantation embryos (i.e., 4/8) are so few that any numerical variation will make a substantial difference to derived proportions. If he had observed 3/8 abnormal embryos, his estimate of pre-implantation loss would have been 13% rather than 30%: for 5/8 it would have been 48%, with a fertilisation rate of 111%, which is clearly impossible. It seems therefore, that Hertig designed his analysis based on a post-hoc examination and selective use of the data. His own caveat about the lack of relevant and necessary data should be taken at least as seriously as his conclusions.

Hertig and Rock’s contribution to human embryology is undeniable and their quantitative conclusions have profoundly influenced our impression of the extent of early human embryo mortality. Regrettably, their estimates have a cripplingly low precision, which undermines their biological credibility or utility. In conclusion, Hertig’s data and flawed analysis cannot be regarded as a reliable quantitative foundation upon which to evaluate and understand natural human reproduction.