In the light of the recent hospital admission of the Duchess of Cambridge for an condition loosely associated with female babies and multiple births, I’ve been asked by the Radio 4 programme “More or Less” to calculate the probability that she is pregnant with more than one embryo. I’m somewhat reluctant to contribute to what is already a topic of rampant media speculation, and the attendant intrusive journalism that often plagues issues like this (which, after I had written this post, led to a sad and particularly tragic outcome). Nevertheless, few media articles seem to give links to solid data sources, and some even give rather misleading information, so I’ve overcome my reluctance in order to put some solid statistical facts into the public domain. Simply put, compared to the average, the probability of a mother having twins given that she has this condition is not quite doubled. However, it’s still likely to be a very low number: something like an increase from about 1.5% to a 2.4% chance. For the gory details, read on.

At the outset, I should point out that I’m by no means an expert on obstetrics! What I’m about to discuss is simply the result of a search of the relevant scientific literature, coupled with a brief chat to a friend who is an obstetrician. Anyone who knows how to look up scientific references (and can access the relevant journals) could do the same.

## The condition

Severe morning sickness – the reason for this hospital admission – is, of course, horrible for the mother: debilitating and potentially dangerous if left untreated. Famously, it’s been suggested as cause of Charlotte Brontë’s death, although that diagnosis has recently been disputed [zotpressInText item=”PCZQTU3I”]. Here’s a short well-written summary paragraph:

At least half of pregnant women experience nausea and vomiting. At the extreme end of the spectrum are women whose symptoms are so severe or so prolonged that they lose weight and develop dehydration and ketosis. This condition, hyperemesis gravidarum, affects 0.5-1% of pregnancies, but the causes are not fully understood. [zotpressInText item=”F9JAC3KA”]

I must admit that, although I know about long-period hospitalization for morning sickness, until writing this I wasn’t familiar with the actual term “hyperemesis gravidarum”. It simply translates as “excessive vomiting of pregnant women” – from the same Greek root as “emetics”: things which make you vomit. I do get annoyed by medical terms in Latin or Greek which are incomprehensible to the uninitiated.

That brings us to the first statistical issue. Using a separate medical name implies that it can be objectively diagnosed as an illness separate from “normal” morning sickness. But that’s not true. Basically, it’s diagnosed when the symptoms are severe enough to warrant admittance to hospital [zotpressInText item=”IQU7NMME”]. Of course, that line may be drawn in different places depending on the doctor, the hospital, and the country. That means pooling data from different times or places may be problematic. Moreover, an analysis based on simple presence/absence of the diagnosis ignores a lot of useful information. In statistical terms, analyses which treat severe morning sickness as a “dichotomous variable” rather than a continuous one, may lack the power to spot anything useful. And in the specific case of the Duchess of Cambridge, the public (quite rightly) don’t know where she lies on this spectrum, so we may not have much to go on.

## Contributing factors

How about the association with twins and female babies? Well, there *are* reasons to expect this. Despite its poorly understood causes, morning sickness does have a loose association with heightened levels of a hormone called “chorionic gonadotrophin“. This hormone is also increased when the child is female, when there are multiple embryos, and in other medical conditions such as molar pregnancies. Finding a robust *statistical* link is more tricky, because of the numbers involved.

Morning sickness extreme enough for hospitalization is (thankfully) very rare. Having multiple births is also a relatively rare occurrence: about 1.6% of pregnancies in England and Wales in 2010. Even if they occur considerably more often together, we’re talking tiny proportions – of the order of 0.01% of all pregnancies. For example, a UK study of almost 10000 pregnancies over a 5 year period in a Norfolk hospital found 209 sufferers, of which only 4 had twins [zotpressInText item=”C7HGSMZJ”]. As I’ve discussed before, extrapolating from such tiny numbers is fraught with difficulty – especially when each number is simply a dichotomous variable.

There are two ways to collect large enough numbers to draw reasonable conclusions. The first is deliberate targeting. For example, using the Hyperemesis Education and Research Foundation, a study published this year managed to solicit information from 395 mothers with the condition [zotpressInText item=”UTDDKANR”]. Or you could target twin births, as was done 20 years ago in Norway [zotpressInText item=”V3UHTJIG”]. But these targeted studies can suffer from sampling bias. What it the mothers with the severest symptoms not only have a greater incidence of twins, but are more likely to volunteer? Or what if the age of the mother influences the probability of taking part?

For this reason, the most trustworthy analyses are from countries where data has been collected for *all* births in the entire country (which also means that “hyperemesis gravidarum” should be diagnosed relatively consistently). The scandinavian countries seem to be particularly good at this. There are datasets providing morning sickness information for all Swedish births from 1987-1995 [zotpressInText item=”SCXWNVAJ”], native Danish births from 1980-1996 [zotpressInText item=”PNSTS4SS”], and all Norwegian births from 1967-2005 [zotpressInText item=”KM948TU4″].

The last two of these studies provide information on multiple births, but the most recent and extensive is the Norwegian one, so I’ll use that as an example. Unfortunately the paper is not open access, so you might need to take my summary on trust. The dataset consists of almost a million datapoints: 900 074 pregnancies. Of those, 8 296 (0.9%^{a}8296/900074 = 0.0092, or 0.92%, slightly at odds with their quoted figure of 0.89%. I’m at a loss to explain this discrepancy) were diagnosed with morning sickness that required hospitalization. Most of those (98.8%) were pregnancies with a single foetus, hence the prevalence of the condition in these mothers is about average: 0.9% (Table III in the paper). However, of the mothers that gave birth to twins or more, 1.3% were diagnosed with severe morning sickness.

### Calculating the probabilities, given a diagnosis

Doctors are mostly interested in figures like this 1.3%: in other words, the probability of the condition, given various predisposing factors – multiple births, age and ethnic origin of mother, and so on. But here we’re interested in calculating the inverse: the probability of multiple babies, given the condition. That’s exactly what Bayes’ theorem does, so brace yourself for some statistics!

The probability of hyperemesis gravidarum given multiple births is written pr{HG|multiple} – which in this study was 1.3% (0.013). Instead, we want the probability of multiple births, given hyperemesis gravidarum: pr{multiple|HG}. Bayes’ theorem says

pr{multiple|HG} = pr{multiple} x pr{HG|multiple} ÷ pr{HG}

In this study, the initial, basic probability of a multiple births is pr{multiple}=1.2%. Bayes allows us to update this knowing that the mother has severe morning sickness: we simply multiplying by pr{HG|multiple} ÷ pr{HG}, or 0.013 ÷ 0.009 = 1.44. So the new probability is 44 percent higher: 1.2% ×1.44 = 1.73%.

However, we can do better than this, because the Norwegian study also adjusted for a variety of other factors (year of birth, birth country, education, marital status, and age of mother, gender of embryo), by using a statistical technique called multiple logistic regression. This is used where a number of factors multiply to give a probability of something happening or not happening. Mathematically, the relationship can be expressed as a statistical model. In this study, the model specifies a relationship between various factors and pr{HG}, the probability of severe morning sickness. More precisely, what are used are not strict probabilities, but “odds” ^{b}Using the odds, then taking the natural logarithm, is a way of linearizing the relationship, which is mathematically convenient. For example, log(odds) can range from -∞ to ∞, rather than probabilities, which inconveniently range from 0 to 1., the probability divided by one-minus-the-probability. In other words, odds(HG) = pr{HG}/(1-pr{HG}). The statistical model can be thought of as something like this

odds(HG) = (background odds) x (modification due to maternal age) x (modification due to number of embryos) x …

These “modifications” are otherwise known as “odds ratios”, or OR for short. These tell us the change in odds relative to some alternative. For example, this study quotes the change in odds due to multiple babies as 1.42 (in Table III of their paper^{c}This is quoting the “adjusted odds ratio”, which takes into account the possibility that e.g. people with twins are also more likely to be older mothers (i.e. confounding, or collinearity between factors). In this case, the crude (unadjusted) odds ratio is 1.43, so there’s actually not much of a difference. A statistician would say that the factors are more-or-less orthogonal.), relative to the alternative odds of a single birth. If this alternative is set to 1 (no change in odds), then to get the odds with multiple babies, we multiply by 1.42. In terms of our model,

odds(HG|single) = (combined odds from all other factors) x 1.0

and

odds(HG|multiple) = (combined odds from other factors) x 1.42

As long as the probabilities from other factors don’t change depending on the number of embryos (something assumed in this study, and which I’ll come back to later), then that means

odds(HG|multiple) = odds(HG|single) x 1.42

Now come two bits of statistical jiggery pokery. Because nearly all births are single ones, odds(HG|single) is basically identical to odds(HG), they are both about 0.0093. Moreover, when the numbers are that small, odds are almost identical to probabilities (because 1-pr{HG} ≈ 1). So we can say that

pr{HG|multiple} ≈ pr{HG} x 1.42

and if we stick this into Bayes theorem, the pr{HG} terms cancel, telling us that to update the basic probability of multiple births, all we have to do is to multiply by the adjusted odds ratio.

pr{multiple|HG} ≈ pr{multiple} ~~x ( pr{HG}~~ x 1.42 ~~)~~~~÷~~ pr{HG}

Phew.

Let me reiterate. In this case, to get the increased probability of multiple births, simply multiply by the adjusted odds ratio. If the background probability of twins is 1.5%, the probability if you are known to suffer from extreme morning sickness is simply 1.5% x 1.42, or 2.13%. Of course, that 1.42 figure has some uncertainty associated with it: the paper gives a 95% confidence interval of 1.20 to 1.68. And for the Danish study [zotpressInText item=”PNSTS4SS”], the odds ratio is 2.0, with a confidence interval from 1.67 to 2.39 (their Table 2). Given that the Danes didn’t adjust for other factors, and had a slightly lower sample size, I’m more inclined towards the Norwegian figure. So I’h happy to assume an odds ratio of about 1.6. If we previously assessed someone’s probability of twins at about 1.5%, in the light of a hospital admission for severe morning sickness. we should now adjust it to about 1.5 x 1.6, or 2.4%. An increase of 60% is a fair way away from the 300% claimed by some newspapers.

This analysis doesn’t account for interactions between different factors. For example, what if Nigerians (some groups of whom have a very high twinning rate) actually don’t get worse morning sickness with twins? That hypothetical possibility is simply ignored in this analysis. In this case, the calculations above would give a misleading answer for the probability of twins, given both that you are Nigerian and suffer acute morning sickness. To calculate the probabilities for a particular woman, we might want to repeat the analysis with these interactions also tested. I don’t have access to the data to do this, and I’m not sure I would want to focus on a particular case in this way.

I haven’t talked about the increased probability of a girl, given severe morning sickness [zotpressInText item=”SCXWNVAJ”], or indeed, other factors which predispose a woman towards multiple births [zotpressInText item=”JXBT426D”]. I’m planning to do so in a follow-up blog post.

## References

Notes

a. | ↑ | 8296/900074 = 0.0092, or 0.92%, slightly at odds with their quoted figure of 0.89%. I’m at a loss to explain this discrepancy |

b. | ↑ | Using the odds, then taking the natural logarithm, is a way of linearizing the relationship, which is mathematically convenient. For example, log(odds) can range from -∞ to ∞, rather than probabilities, which inconveniently range from 0 to 1. |

c. | ↑ | This is quoting the “adjusted odds ratio”, which takes into account the possibility that e.g. people with twins are also more likely to be older mothers (i.e. confounding, or collinearity between factors). In this case, the crude (unadjusted) odds ratio is 1.43, so there’s actually not much of a difference. A statistician would say that the factors are more-or-less orthogonal. |

That was absolutely fascinating, thank you – I followed a link to your blog from your article on the BBC News website.

I can’t add to your analysis of the odds, except to say that hyperemesis gravidum is a truly miserable thing to have – it’s hard to beat throwing up all day, every day for 9 months in the “yuck” stakes!

I suffered from it – and am now the semi-proud mother of an angelic-looking little monkey who is 7 years old – only one, and he’s a boy. So I’m in the 97.6% of non-Nigerian women who have horrible sickness, and only one baby as a result of the pregnancy.

Thanks – and I’m sorry to hear that you suffered from it. My wife had mild morning sickness for about 6 months, and that was bad enough. For it to be severe enough for hospital admission must be yet another order of magnitude of unpleasantness. I’m glad to hear that it all turned out fine for you. I do feel sorry for Kate Middleton, not only because of the sickness, but also being forced to announce this early in the pregnancy. It must be horrible to know that if anything goes wrong, it will be pored over by millions of people who don’t know you.

Interesting and informative blog. My understanding is that not all Nigerians have the same rate of twin pregnancies. The Yoruba have the highest. Having said that I had simply horrible morning, afternoon and night sickness. I’m half Nigerian and had non identical twins.

Yes – that’s what I’ve read too. It’s quite variable within Nigeria, although it’s not just the Yoruba with a raised twinning rate. That’s just what I’ve gleaned from Wikipedia, though. I don’t really think that Nigerians have a different response to bearing twins in terms of morning sickness, though. It was simply an academic point that such possibilities weren’t taken into account in the Norwegian analysis. There are good reasons not to throw all these possibilities into the statistical model, however.