# Mother milk of 6 Corona-positive (COVID-19) women does not contain the virus - can we make a confidence statement about this?

I am asking this question because I believe it would be great if the statistics community could make a contribution to solving this serious puzzle until more evidence is available.

The UK Royal Office of Obstetricians and Gynecologists publishes guidelines on neonatal and prenatal treatment of Corona positive women and babies of Corona positive women. There is insufficient evidence to give an advice whether breastfeeding is safe in the sense of not passing virus from mother to child via the mother milk. There are other infection ways but mother milk could be expressed and fed while keeping distance, e.g. by father.

A source is quoted in the guidelines (p. 27) that 6 Chinese women's breast milk was tested Corona negative; that is, six of out six samples were negative. If we consider these tests independent events, where the event is to sample a woman who passes the virus via the mothermilk, we have a binomial trial.

Which confidence or credibility statements can we make about the probability to pass on the virus to the mother milk? This probability is estimated at zero in this trial, of course.

Some statements can be made; for example if the probability was 0.50 the likelihood of the data would be 0.015. If it approaches 0, the likelihood approaches 1, but it is not differentiable at zero, so standard likelihood inference fails.

Please note: The same guidelines advise in section 4.8.2 to continue breastfeeding by COVID-19 positive women: ' In the light of the current evidence, we advise that the benefits of breastfeeding outweigh any potential risks of transmission of the virus through breastmilk. The risks and benefits of breastfeeding, including the risk of holding the baby in close proximity to the mother, should be discussed with her.' (p. 27)

• It depends on the probability model for the outcome. From a biology perspective, I don't think it makes sense to assume a binomial model. – AdamO Mar 18 at 19:44
• @AdamO Interesting. Which models are alternatives? – tomka Mar 18 at 19:47
• Not to confound a good statistics question with epidemiology but breasts are likely to carry virus from the owner due to their position and necessity of handling, not to say fondling, in the case of providing the said beverage. – Stian Yttervik Mar 19 at 11:48
• @StianYttervik You can do relatively save expression under sterile conditions and feeding by different person, as said in the Q. But I agree this may elevate a de-facto zero probability in practice to non-zero value. – tomka Mar 19 at 13:29
• Sterile conditions and infants don't go well together. – Mast Mar 20 at 9:09

## 2 Answers

There is the rule of three saying

if a certain event did not occur in a sample with $$n$$ subjects, the interval from $$0$$ to $$3/n$$ is a 95% confidence interval for the rate of occurrences in the population.

You have $$n=6$$, so says $$[0, 3/6=0.5]$$ is a 95% confidence interval for the binomial $$p$$ of transmission. In non-technical language: 6 non-events is way too few for any strong conclusions. This data is interesting, suggesting something, but no more.

This rule of three is discussed at CV multiple times Is Rule of Three inappropriate in some cases?, When to use (and not use) the rule of three, Using Rule of Three to obtain confidence interval for a binomial population and certainly more ... A more principled approach is here: Confidence interval around binomial estimate of 0 or 1 and using code from an answer to that question:

binom::binom.confint(0, 6, method=c("wilson", "bayes", "agresti-coull"), type="central")

method x n       mean       lower     upper
1 agresti-coull 0 6 0.00000000 -0.05244649 0.4427808
2         bayes 0 6 0.07142857  0.00000000 0.2641729
3        wilson 0 6 0.00000000  0.00000000 0.3903343
• The rule of 3 holds for large N, say 100. It won't be reliable w/ N=6. – gung - Reinstate Monica Mar 19 at 11:49
• In non-technical language, we can only be fairly sure the virus would be in the breast milk of somewhere between none and half of all infected women. This also assumes no bias in the sample set, which is hard to avoid with only 6 samples. It also doesn't say much about the risk of the child actually contracting the virus (or suffering ill effects if they do) if the milk is infected. – NotThatGuy Mar 19 at 12:54
• binom.test(0, 6) in the base of R can also be used and gives the same interval as binom::binom.confint(0, 6, method = "exact") – G. Grothendieck Mar 19 at 14:40

These tests of mother's milk are not useful to determine a confidence interval for the risk of contamination of children. Kjetil explained this very nicely in his answer that we end up with a large confidence interval of [0, 0.5], and AdamO has mentioned in the comments that this actually makes no sense because the binomial distribution is not a good model (below I argue that, while we may not have a clear idea of a better model, we can be reasonably certain that it will make the situation only/likely worse*).

But these tests are not made in order to make some estimate of the probability that mother's milk might be contaminated. Instead, they are used as a search for a Black Swan and to test the theory that babies do not get sick. We have already good reasons to believe that mother's milk is not contaminated, or at least not such that it makes babies get COVID-19 (at least not with heavy symptoms). This test is a trial for that theory. If for some weird reason mother's milk would be heavily contaminated but babies still do not get sick, then this test would have shown that.

AdamO:

It depends on the probability model for the outcome. From a biology perspective, I don't think it makes sense to assume a binomial model.

Tomka:

Which models are alternatives?

It's difficult to pinpoint a particular model, but at least some effects warn us that the binomial model may be inaccurate

1. Should we consider the six tests independent? Possibly they have some correlation ,e.g. some testing-problem that simultaneously occurred in all samples. (RNA is unstable, maybe the virus makes it to the baby but not to the lab)
2. Is infectivity of breast milk considered a constant? Potentially the breast milk will be more or less infectious depending on the difference in the time that the mother got infected and the time that the breast milk was sampled.

So while there might not be a particular alternative model, we should treat the binomial model with a lot of caution. It is a bit of an ideal postitive situation. Also, the probability that the milk contains the virus is not equal to the probabilitiy that kids get sick from it. If the breast milk of infected mothers only contains virus, say, 10% of the time for each mother, then sampling milk from those mother's may easily give 6 negaŧive results but still, nearly 100% of the babies may get sick

A better measure that we do not need to worry about babies (and breast milk) would be more direct measurements of the outcome (do babies/kids get sick?). Sample a set of babies that have pneumonia for the presence of nCoV-19 virus (or, if the sampling/testing is too expensive or otherwise not possible, then look at the number of cases and see whether it increase like the cases in the elderly population).

In that way, you can see whether the risk of getting pneumonia/SARS increases for babies. What I understand from the stories (but I have not seen raw numbers and I am not sure whether babies are actually actively tested) is that babies do not get the virus.

So, since babies do not get COVID-19 (at least not many), we do not need to worry about nCoV-19 virus in mother's milk (unless we give mother's milk also to the elderly population).

Keep on breastfeeding (as also noted by the RCPCH: the benefits of breastfeeding outweigh the risks) If babies can get sick from nCoV-19 then breastmilk might actually be the thing that protects them and get's them better. We know that breastfeeding is good. But regarding a risk of COVID-19 for babies**... there isn't any data that shows that babies are under considerable risk due to the spread of this new virus. Data, data, data, use real data and not a panic interpretation of the data. Let's not kill the cats to control the plague.

(*)One loophole in my reasoning, that the situation will only be worse when we use alternative models, would be some particular Bayesian models. Say that, based on prior knowledge, the probability that mother's milk is considered either heavily contaminated with probability of contamination p larger than 0.9, or very little contaminated with probability of contamination p smaller than 0.1. Then a posterior distribution for $$p$$ would lean heavily towards $$p<0.1$$. So the evaluation of this test with 6 moms may depend a lot on prior knowledge.

The computed posterior is demonstrated below. On the left when we would have a continuous prior for p<0.1 and p>0.9 ,in which case the posterior scales like $$p^7$$. On the right when we would have a beta distribution as the prior, in which case the posterior is also a beta distribution (and simple to compute by just adding the observations to the parameters of the prior distribution). In both cases a prior distribution that places the probability more to the edges rather than the center (either the contamination probability of breats milk is high or it is low, will make that the result of the 6 cases is stronger than we would imagine from using the confidence interval that has no idea of these prior distribution.

(**) Besides a risk for babies getting sick we actually would also need to consider the risk that babies may not get sick but still pass on the disease. However since babies have not much close contact with many other people this may not be expected to be a main/important route of transmission. And the advise given by the RCPCH (to have other people feed expressed breastmilk to the babies/kids) may actually increase the risk of spread, and get those others sick.

• I added a note to the Q to emphasize the current advise to continue to breastfeed. – tomka Mar 19 at 14:22