What I Learned From Treating Childbirth as Failure, Part II
A couple of years ago, I wrote a blog post titled "What I Learned From Treating Childbirth as Failure" that conveniently ended up getting published the day before my daughter was born. You should read it first, but to summarize it demonstrates how we can predict the odds of an event happening during certain time intervals even when the original data is highly censored.
Since then, several people have asked (two in the comments alone) where I came up with the numbers I stated at the end:
- When should a relative arrive on a 7-day stay to have the greatest chance of being there for the birth? (May 17th)
- What are the odds of the baby being born on a weekend? (28.6%)
- What are the odds of the baby being born on her great-grandmother's birthday, May 14th? (3.7%)
To answer these, let's recap: using data, we have a good estimate of the distribution that natural childbirths follow based on days in gestation. That distribution can be seen here:
I've marked with a reference line 280 days, which corresponds to the "due date." So to keep things in terms of my original post, the due date corresponds to day 280, or May 19. A key concept to understand in calculating the numbers stated above is that this is a continuous scale, so May 19 includes all points starting at 280 and going up to, but not including, 281. So 280.1 and 280.3587465 and 280.9999999 are all part of May 19th.
So the odds that the baby would be born on the due date is the probability of falling between 280 and 281 in the distribution above are about 4.7%, as shown here:
Now on to the specific bullet points from above, which I will cover in a different order for simplicity...
What are the odds of the baby being born on her great-grandmother's birthday, May 14th?
If you grasped what I said above about the odds of being born on the due date, this one is easy. It's the same idea, except May 14th corresponds to day 275, so I just need the odds of being between 275 and 276 in the distribution:
There you have it...3.7%. Another way to find that would be to find the cumulative odds of being born by day 276 (30.3%) and subtract the cumulative odds of being born by day 275 (26.6%):
When should a relative arrive on a 7-day stay to have the greatest chance of being there for the birth?
Thus far we have looked at a single day, but there's no reason we couldn't look at a half-day, a specific hour, a 3-day period, or any other time frame. So this question would pertain to a 7-day period rather than a 1-day period. For simplicity I limited myself to midnight-to-midnight, which is obviously not practical but keeps things conceptually easier to explain. For example, here are the odds of the baby being born during the 7-day period starting day 273 (a week before the due date) and running through day 280 (the due date):
So the odds of the baby being born early but not more than a week early are about 27.5%. To answer the question, I just look at every 7-day period that is reasonable and find the highest odds, which corresponds to two days before the due date (May 17th) and the six days afterward:
This gives a 32.1% chance of being there for the birth.
What are the odds of the baby being born on a weekend?
This is really no different than the earlier questions, except now there are multiple areas of interest. First we need to know which days correspond to weekends. The due date of May 19th—day 280—was a Saturday. So days 280 and 281 were on a weekend, and likewise were days 273 and 274 and days 287 and 288 and so on. Luckily there is a limited area where reasonable probability exists, so we just add up the odds of each weekend:
.0009995 + .007606 + .02827 + .06573 + .09391 + .06904 + .01893 + .001205 = .28569
There you have it—there is a 28.6% chance of the baby being born on a weekend.
I hope this has cleared up how you can convert the estimated distribution to the odds of an event happening during any time interval...in my case, my daughter was ultimately not born on her great-grandmother's birthday, and not on a weekend. But she was born during the most likely 7-day window, to the delight of her visiting grandparents!