Distribution of Actual Births Around Estimated Due Date

Given that we're now in overtime here for our second child, I have become very interested in the question of what our "due date" meant in the first place.

The plotted on the left are data from a study of Canadian births, 1972-1986 (Arbuckle & Sherman 1989). I then started trying to aggregate some data for my own plots.

The first thing I noticed required a little attention was the "fence post" problem relating to recording gestational age. If a study A records births in the 39th week, where exactly is that on the x axis? Well, if we assume that they started counting with 1, then the 39th week is actually 38.5 +/- 0.5 weeks (they are counting fence). But if study B records births from week 39-40, the implication is that they started at 0 (they are counting fence posts), so 39-40 is 39.5 +/- 0.5. That was a little tricky.

Then we have studies that bin over different time intervals. If another study C records births at 37-41 weeks, how do we relate that to A and B? The intelligent way to plot this would be to use the probability density of going into labor, which divides out by the length of time over which the observation was made. So we should be careful to do that.

And then there's just the general problem that a lot of studies list percentages for births in each time bin, but don't list total populations or error bars, so we don't know what the errors in their measurements were. Ugh. So I crossed my fingers and hoped they followed good practices with their significant figures: I assigned an error equal to the last significant digit they list.


I got most of my data from this semi-thorough compilation of census data and going to some of the original sources. The data aren't great, but they are adequate (see plot to left). I fit to the data an increasing exponential tail to the left, plus a normal distribution. The fit isn't great (despite some claims in the literature of it being normally distributed), but it captures enough of the overall distribution.


The width of the normal distribution was 1.5 weeks, centered at 39.5 weeks. This seems consistent with several sources that suggest ~10% of pregnancies would go into the 42nd week if they were allowed to. It also suggests that the "due date" means the mean of the normal portion of the distribution. Yet fully 1/2 of pregnancies will go beyond the expected due date, and 1/6th will go past 41 weeks, according to this coarse fit.

Of course, none of this accounts for biases that we know exist. The growing prevalence of inductions and C-sections move births earlier artificially. Although the statistical significance may questionable owing to systematic biases (self-selection for uncomplicated pregnancies, etc.), it appears that the recorded midwife births go later than the aggregate (presumably hospital-dominated) births at the 2-sigma level. This may potentially indicate that without intervention biases, the distribution of birth dates around the expected due date could be broader and weighted toward later dates.

Don Backer

Sadly, the world and I lost Don Backer on July 25, 2010. In addition to being an inspiring scientist, instrumentalist, and educator, Don was my graduate and post-doctoral advisor, and my close friend. He and I worked closely together from 2004 until last Sunday, when he died suddenly of an apparent heart attack.



Don was well known for discovering the first millisecond pulsar early in his career. More recently, he and I had been working on the Precision Array for Probing the Epoch of Reionization--and experiment for detecting the first stars and galaxies that formed in the universe. We recently had a made a lot of exciting progress with this experiment, and it is especially tragic to lose him at such a pivotal time.

Don was a very warm yet reserved man. He was always extremely busy, and I envied his ability to juggle a huge number of tasks at once. Yet every time I walked into his office, gave me a big welcoming smile, saying "Hi Aaron! Come on in." In that instant between when he looked up and when recognized me, I would sometimes see a hint of displeasure at being interrupted (a lot of people in the department walked into Don's office to hassle him about any of the many projects that he was involved in), but it was always gone the instant he recognized me, and I took pride in being someone from whom Don welcomed interruptions.

Don was always a model to me of how to be and instrumentalist and a scientist. I've long been interested in both building and using scientific instruments, and Don was a shining example of how to do both. I learned a lot from Don that helped guide me professionally, and I owe him a lot for his advice and generosity. It is sobering to consider that my next steps will have to be without Don's quiet support and encouragement. In many respects, though, Don's generosity has already helped pave the next steps for me. I'm sure I will continue to incur debt to him for years to come.

One of my favorite qualities of Don was his grand sense of adventure. Our foray into the Karoo desert to deploy PAPER in South Africa could not have happened without Don's enthusiasm for traveling, roughing it, and flying by the seat of the pants. I loved going on deployment expeditions with Don. He was always bright-eyed and smiling, summoning such energy at 66 years that I, at 29, struggled to keep up. It was not hard to see the Don of the black-and-white photographs, the same wiry energy and wry grin that stood in front of me.

Don was never very forthcoming with advice--he advised me more by example. I'm pretty sure this was a result of a very ingrained sense of humility. Don never said "you're wrong", or "you should". I think he didn't feel it was his place to pass judgment on people. Despite this humility, or probably because of it, Don was an effective leader. Without badgering people or using heavy-handed methods, Don brought people into consensus and helped move projects forward. Unfortunately, his effectiveness, coupled with his self-described "responsibility gene", meant that he was often called upon to bail out troubled projects, and he had a hard time refusing them. I often wished Don spent more time on PAPER. I think he did, too.

Don and I were a great team. I'm not a good multi-tasker. Don insulated me from a lot of project management, logistics, and distractions, carving out a space for me to work effectively toward our goal. Soon, some of the important products of our partnership will bear fruit, and I'm sad that Don won't be there to see it. But he knew it was in the works before he left, and for that I am thankful.

I'm sad to have lost a good friend and mentor. Things are hard now as we try to pick up the pieces of all the many things Don was managing. I'm sad that he's not here to help. He was always good at bailing us out.

Who Dominates Health Care Costs?

There's nothing like a little bout with MRSA to make one pay a little more attention to the state of health care legislation. Two nights in the ER are definitely making me thankful for health insurance. Knowing that it wasn't going to cost me an arm and a leg to get antibiotics through an IV (in fact, it probably saved me the leg) definitely helped me to seek care early, rather than waiting for the infection to get truly life-threatening. And that probably saved in health care costs in the long run.

I've heard it argued many times by the other side that universal health care will drive up the cost of health care for everyone, because so-called "healthy people" will be paying, through their premiums, for the bills of the "unhealthy". Ignoring that:

1. the above is a tautological statement about what insurance is
2. people routinely go from the "healthy" group to the "unhealthy" group and back again
3. we should maybe feel a moral obligation to care for the unhealthy

Yeah, ignoring that, I wanted to know if the underlying assumption was, in fact, true. Who dominates health care costs? Is it the small number of extremely sick people? Or is it the larger number of moderately sick people?

To answer that question, I went searching for the population distribution of health care costs. I found the following publication: Variations in Lifetime Healthcare Costs across a Population (Forget et al. 2008). To the left are reproduced Figs. 4 and 5.

Given all the hype, I was somewhat underwhelmed to see that these curves depict (with the exception of an excess at the lowest cost bin) a gamma distribution. This isn't surprising, because a gamma distribution is supposed to represent the sum of a bunch of exponentially-distributed random variables.

To find the contribution of people in each cost bin to the total health care cost of the population, we simply need to multiply the population of that bin (drawn from a gamma function) by the mean health care cost of that bin (a linearly increasing function). Setting the mode of the gamma distributon to $90k for females, and tweaking the k and theta parameters (I'll chi-by-eye it at k=4.5, theta=1.0) we get the following distributions of fractional population (black) and fractional total health care cost (red), as a function of lifetime healthcare cost:

So who dominates health care costs? Those just slightly above the mode, which is to say, the large number of people who are just a little sicker than most. And that really could be any of us, folks.

Countability and Strong Positive Anymore

Seven or eight years ago, I discovered that I have a linguistic condition called "Positive Anymore." I was in college, chatting with my roommates, and said something like "Anymore, I just lift weights in the Leverett gym." One of my roommates just couldn't take it anymore. "I've heard you say 'anymore' like that for years now. What the hell does it mean?" A quick poll of those present revealed that I was the only one for whom that construction made grammatical sense. My aunt (a linguistics professor) gave me the prognosis: I had Positive Anymore.

I used to think that PA was a linguistic shortcoming of mine, but anymore I'm convinced it's more like a superpower. Whereas most English speakers can only use the word in negative constructions like "I don't drive anymore" I have the uncanny ability to use it positively as per the first sentence in this paragraph. Moreover, I don't just have PA, I have strong PA, which means I can, at will, detach "anymore" and put it anywhere in a sentence. "Anymore, I just take the bus." Astounded yet?
If you're still having trouble parsing that, replace "anymore" with "nowadays"--to me, they mean about the same thing.

I receive no end of flak from friends, relatives, and spouses about my PA, although it's not that uncommon a condition. In fact, I have caught several of my relatives (mostly on my dad's side) using PA even after making fun of me for my PA. They have it and don't even know it.

Anyway, S and I were trying to figure out if my use of PA was inconsistent with my use of "any", which I use according to the standard rules. But it turns out the standard rules are weirder than you might think. For example, "I don't want any spam" is a grammatical negative construction; "I want any spam" is ungrammatical and positive. "Do you want any spam?" is grammatical and seems positive, but it turns out that there is an implied negative in English owing to the uncertainty inherent in questions and subjunctives. Hmph. And then what about "I like any spam I can get?" That seems positive again, but the clause "I can get" is required to make it grammatical. And then the plot thickens. "I feed spam to any dog" is grammatical and did not require a clause to modify "any dog."

Our theory was that using "any" positively without a clause requires the noun modified by "any" to come in quantized units--to be countable. Dogs are countable; spam is a continuum, much like water or space-time. The best example we could think of that illustrated this was "fish". Fish can be countable noun (number of live fishes) or continuum noun (amount of dead fish to eat). If I say "I'll take any fish," the ambiguity in the countability of fish is broken--it's clear I'm talking about a live fish (or, perhaps, a type of fish, which is also countable). But if I say "I'll take any fish that you give me," the ambiguity is preserved.

The implication was that for my use of PA to be consistent with the standard use of "any" (which it may not need to be, since "anymore" is one word, not "any more"), I must be thinking of the time interval referred to by PA (i.e. now and continuing indefinitely into the future) as something countable rather than continuous. Maybe. I don't know. Anymore, I'm just really confused.

PAPER: 8 Station Results

Yesterday I submitted our paper on the 4- and 8-antenna deployments of PAPER to the Astronomical Journal & astro-ph. Although it does not place any meaningful constraints on cosmic reionization (when light from the first stars broke up the bulk of the hydrogen gas that had been sitting around since it originally formed), it nonetheless illustrates a first level of calibration and analysis towards that goal. This paper should have some impact in the community, as it shows that we are fairly well along in our experiment and it provides a first look at some of the astrophysical foregrounds that will interfere with detecting reionization. This paper also will double as the last chapter of my dissertation, so publishing it puts me in the endgame of my doctoral studies... This fall I'm going to take an NSF postdoc back at Berkeley. Basically, that just means I'll be doing the same research (along with some teaching), but with better pay.