Tag Archives: statistics

The Pregnant Mathematician Continues Teaching, Probably

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Recent Events
During the month of June, I’m teaching our “Introductory Calculus (Math 120)” course. We meet five days a week for 145 minutes, with a total of twenty class sessions running June 5th through July 2nd. Our final exam will be held on Wednesday, July 3rd. A few days ago, my very compassionate students expressed interest in knowing the probability their final exam would be cancelled due to an unexpectedly early birth of my daughter. I’m sure their inquiry was based solely on concerns for our health and not at all from them hoping to escape the intellectual challenge of a cumulative final exam!

Also recently, my husband and I went on an “Expectant Parents Tour” of the hospital where I will be delivering. Our son was born in November 2010, but at a different hospital; this time, I will be delivering at a hospital that was still under construction then! Since they have recently opened their doors, their NICU [Neonatal Intensive Care Unit] is still a “Level 1” facility. This means that they are certified to care for healthy infants born at 36 weeks or later (measured from LMP) or 34 weeks gestational age (i.e., since conception). Infants who are born with complications, or who are born before 36 weeks, are usually transferred to another local hospital. It turns out that reaching “Week 36” of my pregnancy and having lots of calculus final exams to grade will coincide perfectly.

Some Probability
While on our tour, I started wondering about the chance that I would go into labor early enough that it would affect my summer class. Not only would this be an unfortunate inconvenience for my students, it would also mean that I would probably have to deliver at a different hospital since I wouldn’t be at 36 weeks yet. What’s the chance this happens?

First, I should point out that I’ve had a relatively uneventful pregnancy — thankfully, both my unborn daughter and I have been in excellent health (if not a little grumpy from having to share the same circulatory system and oxygen supply). Second, my son was born within a few days of his due date, a little on the early side. Third, I’m not carrying multiples, nor am I expecting any major complications as my due date approaches. So we will just assume that this is an average pregnancy as far as the medical issues are concerned.

One of the things I talked about in my original “Pregnant Mathematician” post was how due dates are calculated using Naegele’s rule. Also, there was a rather large (n=427,582) study done in Norway [See Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.] that found the mean gestational length for singleton pregnancies was 281 days, with a standard deviation of 13 days.

Let’s assume a mean of 281 days and a standard deviation of 13 days. What’s the chance a woman goes into labor 251 days or earlier (corresponding to 36 weeks)? Notice that 251 is about 281+(-2.31)*13, so giving birth prior to 36 weeks means you’re about 2.31 standard deviations away from the mean. By the Empirical Rule, I know this would be quite rare: There’s less than a 2.5% chance!

We can use Wolfram|Alpha to compute the exact probability. Our input is the command “CDF[NormalDistribution[mean, stdev], X]”; in this case, we are taking mean=281, stdev=13, and X=251. Wolfram|Alpha returns a result of 0.0105081, meaning there is about a 1.1% probability that I will give birth early enough to impact my current students.

It’s Gonna Be How Hot?!!

According to tomorrow’s weather forecast, the heat index here will reach 110 degrees tomorrow. This is miserably hot for everyone, but especially if you’re pregnant. Although my actual due “date” is in the first few days of August, I’m sincerely hoping that the birth occurs sometime in July. August heat in South Carolina is no joke! Wolfram|Alpha comforted me with its computation that there’s about a 41% chance I will give birth before I have the opportunity to be pregnant in August.

I’m also going to assume my chances of a July delivery are even higher than this, since human gestation doesn’t exactly follow a normal distribution. While a measurable percentage of moms give birth two or more weeks early, nearly none give birth two or more weeks late. By that point, OBs usually induce labor because of declining amounts of amniotic fluid and concerns for the health of the newborn. I’m going to take this and a “fingers crossed” approach and assume that a July birthday is at least 50% possible.

[There’s a name for such a distribution; I thought it was a truncated normal distribution, but that doesn’t seem to be quite right. The statistician who told me the term isn’t in his office at present! Anyone know what it’s called?]

Postscript: Dear Students,
Regardless of when I deliver, I can assure you that your calculus final examination will occur as scheduled. I know lots and lots and lots and lots of people who really enjoy torturing unsuspecting college students with tough calculus exams, and it would be easy for me to cajole one of those people into proctoring your test! So, don’t fear: You will certainly have the opportunity to demonstrate all the calculus you have learned this summer & feel proud of your scholastic achievement upon completing our course — including its final exam. 🙂

The Pregnant Mathematician Drinks Coffee

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In our math department’s faculty lounge, one can often find a liquid-y substance some people refer to as “coffee.” This designation seems questionable to me, so instead I have opted for a Starbucks prepaid card. I don’t drink a lot of coffee, but I do have one cup in the morning.

According to Starbucks, a tall (12 oz) cup of Pike Place Roast contains about 260mg of caffeine. Personally, I prefer the Blonde Roast, but I haven’t found its nutritional data. One of my students inquired at Starbucks and the barista told him that the Blonde Roast contains more caffeine per ounce than the others; apparently, since it is less roasted, less caffeine is lost during the roasting process, leading to more in the final product. I wonder if this is true.

How long does caffeine hang out in your body?

Wikipedia reports that the biological half-life of caffeine in an adult human is around 5 hours. The half-life of a substance is the amount of time required for half of the material present to metabolize. In other words, if the half-life of caffeine in your system is 5 hours and you consume 260mg of caffeine at 8am, then five hours later (at 1pm) we would expect 130mg of caffeine to remain in your system — provided you haven’t consumed any more caffeine since your morning coffee.

The half-life of caffeine in your system is related to lots of factors: Your age, your weight, what medications you’re taking, how well your liver is functioning, and whether or not you’re pregnant.

Maternal Caffeine Consumption & Half-Life
It turns out that if you’re pregnant, the half-life of caffeine increases quite a bit. In other words, it takes your body longer to metabolize caffeine. Today’s quick search yielded these few medical studies that agree about this:

According to Golding’s study, by the 35th week of pregnancy, the half-life of caffeine increase to a high of 18 hours. (For comparison, the Knutti et al. study cites a half-life of 10.5 hours during the last four weeks of pregnancy.) Since I am not yet in my 35th week of pregnancy, let’s assume the half-life of caffeine in my body is 12 hours. How is this different from my (assumed) non-pregnant state, when its half-life is only 5 hours?

Suppose I consume a Pike Roast Tall coffee at 8am that contains 260mg of caffeine. What time will it be when only 50mg of caffeine remain in my system? When not pregnant, it would take my body about 11.9 hours, so by 8pm less than 50mg of caffeine would be found in my system. Meanwhile, a half-life of 12 hours means it would take my body 28.5 hours — that’s over a full day!

We discussed this calculation as part of my PreCalculus course a few semesters ago. One student, who was usually rather quiet and didn’t ask many questions, raised his hand. He asked, “So, Dr. Owens, what you’re telling me is that to save money on Starbucks coffee, I should get pregnant?” Laughter ensued, and I assured my students that getting pregnant as a cost-savings measure was really not an optimal strategy.

As far as the risk to maternal and neonate health, the American Congress of Obstetricians and Gynecologists concluded “Moderate caffeine consumption (less than 200 mg per day) does not appear to be a major contributing factor in miscarriage or preterm birth” in 2010 [1] [2].

My Conclusion: Okay, it’s probably best if I limit my caffeine intake while pregnant. But I also have to think about my overall happiness and my enjoyment of life — as far as caffeine goes, today’s science seems to imply that my occasional Starbucks habit is a net positive (happiness minus risk), even when taking into account its expense (increased work productivity minus $2 per cup).

“The Pregnant Mathematician” Drinks Glucola

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Glucose Challenge Screen for Gestational Diabetes
As I posted about a few days ago, this week I had a one-hour glucose challenge test to screen for Gestational Diabetes (GDM). Today I received a phone call from my OB’s office informing me that my results were back and they were within the “normal” levels. Getting a negative result is comforting, but then I went back to hunting for statistical data on what this result really means.

According to an article I found in Obstetrics & Gynaecology, a 1994 study (“Poor sensitivity of the fifty-gram one-hour glucose screening test for hyperglycemia“) by van Turnhout HE, Lotgering FK, Wallenburg HC reported the sensitivity and specificity of the 1-hour glucose challenge test were 27% and 89%, respectively, with a prevalence rate of 5%.

In statistics, sensitivity and specificity are markers of how good of a test you’re considering. The sensitivity of a test tells you, “Out of all the people who have the condition, what percent of them will test positive?” Similarly, the specificity of a test tells you, “Out of all the people who don’t have the condition, what percent of them will test negative?”

If a test were perfect, we would expect both of these to be 100%. This would mean that 100% of people who have the condition really test positive, and 100% of the people who don’t have the condition really test negative. Of course, in the real world, this never really happens.

What Can I Conclude?
Another way we can gauge the performance of a test is to find its positive predictive value and its negative predictive value. I’m going to assume the sensitivity and specificity in the study cited above are correct. The same study above also gives a positive predictive value of 11% and a negative predictive value of 96%, but what do these numbers mean?

Let’s assume we give the same 1-hour glucose challenge test to 10,000 pregnant women. With a prevalence rate of 5%, we would expect 500 women to have GDM and 9500 not to have GDM. Of the 500 with GDM, since the sensitivity is 27%, we know 27% of 500 would screen positive, for a total of 135 women. These are women who have GDM and whose screening will come back positive. Meanwhile, of the 9500 women without GDM, since the specificity is 89%, we would expect 89% of 9500 or 8455 women to have a true negative result. The status of all of our 10,000 participants is displayed in the table below:

Women with GDM Women without GDM
Women who test positive 135 1045
Women who test negative 365 8455
Total 500 9500

According to this table, a total of 135+1045=1180 women would test positive. Of the women who get a positive result, only 135 of them really have GDM; this is the positive predictive value and, in this case, it’s 135/1180 = 11.44%.

What about the women who, like me, get a negative result? There are 8820 of us, and 8455 of us don’t have GDM. This gives a negative predictive value of 8455/8820 = 95.86%.

Was This Worth $40?
My results were negative, so I am one of the women with a negative result. The values above tell me that since I got a negative result on my glucose screening, I can assume there’s about a 96% chance I don’t have GDM. I’m waiting to be billed for this screening, but I’ll go with my initial $40 estimate. Even after this analysis, I’m still wondering if the knowledge I gained was worth the $40 I paid for it. (“Everything is worth what its purchaser will pay for it,” so I suppose this must be too.)

I feel unqualified to answer the “Worth it?” question because I don’t know a way to quantify the importance of this test. It seems clear that if a condition is really, really awful, then finding out you’ve got it is probably worth $40, and so is finding out you’re home free.

Is GDM really, really awful? It certainly has the potential to affect both my health and the health of my unborn child, so it seems better to know about it than not. But there are lots and lots of things that could affect our health that I don’t know about, and won’t be screened for, and probably won’t ever hear about.

One thing I wish I did have, for this screening and all of the others I have been (or will be) offered, is data ahead of time. I want to know the false positive and false negative rates. I want to know the sensitivity and the specificity and the predictive values. And I want to know how much money it’s going to cost me, and how much of a hassle it’s going to be. Lastly, I would like to know more about the medical significance of the condition, and since I’m not a medical doctor, I need it in some kind of quantifiable metric for when I do these kinds of calculations.

“The Pregnant Mathematician”

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I’d like to start by pointing out I know nearly nothing about medicine or obstetrics. I wouldn’t call myself an expert in statistics. I am a mathematician: In my life, what this means is that I was trained to approach problems from a very particular viewpoint. When confronted with choices in my own life, I can’t help but think of them as a math professor would. I would love to find someone trained in obstetrics or medicine to collaborate with on the issues I’ve written about below! Do you know of anyone who would be interested?

“The Pregnant Mathematician”
There isn’t a lot written about what it’s like being a pregnant mathematician. While I would guess that the overlap of these two population groups is small, a better reason might be that it is exhausting to be both a mathematician and pregnant at the same time, and the idea of adding “blogger” to that list seems insane.

Nevertheless, it’s difficult for me to think about having either description without the other. I Tweeted yesterday one of my “daydream goals”:

What would I write about? Well, here’s how my pregnancy has become mathematized over the last couple of months.

Glucola for Breakfast
This morning I was screened for Gestational Diabetes (GDM). This required fasting overnight (nothing but water), arriving at the lab, drinking a very sugary drink, waiting an hour, and having my blood drawn. I’m not exactly sure how much I’ll be billed for this test, but my guess is that it will cost about $40 out-of-pocket.

One of the things I’m always interested in is what the false positive rate for these types of screenings is. In other words, if my doctor phones me in a few days and tells me that my screening test came back positive, what is the chance I really have the underlying condition?

During my hour wait in my doctor’s office, I spent some time trying to find out this answer. According to this New Zealand-based maternity site,

“Approximately 15 – 20% of pregnant women test positive on the [glucose screening] test although only 2 – 5% will have any form of diabetes.”

In other words, if we give the same test I took this morning to 100 pregnant women, we should expect 15-20 of them to have a positive result. However, it will turn out (after further screening) that only 2-5 of them will actually have any form of diabetes. So if my OB tells me that today’s screening has come back positive, then this means I am one of the 20 women with positive results; but only 2-5 of us actually have gestational diabetes. Supposing that 5 of us have the condition, that means 5 out of the 20 positives are true positives.

If my screening this morning comes back positive, there’s around a 5/20 = 25% chance I have any form of diabetes. Looking at this another way, without taking the screening test, my “best guess” of my chance of having GDM is between 3% and 10%, based on the incidence rate for the overall population. So I can feel comfortable that there’s between a 90% and 97% chance that I don’t have GDM. But now I have taken the test; if it comes back positive, this means there’s still a 75% chance that I don’t have GDM.

Basically, I think I just paid $40 to find out if my risk is 10%, or if it’s really higher and is 25%.

How valuable is this knowledge? Is it worth fasting overnight? Is it worth taking time off from work to sit in the waiting room for an hour? Is it worth the $40 I’ll be billed? I don’t know. I never know the answers to these questions, but it seems I always choose to follow my OB’s advice anyway — her practice suggests screening of 100% of their maternity patients, so I just trusted their expertise.

Due Date Calculation
One of the things people always ask when they find out you’re pregnant is, “When are you due?” My obstetrician has some date written down on my medical records, labeled EDD (“Expected Date of Delivery”), that will happen this summer. The basic way the EDD is calculated is using Naegele’s Rule: Take the day of your last menstrual period (LMP), add one year, subtract three months, and add seven days. Example: If LMP date was in January this year (say, January 18th, 2013), adding one year gives 1/18/2014, subtracting 3-months gives 10/18/2013, and adding seven days gives a final EDD of October 25, 2013. There are lots of online calculators that will perform this calculation for you, like this one. This calculation gives 280 days post LMP for an estimated delivery date. But how accurate is that?

Suppose we take forty weeks (280 days) as the mean length of human pregnancies, measured from LMP to delivery date. It seems reasonable to expect that even if your actual delivery date isn’t your EDD, at least it’ll probably be in the same 7-day window. Unfortunately, this isn’t true either. Not only is it not very likely for you to deliver on your EDD, but it isn’t very likely you’ll deliver that week.

A study done in Norway (Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.) involving 427,582 singleton pregnancies found a mean gestation length of 281 days, with a standard deviation of 13 days. By the empirical rule, this means more than 30% of women won’t give birth in the 26-days surrounding their EDD!

Example: Your EDD is June 15th. According to the study cited above, there’s more than a 30% chance that your real delivery day will be either before June 2nd or after June 28th. There’s a very real possibility you won’t even give birth in the month of June. (After talking about this with a few colleagues, one of them found a Statistics book that cited a standard deviation of 16 days!)

Here’s a great article about this same topic: http://spacefem.com/pregnant/charts/duedate0.php

Nuchal Translucency Screening
One non-invasive genetic screening test offered to all pregnant women in the United States is the nuchal translucency screening (“NT screening”). This screening is done in the first trimester (between weeks 11 and 14) and involves an ultrasound and a blood test. It screens for Down Syndrome (trisomy 21) and other chromosomal abnormalities caused by extra copies of chromosomes. The ultrasound image is read by measuring the width of the nuchal fold, found at the base of the neck. If the measurement is outside of the normal range, further testing is indicated. The cost of the test varies widely, but is somewhere in the neighborhood of $200.

The risk of carrying a baby with chromosomal abnormalities increases with maternal age. This page has a large table of the risk of Down Syndrome (and other trisomy abnormalities) as a function of maternal age. I’ll use a maternal age of 29, which carries with it a risk factor of 1 in 1000 for Down Syndrome. This means that out of 1000 moms (all age 29), one will have an affected baby.

One of the problems with the NT screening is the false positive risk. This would happen if your test comes back positive, when in fact you do not have the condition. In other words, this is the “false alarm” risk. According to my OB, the false positive rate for the NT scan is 5%.

Let’s go back and talk about our 1000 women (all pregnant and age 29, with no other risk factors). If we screen all of these women, then a 5% false positive rate means that 5% of them will have a positive test but whose babies do not have Down Syndrome. So that’s 50 women who will get a “false alarm” from this test. Also, one woman will have a true positive: She’ll get a positive test and her baby will have the condition. Altogether, out of the 1000 women, 51 of them will get a positive test. And out of these 51 women, only 1 will have an affected baby. The other 50 women will undergo lots of unnecessary worry.

I was offered the NT screening during the early portion of my pregnancy. Given my age, my child’s risk of Down Syndrome was around 1%. I was given the option to pay $200 to have the screening. If it came back negative, then I’d be worry free. If it came back positive, this would mean my child’s risk is about 1/50 or 2%.

In other words, without taking the screening, there was a 99% chance my child does not have Down Syndrome. Or, if I wanted, I could pay $200 for a test that, if positive, still means there’s a 98% chance my child does not have Down Syndrome. Is this test worth $200 to me?

As with the GDM screening, I followed my OB’s advice and had the nuchal translucency test a few months ago. The results were normal, which was comforting. The bill was around $250.

It’s tough having to juggle my emotions as a mother, my knowledge of statistics as a mathematician, and my interest in minimizing unnecessary financial expenses.

Statistics Group Projects: In Progress

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The Elementary Statistics students in my course are wandering the streets of downtown Charleston gathering data for their group projects. There are five groups and here are the questions they are asking (along with their best guess as to what they will find):

  • Do you own a bike? Best guess: At least 30% “Yes” response rate
  • Do you have a passport? At most 40%
  • Are you on vacation? About 30%
  • Do you consume alcohol? At least 80%
  • Have you ever had a fake ID? Around 50%

I look forward to seeing their data!

Statistics Group Project

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Project Motivation
There are two class meetings left in my “Elementary Statistics” summer course. This class time will be devoted to students working together on a group project. Last semester when I taught this course for the first time I really wanted to implement some type of end-of-term project. I wanted the project to be collaborative in nature since both my own experiences and recent research in education have shown that students explaining concepts to each other is as important to their learning process as hearing their instructor’s explanations. I also wanted the project to be somewhat self-designed by the groups themselves. It was my hope that giving them some freedom in their projects would increase their interest level in what they were doing.

The topics we finished covering at the end of the course were about creating confidence intervals and performing hypothesis tests (sometimes called tests of significance). Because we discussed this material so recently, it seemed appropriate to have this be the jumping-off point for the projects.

Project Introduction
I wanted the students to have experience going out into the “real world” to gather data, so the project asks them to conduct interviews with people they find around campus. Since it’s only a week-long project (instead of over an entire semester), to make things easier each group has to agree on a single”Yes” or “No” question to ask their random sample. There are three rules for the question.

  • First, each member of the group must agree with the group’s decision on the question. They have to discuss different ideas, vote on them, and eventually reach consensus.
  • Second, the question must be “interesting.” This is hard to define, but basically I want them to avoid boring questions like “Are you a human being?” or “Have you ever been to Mars?” that will result in boring data.
  • Third, the question must be “appropriate” — it has to be something each group member would feel comfortable asking a perfect stranger or their grandmother or their kid brother. (Hopefully they would know to avoid offensive or disrespectful or inappropriately personal questions, but who knows?)

Once they have chosen their question, each individual is asked to guess (to the nearest 10%) what proportion of interviewees will answer “Yes” to the question. After reaching an individual conclusion, the groups discuss what they expect as a group. I wrote a handout describing the “What” and “How” of prior probability distributions and each group works on creating [a very basic] one before they are allowed to leave to gather data.

Project Report
The groups have the rest of the class time to gather data together. I tried to avoid giving them much direction on who they should interview, or where they should find the people, or what types of people to ask. (For instance, do they want to focus on College of Charleston undergrads, or are tourists okay too?) I suggested to them that they need to keep in mind a lot of the ideas we discussed in the class, like:

  • What’s an appropriate sample size?
  • What sampling method should we use? (Convenience, cluster, stratified, systematic, etc.)
  • Should we expect bias in our data? If so, what types? (Sampling bias, response bias, nonresponse bias, etc.)
  • Can we do anything to eliminate bias?

Eventually the groups must produce a typed project report, outlining their process from how they decided on a question and constructed their prior to where they conducted their interviews. They must use the methods of inferential statistics that we learned in our class to create a confidence interval for the proportion of subjects who said “Yes” and give a correct interpretation of the confidence interval. They also have to perform a one-proportion hypothesis test. They are expected to use their prior probability distribution to formulate a claim to test. They are graded on both their data analysis and interpretation of results.

Project Grading
I created a grading rubric for the project. It’s available as a public Dropbox file: 104-project-rubric.pdf A colleague looked over it in the copy room and commented, “You sure are overly detailed with that thing!” This is probably a fair criticism, but mostly I was trying to avoid hearing lots of student questions that boiled down to, “What is the least I have to do in order to get an A?”

Something that comes up a lot in discussions about graded collaborative assignments is the “slacker problem”, i.e., How do you keep students from getting by doing zero work? I don’t have a good answer for this. I know that when I was a student, I was annoyed by free-loaders, so I have empathy for students who feel the same way. One of the categories on my grading rubric is “Teamwork Assessment.” Each student must individually send me a confidential e-mail discussing how their group functioned as a team and how they contributed to the overall project. They are asked to give their team a grade of how well they worked together. It was my hope that telling them on the project rubric that (a) they are responsible for their group functioning as a team and (b) they are also responsible for ensuring they contribute to the group that it would create a cultural pressure toward equal collaboration. I can’t say for sure how successful this was last semester, however I was happy that out of nearly 100 students, I only had one or two complain about slackers in their groups.

Looking Forward
This summer’s class has a strong group mentality, I think partly because we have been spending ten hours a week together in class. I hope that this will contribute to great collaborative effort toward these projects. I am also excited to see what questions they will ask and what their data will show. I’ll end this post with a few questions I remember from the class projects last spring:

  • Do you have a fake ID?
  • Do you own an iPhone?
  • Will you vote to re-elect Barack Obama?
  • Did you drink alcohol last weekend?
  • Do you have blue eyes?
  • Do you have a car on campus?