Tuesday, November 18, 2014

The World Cup Problem: Germany v. Brazil

Earlier this semester I posed this problem to my Bayesian statistics class at Olin College:
In the 2014 FIFA World Cup, Germany played Brazil in a semifinal match. Germany scored after 11 minutes and again at the 23 minute mark. At that point in the match, how many goals would you expect Germany to score after 90 minutes? What was the probability that they would score 5 more goals (as, in fact, they did)?
Before you can answer a question like this, you have to make some modeling decisions.  As I suggested to my class, scoring in games like soccer and hockey can be well modeled by a Poisson process, which assumes that each team, against a given opponent, will score goals at some goal-scoring rate, λ, and that this rate is stationary; in other words, the probability of scoring a goal is about the same at any point during the game.

If this model holds, we expect the distribution of time between goals to be exponential, and the distribution of goals per game to be Poisson.

My solution to this problem uses the computation framework from my book, Think Bayes.  The framework is described in this notebook.  If you have read Think Bayes or attended one of my workshops, you might want to attempt this problem before you look at my solution.

If you solve this problem analytically, or use MCMC, and you want to share your solution, please let me know and I will post it here.

And when you are ready, you can see my solution in this notebook.

I will post more of the exercises from my class over the next few weeks.  Coming next: The World Cup Problem Part II: Germany v. Argentina.


UPDATE November 19, 2014:  Cameron Davidon-Pilon kindly (and quickly!) responded to my request for a solution to this problem using PyMC.  You can see his solution here.

If you want to learn more about PyMC, an excellent place to start is Cameron's online book Bayesian Methods for Hackers.

Saturday, October 4, 2014

On efficient algorithms for finding the goddamn endnotes

Introduction

In many recent books, the goddamn endnotes are numbered sequentially within each chapter, but chapter numbers seldom appear in the header on each page.  So a reader who wants to find a goddamn endnote typically has to search backward to figure out which chapter they are reading before they can find the goddamn endnote.  Several simple changes can make this process more efficient and less infuriating.

In this paper, we present a novel algorithm for finding a goddamn endnote (FGE) in O(1) time; that is, time bounded by a constant regardless of the size or number of chapters.

Background

The most widely deployed algorithm for FGE requires O(n+m) time; that is, time proportional to either n, the number of pages in a chapter, or m, the number of chapters, whichever is larger.  The following pseudocode sketches this algorithm:

1) Store the endnote number you want to find as e.
2) Figure out what chapter you are reading by searching for the beginning of the chapter, and store the result as c.
3) Find the footnotes at the end of the chapter and find the footnotes for chapter c.
4) Look up footnote e.

The most common implementation of Step 2 is linear in n, the number of pages in the chapter.  Some readers can execute a binary search by comparing chapter titles in the header, reducing search time to O(log n).

Similarly, the most common implementation of Step 3 is linear in m, the number of chapters, but some readers perform a binary search.

Assuming that the number of footnotes in a chapter is bounded by a constant, we consider Step 4 to be constant time.

Alternatives

1) Put the goddamn chapter number (GCN) in the goddamn header.  This simple change immediately reduces Step 2 to constant time.

2) Number the goddamn endnotes from the beginning to the end of the book.  Do not start over at the beginning of each goddamn chapter.  This change also reduces Step 2 to constant time, but with this change Step 4 may no longer be considered constant time.

3) Along with the endnote number, e, also include p, the page where the goddamn endnote appears. This change makes Step 2 constant time; and Step 3, although technically log time, practically constant time for most readers.

4) Put the goddamn endnotes at the bottom of the goddamn page (cf. footnotes).  For a bounded page size, looking at the bottom of the page is a constant time operation.

Recommendations

We recommend that publishers replace existing linear-time algorithms with any of several easy-to-implement, efficient alternatives, goddammit.

Monday, September 29, 2014

Two hour marathon in 2041

Today Dennis Kimetto ran the Berlin Marathon in 2:02:57, shaving 26 seconds off the previous world record.  That means it's time to check in on the world record progression and update my update from last year of  my article from two years ago.  The following is a revised version of that article, including the new data point.

Abstract: I propose a model that explains why world record progressions in running speed are improving linearly, and should continue as long as the population of potential runners grows exponentially.  Based on recent marathon world records, I extrapolate that we will break the two-hour barrier in 2041.

-----

Let me start with the punchline:


The blue points show the progression of marathon records since 1970, including the new mark.  The blue line is a least-squares fit to the data, and the red line is the target pace for a two-hour marathon, 13.1 mph.  The blue line hits the target pace in 2041.

In general, linear extrapolation of a time series is a dubious business, but in this case I think it is justified:

1) The distribution of running speeds is not a bell curve.  It has a long tail of athletes who are much faster than normal runners.  Below I propose a model that explains this tail, and suggests that there is still room between the fastest human ever born and the fastest possible human.

2) I’m not just fitting a line to arbitrary data; there is theoretical reason to expect the progression of world records to be linear, which I present below.  And since there is no evidence that the curve is starting to roll over, I think it is reasonable to expect it to continue for a while.

3) Finally, I am not extrapolating beyond reasonable human performance. The target pace for a two-hour marathon is 13.1 mph, which is slower than the current world record for the half marathon (58:23 minutes, or 13.5 mph). It is unlikely that the current top marathoners will be able to maintain this pace for two hours, but we have no reason to think that it is beyond theoretical human capability.

My model, and the data it is based on, are below.

-----

In April 2011, I collected the world record progression for running events of various distances and plotted speed versus year (here's the data, mostly from Wikipedia). The following figure shows the results:

You might notice a pattern; for almost all of these events, the world record progression is a remarkably straight line. I am not the first person to notice, but as far as I know, no one has proposed an explanation for the shape of this curve.

Until now -- I think I know why these lines are straight.  Here are the pieces:

1) Each person's potential is determined by several factors that are independent of each other; for example, your VO2 max and the springiness of your tendons are probably unrelated.

2) Each runner's overall potential is limited by their weakest factor. For example, if there are 10 factors, and you are really good at 9 of them, but below average on the 10th, you will probably not be a world-class runner.

3) As a result of (1) and (2), potential is not normally distributed; it is long-tailed. That is, most people are slow, but there are a few people who are much faster.

4) Runner development has the structure of a pipeline. At each stage, the fastest runners are selected to go on to the next stage.

5) If the same number of people went through the pipeline each year, the rate of new records would slow down quickly.

6) But the number of people in the pipeline actually grows exponentially.

7) As a result of (5) and (6) the rate of new records is linear.

8) This model suggests that linear improvement will continue as long as the world population grows exponentially.

Let's look at each of those pieces in detail:

Physiological factors that determine running potential include VO2 max, anaerobic capacity, height, body type, muscle mass and composition (fast and slow twitch), conformation, bone strength, tendon elasticity, healing rate and probably more. Psychological factors include competitiveness, persistence, tolerance of pain and boredom, and focus.

Most of these factors have a large component that is inherent, they are mostly independent of each other, and any one of them can be a limiting factor. That is, if you are good at all of them, and bad at one, you will not be a world-class runner. To summarize: there is only one way to be fast, but there are a lot of ways to be slow.

As a simple model of these factors, we can generate a random person by picking N random numbers, where each number is normally-distributed under a logistic transform. This yields a bell-shaped distribution bounded between 0 and 1, where 0 represents the worst possible value (at least for purposes of running speed) and 1 represents the best possible value.

Then to represent the running potential of each person, we compute the minimum of these factors. Here's what the code looks like:

def GeneratePerson(n=10):
    factors = [random.normalvariate(0.0, 1.0) for i in range(n)]
    logs = [Logistic(x) for x in factors]
    return min(logs)


Yes, that's right, I just reduced a person to a single number. Cue the humanities majors lamenting the blindness and arrogance of scientists. Then explain that this is supposed to be an explanatory model, so simplicity is a virtue. A model that is as rich and complex as the world is not a model.

Here's what the distribution of potential looks like for different values of N:


When N=1, there are many people near the maximum value. If we choose 100,000 people at random, we are likely to see someone near 98% of the limit. But as N increases, the probability of large values drops fast. For N=5, the fastest runner out of 100,000 is near 85%. For N=10, he is at 65%, and for N=50 only 33%.

In this kind of lottery, it takes a long time to hit the jackpot. And that's important, because it suggests that even after 7 billion people, we might not have seen anyone near the theoretical limit.

Let's see what effect this model has on the progression of world records. Imagine that we choose a million people and test them one at a time for running potential (and suppose that we have a perfect test). As we perform tests, we keep track of the fastest runner we have seen, and plot the "world record" as a function of the number of tests.

Here's the code:

def WorldRecord(m=100000, n=10):
    data = []
    best = 0.0
    for i in xrange(m):
        person = GeneratePerson(n)
        if person > best:
            best = person
            data.append(i/m, best))
    return data

And here are the results with M=100,000 people and the number of factors N=10:


The x-axis is the fraction of people we have tested. The y-axis is the potential of the best person we have seen so far. As expected, the world record increases quickly at first and then slows down.

In fact, the time between new records grows geometrically. To see why, consider this: if it took 100 people to set the current record, we expect it to take 100 more to exceed the record. Then after 200 people, it should take 200 more. After 400 people, 400 more, and so on. Since the time between new records grows geometrically, this curve is logarithmic.

So if we test the same number of people each year, the progression of world records is logarithmic, not linear. But if the number of tests per year grows exponentially, that's the same as plotting the previous results on a log scale. Here's what you get:


That's right: a log curve on a log scale is a straight line. And I believe that that's why world records behave the way they do.

This model is unrealistic in some obvious ways. We don't have a perfect test for running potential and we don't apply it to everyone. Not everyone with the potential to be a runner has the opportunity to develop that potential, and some with the opportunity choose not to.

But the pipeline that selects and trains runners behaves, in some ways, like the model.  If a person with record-breaking potential is born in Kenya, where running is the national sport, the chances are good that he will be found, have opportunities to train, and become a world-class runner.  It is not a certainty, but the chances are good.

If the same person is born in rural India, he may not have the opportunity to train; if he is in the United States, he might have options that are more appealing.

So in some sense the relevant population is not the world, but the people who are likely to become professional runners, given the talent.  As long as this population is growing exponentially, world records will increase linearly.

That said, the slope of the line depends on the parameter of exponential growth.  If economic development increases the fraction of people in the world who have the opportunity to become professional runners, these curves could accelerate.

So let's get back to the original question: when will a marathoner break the 2-hour barrier?  Before 1970, marathon times improved quickly; since then the rate has slowed.  But progress has been consistent for the last 40 years, with no sign of an impending asymptote.  So I think it is reasonable to fit a line to recent data and extrapolate.  Here are the results [based on 2011 data; see above for the update]:



The red line is the target: 13.1 mph.  The blue line is a least squares fit to the data.  So here's my prediction: there will be a 2-hour marathon in 2043.  I'll be 76, so my chances of seeing it are pretty good.  But that's a topic for another article (see Chapter 1 of Think Bayes).

Thursday, September 25, 2014

Bayesian election forecasting

Last week Nate Silver posted this article explaining how the FiveThirtyEight Senate forecast model works.  If you are familiar with Silver's work, you probably know that (1) he has been notably successful at predicting outcomes of elections, and (2) he is an advocate for Bayesian statistics.  In last week's article, Silver provides a high-level view of his modeling philosophy and some details about how his model works, but he didn't say much that is explicitly Bayesian.

His first principle of modeling hints at some Bayesian ideas:
Principle 1: A good model should be probabilistic, not deterministic.
The FiveThirtyEight model produces probabilistic forecasts as opposed to hard-and-fast predictions.[...] the FiveThirtyEight model might estimate the Democrat has a 20 percent chance of winning the Senate race in Kentucky.  My view is that this is often the most important part of modeling — and often the hardest part. 
Silver suggests that it is more useful to report the probability of winning than the margin of victory.  Bayesian models are generally good at this kind of thing; classical statistical methods are not.  But Silver never makes the connection between this principle and Bayesian methods; in fact the article doesn't mention Bayes at all.

And that's fine; it was not central to the point of his article.  But since I am teaching my Bayesian statistics class this semester, I will take this opportunity to fill in some details.  I don't know anything about Silver's model other than what's in his article, but I think it is a good guess that there is something in there similar to what follows.

But before I get into it, here's an outline:

  1. I present an example problem and formulate a solution using a Bayesian framework.
  2. I develop Python code to compute a solution; if you don't speak Python, you can skip this part.
  3. I show results for an update with a single poll result.
  4. I show how to combine results from a different polls.
My example supports Silver's argument that it is more useful to predict the probability of winning than the margin of victory: after the second update, the predicted margin of victory decreases, but the probability of winning increases.  In this case, predicting only the margin of victory would misrepresent the effect of the second poll.

Formulating the problem


Here's the exercise I presented to my class:
Exercise 1: The polling company Strategic Vision reports that among likely voters, 53% intend to vote for your favorite candidate and 47% intend to vote for the opponent (let's ignore undecided voters for now).  Suppose that, based on past performance, you estimate that the distribution of error for this company has mean 1.1 percentage point (in favor of your candidate) and standard deviation 3.7 percentage points.  What is the probability that the actual fraction of likely voters who favor your candidate is less than 50%?
Strategic Vision is an actual polling company, but other than that, everything about this example is made up.  Also, the standard deviation of the error is probably lower than what you would see in practice.

To solve this problem, we can treat the polling company like a measurement instrument with known error characteristics.  If we knew the actual fraction of the electorate planning to vote for your candidate, which I'll call A for "actual", and we knew the distribution of the error, ε, we could compute the distribution of the measured value, M:

M = A + ε

But in this case we want to solve the inverse problem: given a measurement M and the distribution of ε, compute the posterior distribution of A.  As always with this kind of problem, we need three things:

1) A prior distribution for A,
2) Data that allow us to improve the estimate of A, and
3) A likelihood function that computes the probability of the data for hypothetical values of A.

Once you have these three things, the Bayesian framework does the rest.

Implementing the solution

To demonstrate, I use the Suite class from thinkbayes2, which is a Python module that goes with my book, Think Bayes.  The Suite class is documented here, but I will explain what you need to know below.  You can download the code from this file in this GitHub repository.

I'll start by defining a new class called Electorate that inherits methods from thinkbayes2.Suite:

class Electorate(thinkbayes2.Suite):
    """Represents hypotheses about the state of the electorate."""

As a starting place, I'll create a uniform prior distribution.  In practice this would not be a good choice, for reasons I'll explain soon, but it will allow me to make a point.

    hypos = range(0, 101)
    suite = Electorate(hypos)

hypos is a list of integers from 0 to 100, representing the percentage of the electorate planning to vote for your candidate.

I'll represent the data with a tuple of three values: the estimated bias of the polling company in percentage points, the standard deviation of their previous errors (also in percentage points), and the result of the poll:

    data = 1.1, 3.7, 53
    suite.Update(data)

When we call Update, it loops through the hypotheses and computes the likelihood of the data under each hypothesis; we have to provide a Likelihood function that does this calculation:

class Electorate(thinkbayes2.Suite):
    """Represents hypotheses about the electorate."""

    def Likelihood(self, data, hypo):
        """Likelihood of the data under the hypothesis.

        hypo: fraction of the population
        data: poll results
        """
        bias, std, result = data
        error = result - hypo
        like = thinkbayes2.EvalNormalPdf(error, bias, std)
        return like

Likelihood unpacks the given data into bias, std, and result.  Given a hypothetical value for A, it computes the hypothetical error.  For example, if hypo is 50 and result is 53, that means the poll is off by 3 percentage points.  The resulting likelihood is the probability that we would be off by that much, given the bias and standard deviation of the poll.

We estimate this probability by evaluating the normal/Gaussian distribution with the given parameters.  I am assuming that the distribution of errors is approximately normal, which is probably not a bad assumption when the probabilities are near 50%.

One technical detail: The result of EvalNormalPdf is actually a probability density, not a probability.  But the result from Likelihood doesn't actually have to be a probability; it only has to be proportional to a probability, so a probability density will do the job nicely.

The results

And that's it -- we've solved the problem!  Here are the results:


The prior distribution is uniform from 0 to 100.  The mean of the posterior is 51.9, which makes sense because the result is 53 and the known bias is 1.1, so the posterior mean is (53 - 1.1).  The standard deviation of the posterior is 3.7, the same as the standard deviation of the error.

To compute the probability of losing the election (if it were held today), we can loop through the hypotheses and add up the probability of all values less than 50%.  The Suite class provides ProbLess, which does that calculation.  The result is 0.26, which means your candidate is a 3:1 favorite.

In retrospect we could have computed this posterior analytically with a lot less work, which is the point I wanted to make by using a uniform prior.  But in general it's not quite so simple, as we can see by incorporating a second poll:
Exercise 2: What if another poll comes out from Research 2000 showing that 49% of likely voters intend to vote for your candidate, but past poll show that this company's results tend to favor the opponent by 2.3 points, and their past errors (after correcting for this bias) have standard deviation 4.1 points.  Now what should you believe?
The second update looks just like the first:

    data = -2.3, 4.1, 49
    suite.Update(data)

The bias is negative now because this polling company (in my fabricated world) tends to favor the opponent.  Here are the results after the second update:


The mean of the new posterior is 51.6, slightly lower than the mean after the first update, 51.9.  The two polls are actually consistent with each other after we correct for the biases of the two companies.

The predicted margin of victory is slightly smaller, but the uncertainty of the prediction is also smaller.  Based on the second update, the probability is 0.22, which means your candidate is now nearly a 4:1 favorite.

Again, this example demonstrates Silver's point: predicting the probability of winning is more meaningful that predicting the margin of error.  And that's exactly the kind of thing Bayesian models are good for.

One more technical note:  This analysis is based on the assumption that errors in one poll iare independent of errors in another.  It seems likely that in practice there is correlation between polls; in that case we could extend this solution to model the errors with a joint distribution that includes the correlation. 


Sunday, September 14, 2014

Regression with Python, pandas and StatsModels

I was at Boston Data-Con 2014 this morning, which was a great event.  The organizer, John Verostek, seems to have created this three-day event single-handedly, so I am hugely impressed.

Imran Malek started the day with a very nice iPython tutorial.  The description is here, and his slides are here.  He grabbed passenger data from the MBTA and generated heat maps showing the number of passengers at each stop in the system during each hour.  The tutorial covered a good range of features, and it seemed like many of the participants were able to download the data and follow along in iPython.

And Imran very kindly let me use his laptop to project slides for my talk, which was next.  The description of my talk is here:
Regression is a powerful tool for fitting data and making predictions. In this talk I present the basics of linear regression and logistic regression and show how to use them in Python. I demonstrate pandas, a Python module that provides structures for data analysis, and StatsModels, a module that provides tools for regression and other statistical analysis. 
As an example, I will use data from the National Survey of Family Growth to generate predictions for the date of birth, weight, and sex of an expected baby. This presentation is based on material from the recent revision of Think Stats, an introduction to data analysis and statistics with Python.
This talk is appropriate for people with no prior experience with regression. Basic familiarity with Python is recommended but not required.
 And here are my slides:


The material for this talk is from the second edition of Think Stats, which is in production now and scheduled for release in early November.  My draft is available here, and you can pre-order a paper copy here.

As I expected, I prepared way more material than I could present.  The audience had some excellent questions, so we spent more time on linear regression and did not get to logistic regression.

The nice people at O'Reilly Media sent over 25 copies of my book, Think Python, so we had a book signing after the talk.  I had a chance to chat with everyone who got a book, which is always a lot of fun.

I believe video of the talk will be available soon.  I will post it here when it is.

Many thanks to John Verostek for organizing this conference, and to the sponsors for breakfast and lunch.  Looking forward to next year!

Friday, August 29, 2014

New study: vaccines prevent disease and death

According to an exciting new study, childhood vaccines are almost miraculously effective at preventing suffering and death due to infectious disease.

Sadly, that is not actually a headline, because it doesn't generate clicks.  What does generate clicks?  This: Journal questions validity of autism and vaccine study (CNN Health).

If, at this point, headlines like this make you roll your eyes and click on more interesting things, like "Teen catches one-in-2 million lobster," let me assure you that you are absolutely right.  In fact, if you want to skip the rest of this post, and read something that will make you happy to live in the 21st century, I recommend this list of vaccine-preventable diseases.

But if you have already heard about this new paper and you are curious to know why it is, from a statistical point of view, completely bogus, read on!

Let me start by reviewing the basics of statistical hypothesis testing.  Suppose you see an apparent effect like a difference in risk of autism between groups of children who are vaccinated at different ages.  You might wonder whether the effect you see in your selected sample is likely to exist in the general population, or whether it might appear by chance in your sample only, and not in the general population.

To answer at least part of that question, you can compute a p-value, which is the probability of seeing a difference as big as the one you saw if, in fact, there is no difference between the groups.  If this value is small, you can conclude that the difference is unlikely to be an artifact of random sampling.  And if you are willing to cut a couple of logical corners, you can conclude that the effect is more likely to exist in the general population.

In many fields it is common to use 5% as a magical threshold for statistical significance.  If the p-value is less than 5%, the effect is considered statistically significant and, more importantly, fit for publication.  If it comes in at 6%, it doesn't count.

This is, of course, arbitrary and mostly silly, but it does have one virtue.  If we follow this process with care, it should yield a known and small false positive rate: if, in fact, there is no difference between the groups, we should expect to conclude, incorrectly, that the apparent difference is significant about 5% of the time.

But that expectation only holds if we apply hypothesis tests very carefully.  In practice, people don't, and there is accumulating evidence that the false positive rate in published research is much higher than 5%, possibly higher than 50%.

There are lots of ways to mess up hypothesis testing, but one of the most common is performing multiple tests.  Every time you perform a test you have a 5% chance of generating a false positive, so if you perform 20 independent tests, you should expect about 1 false positive.  This problem is ably demonstrated by this classic xkcd cartoon:

Significant

If you are not already familiar with xkcd, you're welcome.

So let's get back to the paper reported in the CNN article.  Just looking at the results that appear in the tables, this paper reports 35 p-values.  Five of them are identified as significant at the 5% level.  That's more than the expected number of false positives, but they might still be due to chance (especially since the tests are not independent).

Fortunately, there is a simple process that corrects for multiple tests, the Holm-Bonferroni method.  You can get the details from the Wikipedia article, but I'll give you a quick-and dirty version here: if you perform n tests, the lowest p-value needs to be below 0.05 / n to be considered statistically significant.

Since the paper reports 35 tests, their threshold is 0.05 / 35, which is 0.0014.  Since their lowest p-value is 0.0019, it should not be considered statistically significant, and neither should any of the other test results.

And I am being generous by assuming that the authors only performed 35 tests.  It is likely that they performed many more and chose carefully which ones to report.  And I assume that they are doing everything else right as well.

But even with the benefit of the doubt, these results are not statistically significant.  Given how hard the authors apparently tried to find evidence that vaccines cause autism, the fact that they failed should be considered evidence that vaccines do NOT cause autism.

Before I read this paper, I was nearly certain that vaccines do not cause autism.  After reading this paper, I am very slightly more certain that vaccines do not cause autism.  And by the way, I am also nearly certain that vaccines do prevent suffering and death due to infectious disease.

Now go check out the video of that blue lobster.

UPDATE: Or go read this related article by my friend Ted Bunn.




Friday, August 22, 2014

An exercise in hypothesis testing

I've just turned in the manuscript for the second edition of Think Stats.  If you're dying to get your hands on a copy, you can pre-order one here.

Most of the book is about computational methods, but in the last chapter I break out some analytic methods, too.  In the last section of the book, I explain the underlying philosophy:

This book focuses on computational methods like resampling and permutation. These methods have several advantages over analysis:
  • They are easier to explain and understand. For example, one of the most difficult topics in an introductory statistics class is hypothesis testing. Many students don’t really understand what p-values are. I think the approach I presented in Chapter 9—simulating the null hypothesis and computing test statistics—makes the fundamental idea clearer.
  • They are robust and versatile. Analytic methods are often based on assumptions that might not hold in practice. Computational methods require fewer assumptions, and can be adapted and extended more easily.
  • They are debuggable. Analytic methods are often like a black box: you plug in numbers and they spit out results. But it’s easy to make subtle errors, hard to be confident that the results are right, and hard to find the problem if they are not. Computational methods lend themselves to incremental development and testing, which fosters confidence in the results.
But there is one drawback: computational methods can be slow. Taking into account these pros and cons, I recommend the following process:
  1. Use computational methods during exploration. If you find a satisfactory answer and the run time is acceptable, you can stop.
  2. If run time is not acceptable, look for opportunities to optimize. Using analytic methods is one of several methods of optimization.
  3. If replacing a computational method with an analytic method is appropriate, use the computational method as a basis of comparison, providing mutual validation between the computational and analytic results.
For the vast majority of problems I have worked on, I didn’t have to go past Step 1.
The last exercise in the book is based on a question my colleague, Lynn Stein, asked me for a paper she was working on:

 In a recent paper2, Stein et al. investigate the effects of an intervention intended to mitigate gender-stereotypical task allocation within student engineering teams.  Before and after the intervention, students responded to a survey that asked them to rate their contribution to each aspect of class projects on a 7-point scale. 
Before the intervention, male students reported higher scores for the programming aspect of the project than female students; on average men reported a score of 3.57 with standard error 0.28. Women reported 1.91, on average, with standard error 0.32. 
Question 1:  Compute a 90% confidence interval and a p-value for the gender gap (the difference in means). 
After the intervention, the gender gap was smaller: the average score for men was 3.44 (SE 0.16); the average score for women was 3.18 (SE 0.16). Again, compute the sampling distribution of the gender gap and test it. 
Question 2:  Compute a 90% confidence interval and a p-value for the change in gender gap. 
[2] Stein et al. “Evidence for the persistent effects of an intervention to mitigate gender-sterotypical task allocation within student engineering teams,” Proceedings of the IEEE Frontiers in Education Conference, 2014.
In the book I present ways to do these computations, and I will post my "solutions" here soon.  But first I want to pose these questions as a challenge for statisticians and people learning statistics.  How would you approach these problems?

The reason I ask:  Question 1 is pretty much a textbook problem; you can probably find an online calculator to do it for you.  But you are less likely to find a canned solution to Question 2, so I am curious to see how people go about it.  I hope to post some different solutions soon.

By the way, this is not meant to be a "gotcha" question.  If some people get it wrong, I am not going to make fun of them.  I am looking for different correct approaches; I will ignore mistakes, and only point out incorrect approaches if they are interestingly incorrect.

You can post a solution in the comments below, or discuss it on reddit.com/r/statistics, or if you don't want to be influenced by others, send me email at downey at allendowney dot com.

UPDATE August 26, 2014

The discussion of this question on reddit.com/r/statistics was as interesting as I hoped.  People suggested several very different approaches to the problem.  The range of responses extends from something like, "This is a standard problem with a known, canned answer," to something like, "There are likely to be dependencies among the values that make the standard model invalid, so the best you can do is an upper bound."  In other words, the problem is either trivial or impossible!

The approach I had in mind is to compute sampling distributions for the gender gap and the change in gender gap using normal approximations, and then use the sampling distributions to compute standard errors, confidence intervals, and p-values.

I used a simple Python class that represents a normal distribution.  Here is the API:

class Normal(object):
    """Represents a Normal distribution"""

    def __init__(self, mu, sigma2, label=''):
        """Initializes a Normal object with given mean and variance."""

    def __add__(self, other):
        """Adds two Normal distributions."""

    def __sub__(self, other):
        """Subtracts off another Normal distribution."""

    def __mul__(self, factor):
        """Multiplies by a scalar."""

    def Sum(self, n):
        """Returns the distribution of the sum of n values."""

    def Prob(self, x):
        """Cumulative probability of x."""
     
   def Percentile(self, p):
        """Inverse CDF of p (0 - 100)."""

The implementation of this class is here.

Here's a solution that uses the Normal class.  First we make normal distributions that represent the sampling distributions of the estimated means, using the given means and sampling errors.  The variance of the sampling distribution is the sampling error squared:

    male_before = normal.Normal(3.57, 0.28**2)
    male_after = normal.Normal(3.44, 0.16**2)

    female_before = normal.Normal(1.91, 0.32**2)
    female_after = normal.Normal(3.18, 0.16**2)

Now we compute the gender gap before the intervention, and print the estimated difference, p-value, and confidence interval:

    diff_before = female_before - male_before
    print('mean, p-value', diff_before.mu, 1-diff_before.Prob(0))
    print('CI', diff_before.Percentile(5), diff_before.Percentile(95))

The estimated gender gap is -1.66 with SE 0.43, 90% CI (-2.3, -0.96) and p-value 5e-5.  So that's statistically significant.

Then we compute the gender gap after intervention and the change in gender gap:

    diff_after = female_after - male_after
    diff = diff_after - diff_before
    print('mean, p-value', diff.mu, diff.Prob(0))
    print('CI', diff.Percentile(5), diff.Percentile(95))

The estimated change is 1.4 with SE 0.48, 90% CI (0.61, 2.2) and p-value 0.002.  So that's statistically significant, too.

This solution is based on a two assumptions:

1) It assumes that the sampling distribution of the estimated means is approximately normal.  Since the data are on a Likert scale, the variance and skew are small, so the sum of n values converges to normal quickly.  The samples sizes are in the range of 30-90, so the normal approximation is probably quite good.

This claim is based on the Central Limit Theorem, which only applies if the samples are drawn from the population independently.  In this case, there are dependencies within teams: for example, if someone on a team does a larger share of a task, the rest of the team necessarily does less.  But the team sizes are 2-4 people and the sample sizes are much larger, so these dependencies have short enough "range" that I think it is acceptable to ignore them.

2) Every time we subtract two distributions to get the distribution of the difference, we are assuming that values are drawn from the two distributions independently.  In theory, dependencies within teams could invalidate this assumption, but I don't think it's likely to be a substantial effect.

3) As always, remember that the standard error (and confidence interval) indicate uncertainty due to sampling, but say nothing about measurement error, sampling bias, and modeling error, which are often much larger sources of uncertainty.

The approach I presented here is a bit different from what's presented in most introductory stats classes.  If you have taken (or taught) a stats class recently, I would be curious to know what you think of this problem.  After taking the class, would you be able to solve problems like this?  Or if you are teaching, could your students do it?

Your comments are welcome.