Category Archives: bad statistics

Pandemic blog 23: why one published research finding is misleading

I really like John Ioannidis: his famous 2005 article “Why Most Published Research Findings are False” probably did more than any other paper to draw attention to the problems with blind use of p-value certification in medicine.

But he has a preprint up on medrxiv today that is really poorly done, so much so that it made me mad, and when I get mad, I blog.

Ioannidis has been saying for months that the COVID-19 pandemic, while bad, is not as bad as people think. Obviously this is true for some value of “people.” And I think he is right that the infection fatality rate, or IFR, is in most places not going to be as high as the 0.9% figure the March 16 Imperial College model used as an estimate. But Ioannidis has a much stronger claim; he thinks the IFR, in general, is going to be about 1 or 2 in a thousand, and in order to make that case, he has written a paper about twelve studies which show a high prevalence of antibodies in populations where not very many people have died. High prevalence of infection + few deaths = low IFR.

I think I am especially irritated with this paper because I agree that the IFR now looks lower than it looked two months ago, and I think it’s important to have good big-picture analysis to back that intuition up — and this isn’t it. There’s a lot wrong with this paper but I just want to focus on one thing that jumped out at me as especially wrong, and that is Ioannidis’s treatment of the Netherlands antibody study.

That study found that in blood donors, all ages 18-72 (Ioannidis says <70, not sure why), 2.7% showed immunity. Ioannidis reports this, then makes the following computation. About 15m of the 17m people in the Netherlands are under 70, so this suggests roughly 400,000 people in that age group had been infected, of whom only 344 had died at the time of the study, giving an IFR of a mere 0.09%. Some plague! Ioannidis puts this number in his table and counts it among those of which he writes “Seven of the 12 inferred IFRs are in the range 0.07 to 0.20 (corrected IFR of 0.06 to 0.16) which are similar to IFR values of seasonal influenza.”

But of course the one thing we really do know about COVID, in this sea of uncertainty, is that it’s much, much more deadly to old people. The IFR for people under 70 is not going to be a good estimate for the overall IFR.

I hashed out some numbers — it looks to me like, using the original March 16 Imperial College estimates, derived from Wuhan, you would derive an infection fatality rate of about 0.47% among people age 20-70. There are about 10.8m Dutch people in that range (I am taking all this from Wikipedia data on the age distribution of the Netherlands) so if 2.7% of those are infected, that’s about 300,000 infections, and 344 deaths in that group is about 0.11%. Lower than the Imperial estimate! But four times lower, not ten times lower.

What about the overall IFR? That, after all, is what Ioannidis’s paper is about. If you count the old people who died, the toll as of April 15 wasn’t 344, it was over 3100. If the 2.7% prevalence rate were accurate as a population-wide estimate, the total number of infected people would be about 460,000, for an IFR of 0.67%, more than seven times higher than the figure Ioannidis reports (though still a little lower than the 0.9% figure in the Imperial paper.) Now we definitely don’t know that the infection rate among old Dutch people is the same as it is in the overall population! But even if you suppose that every single person over 70 in the country is infected, that gets you to a little over 2 million infections, and an IFR of 0.15%. In other words, the number reported by Ioannidis is substantially lower than the theoretical minimum the IFR could actually be. And of course, it’s not the case that everybody over 70 already had COVID-19 in the middle of April. (For one thing, that would make the IFR for over-70s only slightly higher than the IFR overall, which contradicts the one thing about COVID we really know!)

There’s no fraud here, I hasten to say. Ioannidis tells you exactly what he’s doing. But he’s doing the wrong thing.

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Pandemic blog 5: exponential

When do you go the grocery store? If you’re concerned about your own risk of infection, the logic of exponential growth insists that today is always better than tomorrow. But the community is better served by each person waiting as long as they can, so as to slow the overall exponential constant.

What is the exponential constant? People are constantly graphing the number of confirmed cases in each country, state, locality on a log-linear scale and watching the slope, but I don’t see how, in a principled way, to untangle the effects of increased testing from actual increases in infection. I guess if one hypothesizes that there’s something like a true mean rate you could plot state-by-state nominal cases against tests done and see if you can fit exp(ct)*(tests per capita) to it. But there are state-to-state differences in testing criteria, state-to-state differences in mitigation strategy, etc.

AB and I made chocolate chip cookies today. Dr. Mrs. Q and CJ watched Inside Out. Weather’s warmer and I think we’ll get some driveway basketball in. We listened to “The Gambler” in honor of Kenny Rogers, deceased today. I had forgotten, or didn’t know, what an ice-cold love letter to death it is. “Every hand’s a winner, and every hand’s a loser, and the best that you can hope for is to die in your sleep.” Damn.

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The Americans can retire at 42

On the front page of my New York Times today, this capsule summary:

The French can retire at 62. Or 52. Sometimes 42. President Emmanuel Macron calls the tangle unsustainable. A million protesters disagree.

In the actual article, we learn that the retirement age of 42 applies to one group of workers; dancers in the national ballet. I find it very annoying when an article is teased with a number presented as normal when it’s actually extremely atypical. You could write the same teaser about the United States, having New York City firefighters in mind. But you would be misleading your audience even though the claim would be, I suppose, technically correct.

Nebraska isn’t poor and Nebraskans aren’t naive

David Brooks writes in the New York Times that we should figure out how to bottle the civic health southwest Nebraska enjoys:

Everybody says rural America is collapsing. But I keep going to places with more moral coherence and social commitment than we have in booming urban areas. These visits prompt the same question: How can we spread the civic mind-set they have in abundance?

For example, I spent this week in Nebraska, in towns like McCook and Grand Island. These places are not rich. At many of the schools, 50 percent of the students receive free or reduced-cost lunch. But they don’t have the pathologies we associate with poverty.

Maybe that’s because those places aren’t high in poverty! The poverty rate in McCook is 9.6%; in Grand Island it’s 15%. The national rate is 12.3%. Here’s a Census page with those numbers. What about the lunches? 50 percent of students receiving free or reduced-price lunch sounds like a lot, unless you know that slightly more than half of all US public school students are eligible for free and reduced-price lunch. (Brooks says “receive,” not “are eligible for,” but it’s the latter statistics that are widely reported and I’m guessing that’s what he means; apologies if I’m wrong.)

Crime is low. Many people leave their homes and cars unlocked.

Is it? And do they? I didn’t immediately find city-level crime data that looked rock solid to me, but if you trust, crime in Grand Island roughly tracks national levels while crime in McCook is a little lower. And long-time Grand Island resident Gary Christensen has a different take than Brooks does:

Gary Christensen, a Grand Island resident for over 68 years says times are changing.
“It was a community that you could leave you doors open leave the keys in your car and that kind of thing, and nobody ever bothered it. But those days are long gone,” said Gary Christensen, resident.

One way you can respond to this is to say I’m missing the point of Brooks’s article. Isn’t he just saying civic involvement is important and it’s healthy when people feel a sense of community with their neighbors? Are the statistics really that important?

Yes. They’re important. Because what Brooks is really doing here is inviting us to lower ourselves into a warm comfortable stereotype; that where the civic virtues are to be found in full bloom, where people are “just folks,” are in the rural parts of Nebraska, not in New Orleans, or Seattle, or Laredo, or Madison, and most definitely not in Brooklyn or Brookline or Bethesda. But he can’t just say “you know how those people are.” There needs to be some vaguely evidentiary throat-clearing before you launch into what you were going to say anyway.

Which is that Nebraska people are simple dewy real Americans, not like you, urbanized coastal reader of the New York Times. I don’t buy it. McCook, Nebraska sounds nice; but it sounds nice in the same way that urbanized coastal communities are nice. You go someplace and talk to a guy who’s on the city council, you’re gonna be talking to a guy who cares about his community and thinks a lot about how to improve it. Even in Bethesda.

Constantly they are thinking: Does this help my town or hurt it? And when you tell them that this pervasive civic mind-set is an unusual way to be, they look at you blankly because they can’t fathom any other.

There’s Brooks in a nutshell. The only good people are the people who don’t know any better than to be good. By saying so, he condescends to his subjects, his readers, and himself all at once. I don’t buy it. I’ll bet people in southwest Nebraska can fathom a lot more than Brooks thinks they can. I think they probably fathom David Brooks better than he fathoms them.

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Ocular regression

A phrase I learned from Aaron Clauset’s great colloquium on the non-ubiquity of scale-free networks.  “Ocular regression” is the practice of squinting at the data until it looks linear.


“Worst of the worst maps”: a factual mistake in Gill v. Whitford

The oral arguments in Gill v. Whitford, the Wisconsin gerrymandering case, are now a month behind us.  But there’s a factual error in the state’s case, and I don’t want to let it be forgotten.  Thanks to Mira Bernstein for pointing this issue out to me.

Misha Tseytlin, Wisconsin’s solicitor general, was one of two lawyers arguing that the state’s Republican-drawn legislative boundaries should be allowed to stand.  Tseytlin argued that the metrics that flagged Wisconsin’s maps as drastically skewed in the GOP’s favor were unreliable:

And I think the easiest way to see this is to take a look at a chart that plaintiff’s own expert created, and that’s available on Supplemental Appendix 235. This is plain — plaintiff’s expert studied maps from 30 years, and he identified the 17 worst of the worst maps. What is so striking about that list of 17 is that 10 were neutral draws.  There were court-drawn maps, commission-drawn maps, bipartisan drawn maps, including the immediately prior Wisconsin drawn map.

That’s a strong claim, which jumped out at me when I read the transcripts–10 of the 17 very worst maps, according to the metrics, were drawn by neutral parties!  That really makes it sound like whatever those metrics are measuring, it’s not partisan gerrymandering.

But the claim isn’t true.

(To be clear, I believe Tseytlin made a mistake here, not a deliberate misrepresentation.)

The table he’s referring to is on p.55 of this paper by Simon Jackman, described as follows:

Of these, 17 plans are utterly unambiguous with respect to the sign of the efficiency gap estimates recorded over the life of the plan:

Let me unpack what Jackman’s saying here.  These are the 17 maps where we can be sure the efficiency gap favored the same party, three elections in a row.  You might ask: why wouldn’t we be sure about which side the map favors?  Isn’t the efficiency gap something we can compute precisely?  Not exactly.  The basic efficiency gap formula assumes both parties are running candidates in every district.  If there’s an uncontested race, you have to make your best estimate for what the candidate’s vote shares would have been if there had been candidates of both parties.  So you have an estimate for the efficiency gap, but also some uncertainty.  The more uncontested races, the more uncertain you are about the efficiency gap.

So the maps on this list aren’t the 17 “worst of the worst maps.”  They’re not the ones with the highest efficiency gaps, not the ones most badly gerrymandered by any measure.  They’re the ones in states with so few uncontested races that we can be essentially certain the efficiency gap favored the same party three years running.

Tseytlin’s argument is supposed to make you think that big efficiency gaps are as likely to come from neutral maps as partisan ones.  But that’s not true.  Maps drawn by Democratic legislatures have average efficiency gap favoring Democrats; those by GOP on average favor the GOP; neutral maps are in between, and have smaller efficiency gaps overall.

That’s from p.35 of another Jackman paper.  Note the big change after 2010.  It wasn’t always the case that partisan legislators automatically thumbed the scales strongly in their favor when drawing the maps.  But these days, it kind of is.  Is that because partisanship is worse now?  Or because cheaper, faster computation makes it easier for one-party legislatures to do what they always would have done, if they could?  I can’t say for sure.

Efficiency gap isn’t a perfect measure, and neither side in this case is arguing it should be the single or final arbiter of unconstitutional gerrymandering.  But the idea that efficiency gap flags neutral maps as often as partisan maps is just wrong, and it shouldn’t have been part of the state’s argument before the court.

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How many books do I read in a year?


2006: 27

2007: 19

2008: 22

2009: 30

2010: 23

2011: 19

2012: 27

2013: 35

2014: 31

2015: 38

2016: 29

Don’t quite know what to make of this.  I’m sort of surprised there’s so much variation!  I’d have thought I’d have read less when my kids were infants, or when I was writing my own book, but it seems pretty random.   I do see that I’ve been clearly reading more books the last few years than I did in 2012 and before.

Lists, as always, are here (2011 on) and here (2006-2010.)


Bad Bayesian

Couldn’t find my phone yesterday morning.  I definitely remembered having it in the car on the way home from the kids’ swim lesson, so I knew I hadn’t left it.  “Find my iPhone” told me the phone was on the blacktop of the elementary school, about 1000 feet from my house.  What?  Why?  Then a few minutes later the location updated to the driveway of a bank, closer to my house but in the other direction.  So I went over to the bank and looked around in the driveway, even peering into the garbage shed and seeing if my phone was in their dumpster.

But why did I do that?  It was terrible reason.  There was no chain of events leaving my phone at the bank, or at the school, which wasn’t incredibly a prior unlikely.  I should have reasoned:  “The insistence of Find my iPhone that my phone is at the bank drastically increases the probability my phone is at the bank, but that probability started out so tiny that it remains tiny, and the highest-expected-utility use of my time is to keep looking around my house and my car until I find it.”

Anyway, it was in the basement.




91 white friends

Just ran across this hunk of data journalism from the Washington Post:

In a 100-friend scenario, the average white person has 91 white friends; one each of black, Latino, Asian, mixed race, and other races; and three friends of unknown race. The average black person, on the other hand, has 83 black friends, eight white friends, two Latino friends, zero Asian friends, three mixed race friends, one other race friend and four friends of unknown race.

Going back to Chris Rock’s point, the average black person’s friend network is eight percent white, but the average white person’s network is only one percent black. To put it another way: Blacks have ten times as many black friends as white friends. But white Americans have an astonishing 91 times as many white friends as black friends.

100 friends and only one black person!  That’s pretty white!

It’s worth taking a look at the actual study they’re writing about.  They didn’t ask people to list their top 100 friends.  They said to list at most seven people, using this prompt:

From time to time, most people discuss important matters with other people. Looking back over the last six months – who are the people with whom you discussed matters important to you?

The white respondents only named 3.3 people on average, of whom 1.9 were immediate family members.  So a better headline wouldn’t be “75% of white people have no black friends,” but “75% of whites are married to another white person, have two white parents, and have a white best friend, if they have a best friend”  As for the quoted paragraph, it should read

In a 100-friend scenario, the average white person has 57 immediate family members.

Who knew?

(Note:  I just noticed that Emily Swanson at Huffington Post made this point much earlier.)







Ranking mathematicians by hinge loss

As I mentioned, I’m reading Ph.D. admission files.  Each file is read by two committee members and thus each file has two numerical scores.

How to put all this information together into a preliminary ranking?

The traditional way is to assign to each applicant their mean score.  But there’s a problem: different raters have different scales.  My 7 might be your 5.

You could just normalize the scores by subtracting that rater’s overall mean.  But that’s problematic too.  What if one rater actually happens to have looked at stronger files?  Or even if not:  what if the relation between rater A’s scale and rater B’s scale isn’t linear?  Maybe, for instance, rater A gives everyone she doesn’t think should get in a 0, while rater A uses a range of low scores to express the same opinion, depending on just how unsuitable the candidate seems.

Here’s what I did last year.  If (r,a,a’) is a triple with r is a rater and a and a’ are two applicants, such that r rated a higher than a’, you can think of that as a judgment that a is more admittable than a’.  And you can put all those judgments from all the raters in a big bag, and then see if you can find a ranking of the applicants (or, if you like, a real-valued function f on the applicants) such that, for every judgment a > a’, we have f(a) > f(a’).

Of course, this might not be possible — two raters might disagree!  Or there might be more complicated incompatibilities generated by multiple raters.  Still, you can ask:  what if I tried to minimize the number of “mistakes”, i.e. the number of judgments in your bag that your choice of ranking contradicts?

Well, you can ask that, but you may not get an answer, because that’s a highly non-convex minimization problem, and is as far as we know completely intractable.

But here’s a way out, or at least a way part of the way out — we can use a convex relaxation.  Set it up this way.  Let V be the space of real-valued functions on applicants.  For each judgment j, let mistake_j(f) be the step function

mistake_j(f) = 1 if f(a) < f(a’) + 1

mistake_j(f) = 0 if f(a) >= f(a’) + 1

Then “minimize total number of mistakes” is the problem of minimizing

M = sum_j mistake_j(f)

over V.  And M is terribly nonconvex.  If you try to gradient-descend (e.g. start with a random ranking and then switch two adjacent applicants whenever doing so reduces the total number of mistakes) you are likely to get caught in a local minimum that’s far from optimal.  (Or at least that can happen; whether this typically actually happens in practice, I haven’t checked!)

So here’s the move:  replace mistake_j(f) with a function that’s “close enough,” but is convex.  It acts as a sort of tractable proxy for the optimization you’re actually after.  The customary choice here is the hinge loss:

hinge_j(f) = min(0, f(a)-f(a’) -1).

Then H := sum_j hinge_j(f) is a convex function on f, which you can easily minimize in Matlab or python.  If you can actually find an f with H(f) = 0, you’ve found a ranking which agrees with every judgment in your bag.  Usually you can’t, but that’s OK!  You’ve very quickly found a function H which does a decent job aggregating the committee scores. and which you can use as your starting point.

Now here’s a paper by Nihal Shah and Martin Wainwright commenter Dustin Mixon linked in my last ranking post.  It suggests doing something much simpler:  using a linear function as a proxy for mistake_j.  What this amounts to is:  score each applicant by the number of times they were placed above another applicant.  Should I be doing this instead?  My first instinct is no.  It looks like Shah and Wainwright assume that each pair of applicants is equally likely to be compared; I think I don’t want to assume that, and I think (but correct me if I’m wrong!) the optimality they get may not be robust to that?

Anyway, all thoughts on this question — or suggestions as to something totally different I could be doing — welcome, of course.




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