Risk compensation is not a serious problem

Nov 28 JDN 2459547

Risk compensation. It’s one of those simple but counter-intuitive ideas that economists love, and it has been a major consideration in regulatory policy since the 1970s.

The idea is this: The risk we face in our actions is partly under our control. It requires effort to reduce risk, and effort is costly. So when an external source, such as a government regulation, reduces our risk, we will compensate by reducing the effort we expend, and thus our risk will decrease less, or maybe not at all. Indeed, perhaps we’ll even overcompensate and make our risk worse!

It’s often used as an argument against various kinds of safety efforts: Airbags will make people drive worse! Masks will make people go out and get infected!

The basic theory here is sound: Effort to reduce risk is costly, and people try to reduce costly things.

Indeed, it’s theoretically possible that risk compensation could yield the exact same risk, or even more risk than before—or at least, I wasn’t able to prove that for any possible risk profile and cost function it couldn’t happen.

But I wasn’t able to find any actual risk profiles or cost functions that would yield this result, even for a quite general form. Here, let me show you.

Let’s say there’s some possible harm H. There is also some probability that it will occur, which you can mitigate with some choice x. For simplicity let’s say that it’s one-to-one, so that your risk of H occurring is precisely 1-x. Since probabilities must be between 0 and 1, thus so must x.

Reducing that risk costs effort. I won’t say much about that cost, except to call it c(x) and assume the following:

(1) It is increasing: More effort reduces risk more and costs more than less effort.

(2) It is convex: Reducing risk from a high level to a low level (e.g. 0.9 to 0.8) costs less than reducing it from a low level to an even lower level (e.g. 0.2 to 0.1).

These both seem like eminently plausible—indeed, nigh-unassailable—assumptions. And they result in the following total expected cost (the opposite of your expected utility):

(1-x)H + c(x)

Now let’s suppose there’s some policy which will reduce your risk by a factor r, which must be between 0 and 1. Your cost then becomes:

r(1-x)H + c(x)

Minimizing this yields the following result:

rH = c'(x)

where c'(x) is the derivative of c(x). Since c(x) is increasing and convex, c'(x) is positive and increasing.

Thus, if I make r smaller—an external source of less risk—then I will reduce the optimal choice of x. This is risk compensation.

But have I reduced or increased the amount of risk?

The total risk is r(1-x); since r decreased and so did x, it’s not clear whether this went up or down. Indeed, it’s theoretically possible to have cost functions that would make it go up—but I’ve never seen one.

For instance, suppose we assume that c(x) = axb, where a and b are constants. This seems like a pretty general form, doesn’t it? To maintain the assumption that c(x) is increasing and convex, I need a > 0 and b > 1. (If 0 < b < 1, you get a function that’s increasing but concave. If b=1, you get a linear function and some weird corner solutions where you either expend no effort at all or all possible effort.)

Then I’m trying to minimize:

r(1-x)H + axb

This results in a closed-form solution for x:

x = (rH/ab)^(1/(b-1))

Since b>1, 1/(b-1) > 0.

Thus, the optimal choice of x is increasing in rH and decreasing in ab. That is, reducing the harm H or the overall risk r will make me put in less effort, while reducing the cost of effort (via either a or b) will make me put in more effort. These all make sense.

Can I ever increase the overall risk by reducing r? Let’s see.

My total risk r(1-x) is therefore:

r(1-x) = r[1-(rH/ab)^(1/(b-1))]

Can making r smaller ever make this larger?

Well, let’s compare it against the case when r=1. We want to see if there’s a case where it’s actually larger.

r[1-(rH/ab)^(1/(b-1))] > [1-(H/ab)^(1/(b-1))]

r – r^(1/(b-1)) (H/ab)^(1/(b-1)) > 1 – (H/ab)^(1/(b-1))

For this to be true, we would need r > 1, which would mean we didn’t reduce risk at all. Thus, reducing risk externally reduces total risk even after compensation.

Now, to be fair, this isn’t a fully general model. I had to assume some specific functional forms. But I didn’t assume much, did I?

Indeed, there is a fully general argument that externally reduced risk will never harm you. It’s quite simple.

There are three states to consider: In state A, you have your original level of risk and your original level of effort to reduce it. In state B, you have an externally reduced level of risk and your original level of effort. In state C, you have an externally reduced level of risk, and you compensate by reducing your effort.

Which states make you better off?

Well, clearly state B is better than state A: You get reduced risk at no cost to you.

Furthermore, state C must be better than state B: You voluntarily chose to risk-compensate precisely because it made you better off.

Therefore, as long as your preferences are rational, state C is better than state A.

Externally reduced risk will never make you worse off.

QED. That’s it. That’s the whole proof.

But I’m a behavioral economist, am I not? What if people aren’t being rational? Perhaps there’s some behavioral bias that causes people to overcompensate for reduced risks. That’s ultimately an empirical question.

So, what does the empirical data say? Risk compensation is almost never a serious problem in the real world. Measures designed to increase safety, lo and behold, actually increase safety. Removing safety regulations, astonishingly enough, makes people less safe and worse off.

If we ever do find a case where risk compensation is very large, then I guess we can remove that safety measure, or find some way to get people to stop overcompensating. But in the real world this has basically never happened.

It’s still a fair question whether any given safety measure is worth the cost: Implementing regulations can be expensive, after all. And while many people would like to think that “no amount of money is worth a human life”, nobody does—or should, or even can—act like that in the real world. You wouldn’t drive to work or get out of bed in the morning if you honestly believed that.

If it would cost \$4 billion to save one expected life, it’s definitely not worth it. Indeed, you should still be able to see that even if you don’t think lives can be compared with other things—because \$4 billion could save an awful lot of lives if you spent it more efficiently. (Probablyover a million, in fact, as current estimates of the marginal cost to save one life are about \$2,300.) Inefficient safety interventions don’t just cost money—they prevent us from doing other, more efficient safety interventions.

And as for airbags and wearing masks to prevent COVID? Yes, definitely 100% worth it, as both interventions have already saved tens if not hundreds of thousands of lives.

What’s wrong with police unions?

Nov 14 JDN 2459531

In a previous post I talked about why unions, even though they are collusive, are generally a good thing. But there is one very important exception to this rule: Police unions are almost always harmful.

Most recently, police unions have been leading the charge to fight vaccine mandates. This despite the fact that COVID-19 now kills more police officers than any other cause. They threatened that huge numbers of officers would leave if the mandates were imposed—but it didn’t happen.

But there is a much broader pattern than this: Police unions systematically take the side of individual police offers over the interests of public safety. Even the most incompetent, negligent, or outright murderous behavior by police officers will typically be defended by police unions. (One encouraging development is that lately even some police unions have been reluctant to defend the most outrageous killings by police officers—but this very much the exception, not the rule.)

Police unions are also unusual among unions in their political ties. Conservatives generally oppose unions, but are much friendlier toward police unions. At the other end of the spectrum, socialists normally love unions, but have distanced themselves from police unions for a long time. (The argument in that article that this is because “no other job involves killing people” is a bit weird: Ostensibly, the circumstances in which police are allowed to kill people are not all that different from the circumstances in which private citizens are. Just like us, they’re only supposed to use deadly force to prevent death or grievous bodily harm to themselves or others. The main thing that police are allowed to do that we aren’t is imprison people. Killing isn’t supposed to be a major part of the job.)

Police union also have some other weird features. The total membership of all police unions exceeds the total number of police officers in the United States, because a single officer is often affiliated with multiple unions—normally not at all how unions work. Police unions are also especially powerful and well-organized among unions. They are especially well-funded, and their members are especially loyal.

If we were to adopt a categorical view that unions are always good or always bad—as many people seem to want to—it’s difficult to see why police unions should be different from teachers’ unions or factory workers’ unions. But my argument was very careful not to make such categorical statements. Unions aren’t always or inherently good; they are usually good, because of how they are correcting a power imbalance between workers and corporations.

But when it comes to police, the situation is quite different. Police unions give more bargaining power to government officers against… what? Public accountability? The democratic system? Corporate CEOs are accountable only to their shareholders, but the mayors and city councils who decide police policy are elected (in most of the UK, even police commissioners are directly elected). It’s not clear that there was an imbalance in bargaining power here we would want to correct.

A similar case could be made against all public-sector unions, and indeed that case often is extended to teachers’ unions. If we must sacrifice teachers’ unions in order to destroy police unions, I’d be prepared to bite that bullet. But there are vital differences here as well. Teachers are not responsible for imprisoning people, and bad teachers almost never kill people. (In the rare cases in which teachers have committed murder, they have been charged to the full extent of the law, just as they would be in any other profession.) There surely is some misconduct by teachers that some unions may be protecting, but the harm caused by that misconduct is far lower than the harm caused by police misconduct. Teacher unions also provide a layer of protection for teachers to exercise autonomy, promoting academic freedom.

The form of teacher misconduct I would be most concerned about is sexual abuse of students. And while I’ve seen many essays claiming that teacher unions protect sexual abusers, the only concrete evidence I could find on the subject was a teachers’ union publicly complaining that the government had failed to pass stricter laws against sexual abuse by teachers. The research on teacher misconduct mainly focuses on other casual factors aside from union representation.

Even this Fox News article cherry-picking the worst examples of unions protecting abusive teachers includes line after line like “he was ultimately fired”, “he was pressured to resign”, and “his license was suspended”. So their complaint seems to be that it wasn’t done fast enough? But a fair justice system is necessarily slow. False accusations are rare, but they do happen—we can’t just take someone’s word for it. Ensuring that you don’t get fired until the district mounts strong evidence of misconduct against you is exactly what unions should be doing.

Whether unions are good or bad in a particular industry is ultimately an empirical question. So let’s look at the data, shall we? Teacher unions are positively correlated with school performance. But police unions are positively correlated with increased violent misconduct. There you have it: Teacher unions are good, but police unions are bad.

An unusual recession, a rapid recovery

Jul 11 JDN 2459407

It seems like an egregious understatement to say that the last couple of years have been unusual. The COVID-19 pandemic was historic, comparable in threat—though not in outcome—to the 1918 influenza pandemic.

At this point it looks like we may not be able to fully eradicate COVID. And there are still many places around the world where variants of the virus continue to spread. I personally am a bit worried about the recent surge in the UK; it might add some obstacles (as if I needed any more) to my move to Edinburgh. Yet even in hard-hit places like India and Brazil things are starting to get better. Overall, it seems like the worst is over.

This pandemic disrupted our society in so many ways, great and small, and we are still figuring out what the long-term consequences will be.

But as an economist, one of the things I found most unusual is that this recession fit Real Business Cycle theory.

Real Business Cycle theory (henceforth RBC) posits that recessions are caused by negative technology shocks which result in a sudden drop in labor supply, reducing employment and output. This is generally combined with sophisticated mathematical modeling (DSGE or GTFO), and it typically leads to the conclusion that the recession is optimal and we should do nothing to correct it (which was after all the original motivation of the entire theory—they didn’t like the interventionist policy conclusions of Keynesian models). Alternatively it could suggest that, if we can, we should try to intervene to produce a positive technology shock (but nobody’s really sure how to do that).

For a typical recession, this is utter nonsense. It is obvious to anyone who cares to look that major recessions like the Great Depression and the Great Recession were caused by a lack of labor demand, not supply. There is no apparent technology shock to cause either recession. Instead, they seem to be preciptiated by a financial crisis, which then causes a crisis of liquidity which leads to a downward spiral of layoffs reducing spending and causing more layoffs. Millions of people lose their jobs and become desperate to find new ones, with hundreds of people applying to each opening. RBC predicts a shortage of labor where there is instead a glut. RBC predicts that wages should go up in recessions—but they almost always go down.

But for the COVID-19 recession, RBC actually had some truth to it. We had something very much like a negative technology shock—namely the pandemic. COVID-19 greatly increased the cost of working and the cost of shopping. This led to a reduction in labor demand as usual, but also a reduction in labor supply for once. And while we did go through a phase in which hundreds of people applied to each new opening, we then followed it up with a labor shortage and rising wages. A fall in labor supply should create inflation, and we now have the highest inflation we’ve had in decades—but there’s good reason to think it’s just a transitory spike that will soon settle back to normal.

The recovery from this recession was also much more rapid: Once vaccines started rolling out, the economy began to recover almost immediately. We recovered most of the employment losses in just the first six months, and we’re on track to recover completely in half the time it took after the Great Recession.

This makes it the exception that proves the rule: Now that you’ve seen a recession that actually resembles RBC, you can see just how radically different it was from a typical recession.

Moreover, even in this weird recession the usual policy conclusions from RBC are off-base. It would have been disastrous to withhold the economic relief payments—which I’m happy to say even most Republicans realized. The one thing that RBC got right as far as policy is that a positive technology shock was our salvation—vaccines.

Indeed, while the cause of this recession was very strange and not what Keynesian models were designed to handle, our government largely followed Keynesian policy advice—and it worked. We ran massive government deficits—over \$3 trillion in 2020—and the result was rapid recovery in consumer spending and then employment. I honestly wouldn’t have thought our government had the political will to run a deficit like that, even when the economic models told them they should; but I’m very glad to be wrong. We ran the huge deficit just as the models said we should—and it worked. I wonder how the 2010s might have gone differently had we done the same after 2008.

Perhaps we’ve learned from some of our mistakes.

A prouder year for America, and for me

Jul 4 JDN 2459380

Living under Trump from 2017 to 2020, it was difficult to be patriotic. How can we be proud of a country that would put a man like that in charge? And then there was the COVID pandemic, which initially the US handled terribly—largely because of the aforementioned Trump.

But then Biden took office, and almost immediately things started to improve. This is a testament to how important policy can be—and how different the Democrats and Republicans have become.

The US now has one of the best rates of COVID vaccination in the world (though lately progress seems to be stalling and other countries are catching up). Daily cases in the US are now the lowest they have been since March 2020. Even real GDP is almost back up to its pre-pandemic level (even per-capita), and the surge of inflation we got as things began to re-open already seems to be subsiding.

I can actually celebrate the 4th of July with some enthusiasm this year, whereas the last four years involved continually reminding myself that I was celebrating the liberating values of America’s founding, not the current terrible state of its government. Of course our government policy still retains many significant flaws—but it isn’t the utter embarrassment it was just a year ago.

This may be my last 4th of July to celebrate for the next few years, as I will soon be moving to Scotland (more on that in a moment).

2020 was a very bad year, but even halfway through it’s clear that 2021 is going to be a lot better.

This was true for just about everyone. I was no exception.

The direct effects of the pandemic on me were relatively minor.

Transitioning to remote work was even easier than I expected it to be; in fact I was even able to run experiments online using the same research subject pool as we’d previously used for the lab. I not only didn’t suffer any financial hardship from the lockdowns, I ended up better off because of the relief payments (and the freezing of student loan payments as well as the ludicrous stock boom, which I managed to buy in near the trough of). Ordering groceries online for delivery is so convenient I’m tempted to continue it after the pandemic is over (though it does cost more).

I was careful and/or fortunate enough not to get sick (now that I am fully vaccinated, my future risk is negligible), as were most of my friends and family. I am not close to anyone who died from the virus, though I do have some second-order links to some who died (grandparents of a couple of my friends, the thesis advisor of one of my co-authors).

It was other things, that really made 2020 a miserable year for me. Some of them were indirect effects of the pandemic, and some may not even have been related.

For me, 2020 was a year full of disappointments. It was the year I nearly finished my dissertation and went on the job market, applying for over one hundred jobs—and got zero offers. It was the year I was scheduled to present at an international conference—which was then canceled. It was the year my papers were rejected by multiple journals. It was the year I was scheduled to be married—and then we were forced to postpone the wedding.

But now, in 2021, several of these situations are already improving. We will be married on October 9, and most (though assuredly not all) of the preparations for the wedding are now done. My dissertation is now done except for some formalities. After over a year of searching and applying to over two hundred postings in all, I finally found a job, a postdoc position at the University of Edinburgh. (A postdoc isn’t ideal, but on the other hand, Edinburgh is more prestigious than I thought I’d be able to get.) I still haven’t managed to publish any papers, but I no longer feel as desperate a need to do so now that I’m not scrambling to find a job. Now of course we have to plan for a move overseas, though fortunately the university will reimburse our costs for the visa and most of the moving expenses.

Of course, 2021 isn’t over—neither is the COVID pandemic. But already it looks like it’s going to be a lot better than 2020.

Good news for a change

Mar 28 JDN 2459302

When President Biden made his promise to deliver 100 million vaccine doses to Americans within his first 100 days, many were skeptical. Perhaps we had grown accustomed to the anti-scientific attitudes and utter incompetence of Trump’s administration, and no longer believed that the US federal government could do anything right.

The skeptics were wrong. For the promise has not only been kept, it has been greatly exceeded. As of this writing, Biden has been President for 60 days and we have already administered 121 million vaccine doses. If we continue at the current rate, it is likely that we will have administered over 200 million vaccine doses and fully vaccinated over 100 million Americans by Biden’s promised 100-day timeline—twice as fast as what was originally promised. Biden has made another bold promise: Every adult in the United States vaccinated by the end of May. I admit I’m not confident it can be done—but I wasn’t confident we’d hit 100 million by now either.

In fact, the US now has one of the best rates of COVID vaccination in the world, with the proportion of our population vaccinated far above the world average and below only Israel, UAE, Chile, the UK, and Bahrain (plus some tiny countries like Monaco). In fact, we actually have the largest absolute number of vaccinated individuals in the world, surpassing even China and India.

It turns out that the now-infamous map saying that the US and UK were among the countries best-prepared for a pandemic wasn’t so wrong after all; it’s just that having such awful administration for four years made our otherwise excellent preparedness fail. Put someone good in charge, and yes, indeed, it turns out that the US can deal with pandemics quite well.

The overall rate of new COVID cases in the US began to plummet right around the time the vaccination program gained steam, and has plateaued around 50,000 per day for the past few weeks. This is still much too high, but it is is a vast improvement over the 200,000 cases per day we had in early January. Our death rate due to COVID now hovers around 1,500 people per day—that’s still a 9/11 every two days. But this is half what our death rate was at its worst. And since our baseline death rate is 7,500 deaths per day, 1,800 of them by heart disease, this now means that COVID is no longer the leading cause of death in the United States; heart disease has once again reclaimed its throne. Of course, people dying from heart disease is still a bad thing; but it’s at least a sign of returning to normalcy.

Worldwide, the pandemic is slowing down, but still by no means defeated, with over 400,000 new cases and 7,500 deaths every day. The US rate of 17 new cases per 100,000 people per day is about 3 times the world average, but comparable to Germany (17) and Norway (18), and nowhere near as bad as Chile (30), Brazil (35), France (37), or Sweden (45), let alone the very hardest-hit places like Serbia (71), Hungary (78), Jordan (83), Czechia (90), and Estonia (110). (That big gap between Norway and Sweden? It’s because Sweden resisted using lockdowns.) And there is cause for optimism even in these places, as vaccination rates already exceed total COVID cases.

I can see a few patterns in the rate of vaccination by state: very isolated states have managed to vaccinate their population fastest—Hawaii and Alaska have done very well, and even most of the territories have done quite well (though notably not Puerto Rico). The south has done poorly (for obvious reasons), but not as poorly as I might have feared; even Texas and Mississippi have given at least one dose to 21% of their population. New England has been prioritizing getting as many people with at least one dose as possible, rather than trying to fully vaccinate each person; I think this is the right strategy.

We must continue to stay home when we can and wear masks when we go out. This will definitely continue for at least a few more months, and the vaccine rollout may not even be finished in many countries by the end of the year. In the worst-case scenario, COVID may become an endemic virus that we can’t fully eradicate and we’ll have to keep getting vaccinated every year like we do for influenza (though the good news there is that it likely wouldn’t be much more dangerous than influenza at that point either—though another influenza is nothing to, er, sneeze at).

Yet there is hope at last. Things are finally getting better.

Love in a time of quarantine

Feb 14JDN 2459260

This is our first Valentine’s Day of quarantine—and hopefully our last. With Biden now already taking action and the vaccine rollout proceeding more or less on schedule, there is good reason to think that this pandemic will be behind us by the end of this year.

Yet for now we remain isolated from one another, attempting to substitute superficial digital interactions for the authentic comforts of real face-to-face contact. And anyone who is single, or forced to live away from their loved ones, during quarantine is surely having an especially hard time right now.

I have been quite fortunate in this regard: My fiancé and I have lived together for several years, and during this long period of isolation we’ve at least had each other—if basically no one else.

But even I have felt a strong difference, considerably stronger than I expected it would be: Despite many of my interactions already being conducted via the Internet, needing to do so with all interactions feels deeply constraining. Nearly all of my work can be done remotely—but not quite all, and even what can be done remotely doesn’t always work as well remotely. I am moderately introverted, and I still feel substantially deprived; I can only imagine how awful it must be for the strongly extraverted.

As awkward as face-to-face interactions can be, and as much as I hate making phone calls, somehow Zoom video calls are even worse than either. Being unable to visit someone’s house for dinner and games, or go out to dinner and actually sit inside a restaurant, leaves a surprisingly large emotional void. Nothing in particular feels radically different, but the sum of so many small differences adds up to a rather large one. I think I felt it the most when we were forced to cancel our usual travel back to Michigan over the holiday season.

Make no mistake: Social interaction is not simply something humans enjoy, or are good at. Social interaction is a human need. We need social interaction in much the same way that we need food or sleep. The United Nations considers solitary confinement for more than two weeks to be torture. Long periods in solitary confinement are strongly correlated with suicide—so in that sense, isolation can kill you. Think about the incredibly poor quality of social interactions that goes on in most prisons: Endless conflict, abuse, racism, frequent violence—and then consider that the one thing that inmates find most frightening is to be deprived of that social contact. This is not unlike being fed nothing but stale bread and water, and then suddenly having even that taken away from you.

Even less extreme forms of social isolation—like most of us are feeling right now—have as detrimental an effect on health as smoking or alcoholism, and considerably worse than obesity. Long-term social isolation increases overall mortality risk by more than one-fourth. Robust social interaction is critical for long-term health, both physically and mentally.

This does not mean that the quarantines were a bad idea—on the contrary, we should have enforced them more aggressively, so as to contain the pandemic faster and ultimately need less time in quarantine. Timing is critical here: Successfully containing the pandemic early is much easier than trying to bring it back under control once it has already spread. When the pandemic began, lockdown might have been able to stop the spread. At this point, vaccines are really our only hope of containment.

But it does mean that if you feel terrible lately, there is a very good reason for this, and you are not alone. Due to forces much larger than any of us can control, forces that even the world’s most powerful governments are struggling to contain, you are currently being deprived of a basic human need.

And especially if you are on your own this Valentine’s Day, remember that there are people who love you, even if they can’t be there with you right now.

On the accuracy of testing

Jan 31 JDN 2459246

One of the most important tools we have for controlling the spread of a pandemic is testing to see who is infected. But no test is perfectly reliable. Currently we have tests that are about 80% accurate. But what does it mean to say that a test is “80% accurate”? Many people get this wrong.

First of all, it certainly does not mean that if you have a positive result, you have an 80% chance of having the virus. Yet this is probably what most people think when they hear “80% accurate”.

So I thought it was worthwhile to demystify this a little bit, an explain just what we are talking about when we discuss the accuracy of a test—which turns out to have deep implications not only for pandemics, but for knowledge in general.

There are really two key measures of a test’s accuracy, called sensitivity and specificity, The sensitivity is the probability that, if the true answer is positive (you have the virus), the test result will be positive. This is the sense in which our tests are 80% accurate. The specificity is the probability that, if the true answer is negative (you don’t have the virus), the test result is negative. The terms make sense: A test is sensitive if it always picks up what’s there, and specific if it doesn’t pick up what isn’t there.

These two measures need not be the same, and typically are quite different. In fact, there is often a tradeoff between them: Increasing the sensitivity will often decrease the specificity.

This is easiest to see with an extreme example: I can create a COVID test that has “100% accuracy” in the sense of sensitivity. How do I accomplish this miracle? I simply assume that everyone in the world has COVID. Then it is absolutely guaranteed that I will have zero false negatives.

I will of course have many false positives—indeed the vast majority of my “positive results” will be me assuming that COVID is present without any evidence. But I can guarantee a 100% true positive rate, so long as I am prepared to accept a 0% true negative rate.

It’s possible to combine tests in ways that make them more than the sum of their parts. You can first run a test with a high specificity, and then re-test with a test that has a high sensitivity. The result will have both rates higher than either test alone.

For example, suppose test A has a sensitivity of 70% and a specificity of 90%, while test B has the reverse.

Then, if the true answer is positive, test A will return true 70% of the time, while test B will return true 90% of the time. So there is a 70% + (30%)(90%) = 97% chance of getting a positive result on the combined test.

If the true answer is negative, test A will return false 90% of the time, while test B will return false 70% of the time. So there is a 90% + (10%)(70%) = 97% chance of getting a negative result on the combined test.

Actually if we are going to specify the accuracy of a test in a single number, I think it would be better to use a much more obscure term, the informedness. Informedness is sensitivity plus specificity, minus one. It ranges between -1 and 1, where 1 is a perfect test, and 0 is a test that tells you absolutely nothing. -1 isn’t the worst possible test; it’s a test that’s simply calibrated backwards! Re-label it, and you’ve got a perfect test. So really maybe we should talk about the absolute value of the informedness.

It’s much harder to play tricks with informedness: My “miracle test” that just assumes everyone has the virus actually has an informedness of zero. This makes sense: The “test” actually provides no information you didn’t already have.

Surprisingly, I was not able to quickly find any references to this really neat mathematical result for informedness, but I find it unlikely that I am the only one who came up with it: The informedness of a test is the non-unit eigenvalue of a Markov matrix representing the test. (If you don’t know what all that means, don’t worry about it; it’s not important for this post. I just found it a rather satisfying mathematical result that I couldn’t find anyone else talking about.)

But there’s another problem as well: Even if we know everything about the accuracy of a test, we still can’t infer the probability of actually having the virus from the test result. For that, we need to know the baseline prevalence. Failing to account for that is the very common base rate fallacy.

Here’s a quick example to help you see what the problem is. Suppose that 1% of the population has the virus. And suppose that the tests have 90% sensitivity and 95% specificity. If I get a positive result, what is the probability I have the virus?

If you guessed something like 90%, you have committed the base rate fallacy. It’s actually much smaller than that. In fact, the true probability you have the virus is only 15%.

In a population of 10000 people, 100 (1%) will have the virus while 9900 (99%) will not. Of the 100 who have the virus, 90 (90%) will test positive and 10 (10%) will test negative. Of the 9900 who do not have the virus, 495 (5%) will test positive and 9405 (95%) will test negative.

This means that out of 585 positive test results, only 90 will actually be true positives!

If we wanted to improve the test so that we could say that someone who tests positive is probably actually positive, would it be better to increase sensitivity or specificity? Well, let’s see.

If we increased the sensitivity to 95% and left the specificity at 95%, we’d get 95 true positives and 495 false positives. This raises the probability to only 16%.

But if we increased the specificity to 97% and left the sensitivity at 90%, we’d get 90 true positives and 297 false positives. This raises the probability all the way to 23%.

But suppose instead we care about the probability that you don’t have the virus, given that you test negative. Our original test had 9900 true negatives and 10 false negatives, so it was quite good in this regard; if you test negative, you only have a 0.1% chance of having the virus.

Which approach is better really depends on what we care about. When dealing with a pandemic, false negatives are much worse than false positives, so we care most about sensitivity. (Though my example should show why specificity also matters.) But there are other contexts in which false positives are more harmful—such as convicting a defendant in a court of law—and then we want to choose a test which has a high true negative rate, even if it means accepting a low true positive rate.

In science in general, we seem to care a lot about false positives; a p-value is simply one minus the specificity of the statistical test, and as we all know, low p-values are highly sought after. But the sensitivity of statistical tests is often quite unclear. This means that we can be reasonably confident of our positive results (provided the baseline probability wasn’t too low, the statistics weren’t p-hacked, etc.); but we really don’t know how confident to be in our negative results. Personally I think negative results are undervalued, and part of how we got a replication crisis and p-hacking was by undervaluing those negative results. I think it would be better in general for us to report 95% confidence intervals (or better yet, 95% Bayesian prediction intervals) for all of our effects, rather than worrying about whether they meet some arbitrary threshold probability of not being exactly zero. Nobody really cares whether the effect is exactly zero (and it almost never is!); we care how big the effect is. I think the long-run trend has been toward this kind of analysis, but it’s still far from the norm in the social sciences. We’ve become utterly obsessed with specificity, and basically forgot that sensitivity exists.

Above all, be careful when you encounter a statement like “the test is 80% accurate”; what does that mean? 80% sensitivity? 80% specificity? 80% informedness? 80% probability that an observed positive is true? These are all different things, and the difference can matter a great deal.

This attack on the postal service must not stand

Aug 23 JDN 2459085

Trump has done so many unprecedented and terrible things that we can become numbed by it all, unable to process each new offense because we are already overwhelmed by the others. Perhaps this is a kind of strategy on his part: Keep doing so many outrageous things that we lose our capacity to be outraged. Already it is fair to say that at least half of the 160,000 (and counting) Americans killed by COVID-19 would still be alive if a better President had been in office.

But the attack on the US Postal Service deserves particular attention, because the disruption of mail-in voting during a pandemic could radically alter the results of the election. Indeed, Trump has all but said that this was his goal in defunding the post office.

Trump has long hated the postal service (perhaps because it is a clear example of federal government doing things well and helping people), but his full-scale war upon it started with the appointment of Louis DeJoy as Postmaster General, whose main qualifications appear to be that he has given millions of dollars to Republican campaigns and hates everything the post office stands for. I am quite certain that if there were a Director of Henhouse Affairs, Trump would appoint the Fantastic Mr. Fox.

The White House chief of staff claims that there have been no mail sorting machines decommissioned aside from those that were normally scheduled for replacement. Yet it’s easy to find a number of different sources claiming that there have been far more machines shut down than usual. Postal workers have also spoken out about other kinds of restructuring in the postal system that claim to be about “reducing costs” but seem to be systematically impairing the speed and reliability of service.

Trump claims that mail-in voting is insecure, which has a kernel of truth: Mail-in voting certainly doesn’t have the ironclad security against fraud that in-person voting has. (Unlike in-person voter fraud, mail-in voter fraud actually exists.) But not only is his concern obviously overblown, the USPS has even taken measures to upgrade their security using blockchain encryption. Bitcoin has always been a stupid idea (though a very lucrative one for anyone who bought in early), but blockchain does have some major advantages for voting security, because it is one of the few ways to make a remote system that is simultaneously secure and anonymous. Indeed, I think blockchain encryption (combined with more standard SSL encryption that most web pages already use) might well be a way to implement full-scale online voting—though surely not in time for this election.

The US Postal Service is the most popular federal agency in the United States, followed by the CDC, the Census Bureau, and the Department of Health and Human Services, all of which deservedly have strong bipartisan majority support among voters. It may surprise you to learn that the Department of Homeland Security, the IRS, and the Department of Justice also have strong majority support—though with substantial partisan differences. The most divisive federal agency is ICE, which is beloved by Republicans but hated by Democrats.

Some 91% of Americans approve of the USPS—and why shouldn’t they? It is objectively rated one of the best postal systems in the world—and if anything this isn’t even fair, because most of the other top-rated postal services, particularly Switzerland, the Netherlands, and Singapore, have far smaller areas to cover than the US does. If we restrict ourselves to countries of at least 10 million people and territory of at least 100,000 square kilometers, there are only four postal services rated higher than the US: Japan, Germany, France, and Poland. If we restrict to countries of at least 100 million people, only Japan remains.

Thus, attacking the postal service is clearly not a winning proposition if your goal is to advance the interests of your constituents or even gain more votes. But during a pandemic, mail-in voting is likely to be—and well should be—a very large proportion of all votes. Sabotaging the mail system is a highly effective way to make it much harder to vote in general. And that seems to very much be Trump’s intention.

It is a general pattern that when voting gets harder, Republicans become more likely to win. Liberal voters are more likely to be young adults, poor people, or people of color, all of whom generally have a harder time making it to the polls. This may be less true in this election in particular, because against Trump in particular people who are highly educated and live in cities have been far more likely to vote against Trump—and these are groups of people with particularly high voter turnout. Empirical estimates of how a switch to mail-in voting will affect the election results have been highly ambiguous.

Indeed, perhaps this makes the Republican vote suppression campaign even more sinister: Perhaps they have moved beyond simply trying to tilt the scales in elections and are now willing to actively suppress democracy itself. It sounds radical, if not outright crazy, to assert such a thing—but many of the things that Trump and his Republican lackeys have done would have sounded crazy to me just a few years ago. I can’t believe I’m saying this, but I honestly don’t know that Trump will concede defeat when he loses the election—he may refuse to accept the election results and try to stay in office via some sort of coup d’etat. Why do I think this could happen? Because he said so himself on national television. Vladimir Putin must be so embarrassed; his protege doesn’t even know how to be subtle about his authoritarianism.

FiveThirtyEight is currently giving Biden a 72% chance of victory, which is about 27% too low for my taste. That isn’t much better than the margin Hillary Clinton had four years ago. We can only hope that Trump attacking the most popular agency in our federal government will tilt those odds a little further.

This is not just about selfishness

Aug 2 JDN 2459064

The Millennial term is “Karen”: someone (paradigmatically a middle-aged White woman) who is so privileged, so self-centered, and has such an extreme sense of entitlement, that they are willing to make others suffer in order to avoid the slightest inconvenience.

I recently saw a tweet (which for some reason has been impossible to find; I think I must have misremembered its precise wording, because putting that in quotes in Google yields nothing) saying that Americans are not simply selfish, we are so selfish that we would gladly let others die to avoid mildly inconveniencing ourselves. Searching Twitter for “Americans are selfish” certainly yields plenty of results.

And it is tempting to agree with this, when it seems that re-opening the economy and so many people refusing to wear masks has given us far worse outcomes from COVID-19 than most other countries.

But this can’t be the whole story. Perhaps Americans are a bit more self-centered than other cultures, because of our history of libertarian individualism. But if we were truly so selfish we’d gladly let others die to avoid inconvenience, whence the fact that we donate more to charity than any other country in the world? I don’t simply mean total amount or per-capita dollars (though both of those are also true); I mean as a fraction of GDP Americans give more to charity than any other country, and by a wide margin.

How then do we explain that so many Americans are not wearing masks?

Well, first of all, most of us are wearing masks. The narrative about people not wearing masks has been exaggerated; the majority of Americans, including the majority of Republicans, agree that wearing masks is a matter of public health rather than personal choice. There are some people who refuse to wear masks, and each one adds a little bit more risk to us all; but it’s really not the case that Americans in general are refusing to wear masks.

But I think the most important failings here come from the top down. The Trump administration has handled the pandemic in an astonishingly poor way. First, they denied that it was even a serious problem. Then, they implemented only a half-hearted response. Then, they turned masks into a culture war. Then, they resisted the economic relief package and prevented it from being as large as it needed to me. At every step of the way, they have been at best utterly incompetent and at worst guilty of depraved indifference murder.

From denying it was a problem, to responding too slowly, to disparaging mask use, to pushing to re-open the economy too soon, at every step of the way our government has made things worse. Above all, a better economic relief package—like what most other First World countries have done—would have done a great deal to reduce the harm of lockdowns, and would have made re-opening the economy far less popular.

Republican-led states have followed the President’s lead, refusing to implement even basic common-sense protections. But even Democrat-led states have suffered greatly as well. New York and California have some of the most cases, though this is surely in part because they are huge states with highly urbanized populations that get a lot of visitors and trade from other places. The trajectory of infections looks worst in Lousiana and Missouri, surely among the most conservative of states; but it also looks quite bad in New Jersey and Hawaii, which are among the most liberal.

I think what this shows us is that America lacks coordination. Despite having United in our name and E pluribus unum as our motto (“In God We Trust” was a Cold War change to spite the Soviets), what we lack most of all is unity. Viruses do not respect borders or jurisdictions. More than perhaps any other issue aside from climate change, fighting a pandemic requires a unified, coordinated response—and that is precisely what we did not have.

In some ways the pluralism of the United States can be a great strength; but this year, it was very much a weakness. And as the many crises around us continue, I fear we grow only more divided.

No, unemployment doesn’t kill people

Jun 14 JDN 2459015

Some people have argued that lockdown measures were unnecessary, or ineffective. The data definitely leans the other direction, but there’s enough uncertainty in all this that I can at least consider that a serious possibility. That doesn’t mean we were wrong to use them; in the presence of high uncertainty, assuming the worst-case scenario is often the best strategy. Far better to overreact than underreact. And indeed, I’d say that right now we still can’t be confident enough that things are safe to really re-open most of the economy. Re-opening too early could make things far worse.

There’s another argument for re-opening the economy which seems far more seductive: What about the people harmed by the lockdowns? This massive unemployment is terrible too, isn’t it? In fact, what if we’re killing more people by unemployment than we are saving from the virus? The Mises Institute warns: “Unemployment Kills”. Others have speculated that the recession could cause more deaths than the virus.

But in fact, unemployment does not kill. The evidence on this is quite clear. Even in the Great Depression, with massive unemployment, terrible monetary policy, and only the most minimal social welfare measures in place, death rates did not increase. In fact, for all causes except suicide, death rates decrease during recessions—probably because pollution, traffic accidents, and work-related injury and illness go down. And the suicide rate increase isn’t enough to increase the overall death rate.

Of course, dying by suicide is not the same thing as dying from cancer—and indeed, they are most likely different people being affected in each case. So in that sense unemployment can kill people; but it typically saves more people than it kills. Almost any policy choice will cause some deaths and prevent others, so really the best we can do is look at the overall aggregate and see whether our QALY have gone up or down.

This doesn’t mean that we should go out of our way to have recessions in order to save lives; the number of lives saved is small and the loss in quality of life is probably large enough to compensate for it. (That’s why we use quality-adjusted life years after all.) But this recession isn’t arbitrary; it’s the result of trying to stop a global pandemic, so that we don’t have a repeat of what influenza did in 1918.

When the CDC says it’s okay to open back up, by all means, let’s do that. They have issued guidelines for what we need to do in order to make that happen. But until then, let’s trust in the experts—the epidemiologists who say that we still need lockdown measures, and the economists who agree that it’s worth the cost.