Correlation is correlated with causation

My old friend Allen Downey has a thoroughly sensible post about correlation and causation.

It is true that correlation doesn’t imply causation in the mathematical sense of “imply;” that is, finding a correlation between A and B does not prove that A causes B. However, it does provide evidence that A causes B. It also provides evidence that B causes A, and if there is a hypothetical C that might cause A and B, the correlation is evidence for that hypothesis, too.

If you don’t understand what he means by this, or if you don’t believe it, read the whole thing.

I do think he’s guilty of a bit of rhetorical excess when he says that “the usual mantra, ‘Correlation does not imply causation,’ is true only in a trivial sense.” I think that the mantra means something quite specific and valid, namely something like “correlation does not provide evidence for causation that’s as strong as you seem to think.” One often sees descriptions of measured correlations that imply that the correlation supports the hypothesis of causation to the exclusion of all others, and when that’s wrong it’s convenient to have a compact way of saying so.

But that’s a small quibble, which gets even smaller if I include the next phrase in that quote from Allen,

The point I was trying to make (and will elaborate here) is that the usual mantra, “Correlation does not imply causation,” is true only in a trivial sense, so we need to think about it more carefully [emphasis added].

I couldn’t agree more, and everything Allen goes on to say after this is 100% correct.




Sean Carroll has a post about a report by Laurence Yaffe on the state of funding for theoretical particle physics in the US, or more specifically in the Department of Energy, which historically has been the big funding source in this field. They both use the word “calamity” to describe the situation, but as far as I can tell the evidence cited doesn’t support that conclusion.

(For what it’s worth, I see that Peter Woit views the situation the same way I do. For some people, that will increase my credibility; for others it’ll decrease it.)

Yaffe’s abstract:

A summary is presented of data obtained from a grass-roots effort to understand the effects of the FY13 and FY14 comparative review cycles on the DOE-funded portion of the US high energy theory community and, in particular, on graduate students and postdoctoral researchers who are beginning their careers. For a sample comprised of nearly all of the larger groups undergoing comparative review, total funding declined by an average of 23%, with numerous major groups receiving reductions in the 30–55% range. Funding available for postdoc or graduate student support declined over 30%, with many reductions in the 40–65% range. The total number of postdoc positions in this large sample of theory groups is declining by over 40%. The impacts on young researchers raise grave concerns regarding continued U.S. leadership in high energy theory.


Obviously this is unsustainable, unless as a society we make the decision that particle physics just isn’t worth doing. But hopefully things can be rectified at least a bit, to restore some of that money.

But the striking thing about Yaffe’s report is that it says precisely nothing about the total level of DOE funding in this field. What it says instead is that existing large particle theory research groups have had their funding cut. Is that because the funding is going away or because it’s going to other, smaller groups? As far as I can tell, Yaffe and Carroll assume the former, but they provide no evidence for it.

DOE did recently change their procedure for evaluating grants in various ways. According to Yaffe,

Three years ago, the Office of High Energy Physics (OHEP) within the Department of Energy made significant changes in how university-based research proposals are reviewed, switching to a comparative review process and synchronizing all new grants. Overt goals included decreasing the effects of historical inertia on funding levels for different groups, ensuring a level playing field, and moving to a start date for grants mid-way through the federal fiscal year by which time, it was hoped, Congressional funding decisions would normally be known.

The first two of those goals, it seems to me, pretty much say that the DOE is aiming to redistribute funds away from previously-large research groups (those that have benefitted from “historical inertia”). Yaffe gathered data on large research groups and showed they got smaller, precisely as you’d expect. So it’s not at all clear to me that the alarmist response to this information is warranted.

What we really need to know is simply how much funding high energy theory is getting in comparison with past years. That information isn’t as easy to find as you might think. The most recent DOE budget request does show a drop in high energy theory funding, but a more modest one than Yaffe’s figures, and in any case that wouldn’t have shown up in the figures yet. Over the few previous years, things seem pretty stable. Of course, “stable” in nominal terms is a modest de facto decline in real terms, but nothing like the proclaimed “calamity.”

I’m not a particle theorist, and I don’t deal with DOE, so I haven’t paid close attention to DOE funding levels over the years. It’s certainly possible that I’m missing something here. If anyone knows what it is, I’d be interested to hear.


  1. One can take the view that the government has no business funding pure curiosity-driven research like particle theory anyway. I don’t agree with that view, although I do understand it.
  2. One can take the more moderate view that, even if the government should be funding things like particle theory, previous funding levels were too high and so a cut isn’t a “calamity.” I don’t have that great a sense of what funding levels are like in particle theory, so it’s hard for me to say for sure what I think about that. In general, I think we should be funding more science, not less, but then as a (modest) beneficiary of government research grants, I would think that, wouldn’t I?
Update: Joanne Hewett’s comment on Sean’s post is by far the most informative thing I’ve seen on this subject.


Nature on p-values

Nature has a depressing piece about how to interpret p-values (i.e., the numbers people generally use to describe “statistical significance”). What’s depressing about it? Sentences like this:

Most scientists would look at his original P value of 0.01 and say that there was just a 1% chance of his result being a false alarm.

If it’s really true that “most scientists” think this, then we’re in deep trouble.

Anyway, the article goes on to give a good explanation of why this is wrong:

But they would be wrong. The P value cannot say this: all it can do is summarize the data assuming a specific null hypothesis. It cannot work backwards and make statements about the underlying reality. That requires another piece of information: the odds that a real effect was there in the first place. To ignore this would be like waking up with a headache and concluding that you have a rare brain tumour — possible, but so unlikely that it requires a lot more evidence to supersede an everyday explanation such as an allergic reaction. The more implausible the hypothesis — telepathy, aliens, homeopathy — the greater the chance that an exciting finding is a false alarm, no matter what the P value is.

The main point here is the standard workhorse idea of Bayesian statistics: the experimental evidence gives you a recipe for updating your prior beliefs about the probability that any given statement about the world is true. The evidence alone does not tell you the probability that a hypothesis is true. It cannot do so without folding in a prior.

To rehash the old, standard example, suppose that you take a test to see if you have kuru. The test gives the right answer 99% of the time. You test positive. That test “rules out” the null hypothesis that you’re disease-free with a p-value of 1%. But that doesn’t mean there’s a 99% chance you have the disease. The reason is that the prior probability that you have kuru is very low. Say one person in 100,000 has the disease. When you test 100,000 people, you’ll get  roughly one true positive and 1000 false positives. Your positive test is overwhelmingly likely to be one of the false ones, low p-value notwithstanding.

For some reason, people regard “Bayesian statistics” as something controversial and heterodox. Maybe they wouldn’t think so if it were simply called “correct reasoning,” which is all it is.

You don’t have to think of yourself as “a Bayesian” to interpret p-values in the correct way. Standard statistics textbooks all state clearly that a p-value is not the probability that a hypothesis is true, but rather the probability that, if the null hypothesis is true, a result as extreme as the one actually found would occur.

Here’s a convenient Venn diagram to help you remember this:

(Confession: this picture is a rerun.)

If Nature‘s readers really don’t know this, then something’s seriously wrong with the way we train scientists.

Anyway, there’s a bunch of good stuff in this article:

The irony is that when UK statistician Ronald Fisher introduced the P value in the 1920s, he did not mean it to be a definitive test. He intended it simply as an informal way to judge whether evidence was significant in the old-fashioned sense: worthy of a second look.

Fisher’s got this exactly right. The standard in many fields for “statistical significance” is a p-value of 0.05. Unless you set the value far, far lower than that, a very large number of “significant” results are going to be false. That doesn’t necessarily mean that you shouldn’t use p-values. It just means that you should regard them (particularly with this easy-to-cross 0.05 threshold) as ways to decide which hypotheses to investigate further.

Another really important point:

Perhaps the worst fallacy is the kind of self-deception for which psychologist Uri Simonsohn of the University of Pennsylvania and his colleagues have popularized the term P-hacking; it is also known as data-dredging, snooping, fishing, significance-chasing and double-dipping. “P-hacking,” says Simonsohn, “is trying multiple things until you get the desired result” — even unconsciously.

I didn’t know the term P-hacking, although I’d heard some of the others. Anyway, it’s a sure-fire way to generate significant-looking but utterly false results, and it’s unfortunately not at all unusual.

Grammar and usage advice for my students

Some issues of grammar and usage come up fairly regularly in students’ writing. These are a few I’ve noticed in past semesters. I’ll illustrate them with (anonymous) excerpts from past student essays.

This is definitely not meant to be a complete list. It’s just issues that have struck me particularly often.

The comma splice.

This is by far the most common grammar or punctuation error in my students’ writing.

A comma splice is the joining together of two complete clauses by just a comma. For example,

Ptolemy has made it so Earth is not in the exact center of his model, it is displaced slightly.

The part before the comma is a grammatically complete sentence, and so is the second part. Joining them together with just a comma is an error.

There are several ways to fix a comma splice:

  • Just make the two halves separate sentences.
  • If you want to emphasize that they’re closely related, you can separate them by a semicolon (;) or maybe a colon (:). The semicolon is more common; the colon usually suggests that the second half explains or provides an example of whatever’s in the first half.
  • Put a conjunction (and, but, etc.) in between the two clauses.

There’s an important addendum to the last rule: “however” is not a conjunction, so it doesn’t fix a comma splice. For example:

All the above theories are strong candidates for the explanation of how we observe the heavens, however, only one of them soundly holds the upper-hand beyond the other two and that is the Tychonic model.

This contains a comma splice error. Replacing the comma after “heavens” with a semicolon solves the problem. (The comma after “however” is correct and necessary.)

The subjunctive.

There was a popular song in the 1990s called “What if God was one of us?” This song drove grammar nerds crazy, because it should be “What if God were one of us?” The old musical Fiddler on the Roof got this right, in the name of the song “If I were a rich man.”

The quick-and-dirty rule is this: if you write an “if” clause, and the thing in that clause is something known to be false, then use “were” instead of “was”. In this situation, “were” is the subjunctive form of “was”.

For example:

If the earth was really rotating once a day, we would have been hurled into the universe by now because of the speed.

Change “was” to “were” in this sentence.

(In hindsight, this is a poor example, because the Earth actually does rotate once per day! But this essay was written from the point of view of an anti-Copernican, who believed it was false.)

Beg the question.

An example of how this expression is often used:

This begs the question, can a theory be considered correct if it simply cannot be proven wrong?

The meaning here is clear: “begs the question” is being used to mean something like “raises the question” or “requires us to examine the question.”

The problem is that, traditionally, “beg the question” meant something quite different. It referred to a particular form of logical error, in which a person making an argument assumed the very thing he or she was trying to prove. In other words, accusing someone of “begging the question” means accusing them of circular reasoning.

Idiomatic expressions change over time. Nowadays, the original meaning of “beg the question” is quite uncommon. The phrase is much more commonly used in the sense that the student above meant. But at the moment, traditionalists regard the more modern meaning as “incorrect.”

You might think the traditionalists are being stupid to insist on an archaic meaning. You might say that since everybody instantly understands the intended meaning, there’s no problem. Between you and me, I might even agree with you. But some significant fraction of your readers, particularly if you’re writing papers for college professors, will insist that the newer meaning of this expression is “wrong” and will judge you negatively for it.

This puts “beg the question” in a strange position. If you use it in the old-fashioned, correct sense, lots of people won’t understand you. If you use it in the more current sense, traditionalists will sneer at you. There’s only one solution: avoid the phrase entirely. Fortunately, this isn’t hard to do. Often “raises the question” conveys the meaning you want.

As Bryan Garner put it in his very good book A Dictionary of Modern American Usage,

When a word undergoes a marked change from one use to another . . . it’s likely to be the subject of dispute. Some people (Group 1) insist on the traditional use; others (Group 2) embrace the new use. . . . Any use of [the word] is likely to distract some readers. The new use seems illiterate to Group 1; the old use seems odd to Group 2. The word has become “skunked.”

Beginning a sentence with “So.”

An example:

So when reflecting upon the theories of the best astronomical minds of this era, one cannot help but be swayed by the system that takes the best of each theory.

This is more a matter of taste than of correctness, but I recommend that you not start sentences with “So” in formal writing. Traditionally, “so” is a conjunction, so it should join two clauses together (as it just did in this sentence). I won’t say that using it as an adverb to start a sentence is wrong, but it sounds chatty and informal to me. If you’re going for a traditional, formal tone in your writing, I recommend avoiding it. If you’re deliberately trying to sound conversational, that’s a different matter.

In general, use the word “that” even when it’s optional.

Compare these two sentences:

  • If the Earth were in motion as Copernicus claims, the Bible would read the Earth stood still.
  • If the Earth were in motion as Copernicus claims, the Bible would read that the Earth stood still.

Both are 100% grammatically correct, but the second one’s better, for a subtle reason. In the first sentence, when you get to the words “read the Earth,” you have a momentary tendency to misinterpret it: it looks, just for an instant, as if “Earth” were a direct object. That is, you briefly interpret “read the Earth” in the same way as you would interpret “read the book.” When you get to the rest of the sentence, the confusion is cleared up, but that slight hiccup disrupts your attention just a bit. By including the word “that,” even though it’s not grammatically necessary, the author keeps the hiccup from happening.

This advice goes against one of the most common writing maxims, namely to be concise and omit needless words. But I still think you’re usually better off including “that” in situations like this. The best thing to do is to read the sentence aloud both ways and decide which one sounds better.

Let me repeat that this is not a matter of correctness: both versions of the sentence above is correct. All I ask you to do is consider whether one version or the other is better at helping the reader understand what you mean.


The passive voice.

Lots of people will tell you to avoid writing sentences in the passive voice. For instance, instead of saying

The issue of whether light travels instantaneously or at a finite speed has been debated by many great minds.


Many great minds have debated the issue of whether light travels instantaneously or at a finite speed.

For those who don’t remember their grammar terminology, in an active sentence, the subject of the sentence is the person or thing performing the action in the sentence. In the above examples, the action is debating, and the things doing the debating are the minds. The second version puts the actor right up front as the subject of the sentence. That’s active, and the conventional wisdom is that active is better.

Personally, I think that the blanket rule to avoid the passive is often overstated. (I rambled on about this at some length in another post.) There are lots of times when the passive voice is just what you want. But without a doubt there are many cases, including the above sentence, in which the passive voice sounds bad. My advice: whenever you write a passive sentence, try out what it sounds like if you make it active. Read both versions aloud, and see which one sounds better. More often than not, you’ll decide to make it active.

Spherical triangles

I’m teaching a cosmology course at the moment, in which we talk a lot about curved space. As usual in this sort of situation, I’m trying to make my students build up some intuition about what life is like on curved two-dimensional surfaces, which are easier to grasp than curved three-dimensional space. In that spirit, I asked them a question from our textbook (Ryden’s Introduction to Cosmology), which can be paraphrased

What is the area of the largest equilateral triangle you can draw on a sphere of radius R?

The ground rules here is that a triangle is bounded by “straight lines” on the surface of the sphere. A straight line (formally called a geodesic) on a curved surface is a curve that gives the shortest distance between two points (as long as those points aren’t too far apart). On a sphere, the straight lines are great circles (circles whose center is the center of the sphere).

For instance, here’s a triangle each of whose sides is a quarter of a great circle:


It has the funny property that all of its angles are right angles.

You can go bigger than this, though. Here’s a little animation showing equilateral triangles of different sizes:

The biggest one is one with three 180-degree angles, covering half the sphere. You take a single great circle (“straight line”), mark off three equally spaced point, and call those the vertices.

At least, that’s what I intended the answer to be (and I’m pretty sure it’s what the textbook author intended too). But a student in the class argued for a different answer. Ultimately, the difference between his answer and mine is a matter of definition, so you can say that either is “right,” but I have to confess that I like his answer better than mine.

I’ll show you his answer below.

Continue reading Spherical triangles