Why Tim Lambert is Wrong

Tim Lambert of University of New South Wales (and of fighting John Lott fame) seems to misunderstand the meaning of confidence intervals and Gaussian Distributions when discussing the Lancet study,which used cluster sampling to claim in its abstract that 100,000 excess deaths occurred among Iraqi civilians as a result of the war, here and here.

The study reported that the 95% confidence interval for excess deaths is between 8,000 and 194,000. Lancet picked the midpoint when making the 100,000 excess deaths claim and this has been the source of a lot of controversy. Critics of the study say that it is incorrect to merely pick the midpoint as the most likely figure. Lambert writes:

Yes, the 95% confidence interval by itself doesn’t tell us what the probabilities are. But this doesn’t mean that each value is equally likely. We can also construct other confidence intervals. We can be 67% confident that the number is between 50,000 and 150,000. In this sense the end points of the 95% CI are less likely and the middle is most likely.

In the second link I provide above, he writes:

Not all values in the confidence interval are equally likely. The ones in the middle are more likely and 100,000 is the most likely number.

This seriously misunderstands the point of Gaussian distributions (and Lambert admits earlier in the first post I linked to that the distribution is well-approximated by a Gaussian). First, a Gaussian (a.k.a. Normal, bell curve, etc) distribution is a continuous distribution. As such, any single point has probability of 0 of being attained. The reason is because the "width" or measure of any single point is 0. The only things that have positive measure (or positive probability) in a Guassian distribution are intervals. So, yes, while the "mass" of a Gaussian is mostly "concentrated in the center", it does not mean that a single point in the center is more likely than a single point at the edge. They are both equally likely, meaning not likely at all (this is somewhat of a paradox since every single point has probability 0, but adding them all up, you get 100% probability... this is the result of adding an uncountable, rather than just infinite, number of 0's).

Let me demonstrate with a small example. Take a normal distribution with mean 3 and variance 1. If you look in a statistics table (or compute using Mathematica or Matlab), you'll notice that the interval 0 to 1.04 has about 2.3% probability. On the other hand, the interval between 2.975 and 3.025 has only 2% probability, even though these are clearly closer to the mean of 3 than any point between 0 and 1.04. So, no, points closer to the mean are not more likely. Intervals closer to the mean of the same length are. But that's not what is at question here. The question is which number is more likely, 8000 or 100,000. These two are equally likely in the case of continuous Gaussian distributions.

Here's also why taking the midpoint is a bit silly on just a common sense level. We know for 100% certainty that there are between 0 and 100 billion excess deaths as a result of the war. That doesn't mean that 50 billion is the most likely one.

The point to be made with confidence intervals is that what they mean is if you repeat the experiment many many times, 95% of the time, the real number will fall in the confidence interval given by the experiment. I am not a statistician so I can't comment on the regression analysis that Lancet did (which they don't really make clear... they just refer to a software package they used). I assume that the confidence interval they cite is not a Gaussian interval itself but is a transform of the Gaussian intervals for the standard errors. Again, I'm not an expert on regression analysis, so I don't want to venture into that area. But since my research is in probability theory, I had to correct Prof. Lambert's misunderstanding of the likelihood of specific numbers in a confidence interval.

UPDATE: After a number of email exchanges with Daniel Davies (a.k.a. dsquared) of Crooked Timber and Tim Lambert, I suppose I will have to agree that by looking at a discrete version of a "Gaussian," yes, indeed the mean has the highest likelihood of occuring. However, the Lancet study obtained a confidence interval from their bootstrap procedure, in which, I assume, the parameters are continuous. Again, I am not an expert on bootstrap or maximum likelihood estimators, so I will defer to Davies on this one. As for Tim Lambert, a little less snarkiness is in order.

UPDATE II: Tim Lambert points out in the comments below that my correction was ungracious. I apologize to Tim for being ungracious. As a practical matter, his analysis was more apropos than mine. And I thank him for particularly poignant examples in his email and in the comments on his blog that made me change my views.

Leiter's Hyperventilating

Brian Leiter continues his hysterics about "Academic Bill of Rights" initiatives promoted by David Horowitz in various state legislatures. Now, I do not know the specifics of the Florida Bill Leiter lambasts. It may very well be a terrible bill. Leiter claims that it would allow students to sue their professors if the professors teach something with which students disagree. Nowhere, however, does he quote the text of the bill. Again, Leiter may very well be correct that the bill will severely damage the work done at universities. But he is dead wrong in his arguments. A general critique of his lunacy is warranted as well.

In this post, he writes:

Unfortunately, in a right-wing country like the United States, most of the serious attacks on academic freedom--i.e., the ones in which people are threatened with losing their jobs--are directed at those on the left:  this was true in the 1950s, and it is true today.

What planet is Leiter living on? Does he have any evidence? Could he name names? Also, how many conservatives have been denied tenure because "poor scholarship" became a convenient substitute for being conservative in hiring decisions? Does he have any information on this? Has he not heard of the Jack Goldsmith case in Harvard, where faculty attempted to overturn the tenure offer given to the law prof because he supported Administration policies in Guantanamo? Furthermore, given that academia is predominantly liberal, of course, if there are faculty who lost their jobs because of their views, most would likely be liberal. Does he know of relative percentages of liberal and conservative faculty who were threatened with loss of jobs due to their views? I knew of a number of junior faculty at Yale who were conservatives, but would only admit to that privately because they were afraid that their tenure prospects would be impeded if they made their political views public. Juan Non-Volokh of the Volokh Conspiracy has to post pseudonymously because it might hurt his professional career. I'm sorry, but these people are not just paranoid schizophrenics. There is a reason they fear reprisals.

Next, in the original post I linked to, he quotes an article by a philosophy professor at a small college in Florida. She says that students ought not have a voice on par with faculty because faculty are experts whereas students are not:

Nevertheless, college students believe that they have equal status with their professors. And that is how this movement began—with the absurd notion that students’ opinions, no matter how stupid or wrong those opinions may be, have as much validity as academic scholarship....

But is Ward Churchill an expert on Middle Eastern politics/history? He's barely an expert (and a questionable one at that) in the field of Native American Studies. How many faculty start mouthing off in entirely irrelevant contexts on issues in which they have no specific expertise? Brian Leiter is a Nietzsche scholar. What expertise does he have on International Affairs? Why is his opinion on International Affairs any more valuable than an undergraduate's? Leiter and the professor he quotes argue from authority. They have Ph.D.'s (let me remind you that Ward Churchill has a Master's degree from an eighth-rate college, University of Illinois-Springfield), and so by their twisted logic, they are right and the student is wrong on EVERY matter.

Now, in regards to the idea that peer-reviewed academic scholarship can't be wrong...

1) Have Prof. Leiter or his Florida philosopher friend ever heard of Michael Bellesilles? I seem to recall Mr. Bellesilles having snookered the entire academy to believe made-up research.

2) When many psychologists read a new paper in their field (even for peer review), they usually skip the data analysis section because they don't want to read pages and pages of math. I've seen this first-hand. An acquaintence of mine had a psychology professor who claimed that psychology used more valid methods of inquiry than biology, and thus had a more accurate insight into the functions of human beings.

3) In some fields like Ethnic Studies, Women's and Gender Studies, Peace Studies, etc. what often passes for scholarship is rather laughable.

4) Post-Modernist journal Social Text published a hoax piece by NYU Mathematical Physicist Alan Sokal in which he claimed that the revolutionary new mathematical field of Complex Analysis proves that the Enlightenment was wrong. He later revealed his hoax in the now-defunct Lingua Franca.

5) Law reviews are edited by law students, only a couple of years older than undergrads, with no peer-review whatsoever.

So let's not pretend like official academic scholarship is universally good, factual, or intelligent.

Furthermore, isn't part of the Socratic method having your students engage in a debate with you, the professor. Leiter's philosopher lady writes:

This explains why conservatives are now going after college teachers. Given the massive media control, it’s the last arena left where students are introduced to a humane and rational approach to serious moral issues, where they’ll be exposed to critical analysis, such as examining how the Iraqis, students their own age, feel about the U.S. invasion, an evaluation which has been deliberately ignored from the American corporate media reports from day one of this invasion. Not surprising, my students had never considered what it would be like to be in Iraqi civilian shoes, to be occupied by foreign invaders. It was the first time anyone asked them to think about Iraqi families from an empathic angle.
...

Conservative students have complained to each other: “How can she call herself a philosophy teacher when she doesn’t’ allow students to express their opinions?”

Students labor under the false presumption that philosophy is about the expression of “their” opinions and that all opinions are equally valid. Never mind that most students haven’t read a single philosophy book in their entire lives. Never mind that they do not hold a single college degree on the subject....

Does she then think that undergraduate seminars are worthless? Has she also checked whether her students have read philosophy books or not before disregarding their opinions? I have read philosophy books and I disagree with her. Also, what does her having read philosophy books and having degrees in philosophy have to do with views on international relations, military strategy, polling techniques of Iraqi civilians, sociology, etc.? She might be an expert on Plato, Nietzsche, Kant, etc. She might have read every work by Duns Scotus. So what? I can prove the Weyl Equidistribution Theorem. That doesn't mean that I have more expertise on Iraq than someone who did not study Harmonic Analysis in college.

Earth to Prof. Leiter. The NKVD is not about to break down your office door and take you away. So stop hyperbole that does nothing to further the discussion of academic freedom and bias at universities.

It's 1 a.m. Do you know where your professor is?

Welcome. This is a new weblog dedicated to deconstructing (and sometimes praising) the public statements made by academics. In response to the complete nonsense uttered and published by many of this country's college and university faculty as of late, we at AI felt it necessary to start a weblog to call them out on it. Scholars have a right to speak their minds in newspapers, magazines, blogs, scholarly journals, etc. In a similar vein, we have a right to deconstruct them on our blog. We don't look to critique any particular ideology: Just silliness.