Deinonychus antirrhopus

Statistical Incoherence

At the risk of looking a little obsessed I want to discuss the problem I pointed to in this post. Namely, the problem with interpresting confidence intervals (hereafter CIs). The contention by Tim Lambert, dsquared and others is that one can make probabilistic statements about CIs. For example, there is this statement by Daniel Davies (a.k.a. dsquared),

It also addresses the likelihood of the central figure itself. The wider the confidence interval, the less likely is any individual point estimate. The CI attached to 88-108k is only 16%

Less of a debating point than you apparently think it is, and an absolutely textbook example of “Kaplan’s fallacy” - you are trying to pretend that uncertainty about the point estimate is the same thing as reason to believe that the true number is lower than the point estimate. Whatever the confidence interval, the probability that the true number is more than 98k is 50%; if I believe your confidence interval figures (no idea why I would), I can say with 58% confidence that the true figure is greater than 88k.--note: the portion in italics is a quote of another commenter

This view point is incoherent from a Frequentist statistics viewpoint.¹ Prior to observing data, the view expressed by Daniel Davies is correct, but in the context of that post this is irrelevant since the data is already in hand.

For example, suppose a person is sampling from a data generating process (DGP) that is N(1,1)², but you don't know this (hence the reason why you are sampling). Now suppose you obtain a sample that provides the following:

Mean: 1.31674
St. Dev.: 0.95418
CI: (0.96045,1.67303).

Now, according the Daniel Davies, Tim Lambert, et. al. it is perfectly reasonable to say that the interval [1.31674,infinity) contains the mean of the DGP with probability 0.5. But, we already know that this is false by assumption. So if our hypothetical researcher were to make this claim, he'd be incoherent. The probability that the interval [1.31674,infinity) contains the mean of the distribution N(1,1) is zero. Similarly, suppose our hypothetical sampler obtained the following results,

Mean: 0.99138
St. Dev.: 1.212
CI: (0.53886,1.4439).

Now in this case the interval [0.99138,infinity) contains the mean for the DGP. In fact, we know the smaller interval [0.99138,1.4439] also contains the interval. We know the probability for this interval containing the mean is 1, not 0.5 as Daniel Davies, Tim Limbert, et. al. assert.

So this notion of a probability 0.5 for some interval from the midpoint of the CI to (positive) infinity is nonsense. The reason it is nonsense is because once the data is observed the parameter of interest (in this case the mean) is either in the interval or not--i.e. with probability 1 or 0.

One could argue that since our person taking the samples does not know the mean to him the probability is 0.5. This too highlights yet another misunderstanding of (Frequentist) statistics. The Frequentist approach to statistic defines probability as follows,

In words it reads as the Probability of Event A is equal to the limit of the number of occurences of of A--denoted m(N); divided by the number of trials as the number of trials goes to infinity. This view of probability is an "objective" view in that it is independent of the individual. On top of this, the Fequentist view also holds that parameters are fixed constants (e.g. the mean of our data generating process above). As such to suddenly come along and say that since the person taking the sample doesn't know the value of the mean, hence it could be anywhere violates the basic precepts of the Frequentist view of statistics. To do this abandons the objective view of probability in favor of a subjective view. Further, it treats a constant as a random variable.

So the concept of confidence and probability are not interchangable in the Frequentist approach to statistics. To act like they are is incoherent. To start assigning probabilities to various regions of any given CI to contain the parameter of interest is also incoherent. It is a mish-mash of objective and subjective methods that leaves one methodologically on very shakey ground at best.

[Caveat: It should also be noted that while Daniel Davies and Tim Lambert are incorrect about their probabilistic statements/confidence-probability conflation, one cannot conclude that the Lancet study and Davies and Lambert themselves are not necessarily wrong as well. Asserting this based on the above post would constitute an application of the fallacy-fallacy. Or in other words, Davies, Lambert, et. al. could have the right answer...they just got their the "wrong" way.]

Update: In comments Tim Lambert has claimed I am "wilfully misreperesenting" what he has written. While it is true that Tim has not written anything as egregious as Daniel Davies (dsquared) he has,

1. Written comments that strongly imply that making non-trivial probabilitistic statements about regions in a confidence interval are valid (see Tim's comments to this post of mine and see Tim's comments in this post of his concerning David Ujeio.
2. Tim has taken others to task, and done so rather snarkily (see this post of Tim's here).
3. Tim participates quite a bit in the comment section of his blog and is most likely quite aware of what Daniel Davies has written. So why did Daniel Davies get a pass while Tim makes implications that Tim Worstall is a statistical idiot? (Note by the way that Tim Worstall, upon learning of his mistake retracted thos comments and apologized.)

For these reasons I think Tim's (Lambert) claim of "misrepresentation" is meaningless. Even if Tim read Daniel Davies comment (quoted above) and thought to himself the probability was only 28%; it is still irrelevant. The only valid probability statements are 0 and 1.
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¹Note that when in this context we are talking about Frequentist statistics and not Bayesian statistics. The post at Tim Lambert's site deals with the Lancet study claiming 100,000 Iraqi's killed as a result of coalition bombings.
²That is a normal distribution with mean 1 and variance 1.

A Very Sad Day Here at Deinonychus antirrhopus

I learned that Dr. John Ostrom has passed away (thanks to Jeff at Caerdroia). Dr. Ostrom discovered Deinonychus antirrhopus in the early 1960's. Dr. Ostrom was also one of the first to suggest the link between dinosaurs and birds. He was 77.

Also, Lindsay Beyerstein at Washington Monthly notes Dr. Ostrom's passing and has a very cool picture.

The knuckle-dragging commenters at Washington Monthly on the other hand should probably go the way so some of Kos's commenters (i.e. summarily banned). I mean dancing on the grave of a conservative is sick enough, but sheesh Dr. Ostrom isn't exactly a big political figure.

Defending Mann, Bradley & Hughes

One of the defenses that has been used for Mann, Bradley and Hughes is that Ross McKitrick has made mathematical errors in the past. Brad DeLong has used this defense. But we can't trust DeLong either on anything, based on this type of argument. After all, DeLong doesn't understand the concept of Confidence Intervals (if you don't believe me, go ask Tim Lambert¹).

This type of defense is rather dumb, in my view. Can somebody make a mathematical error? Sure. Are we to believe that Tim Lambert, Brad DeLong and others have never ever made dumb mathematical errors (oh wait we already know DeLong has made a classic one)? Unless the answer is yes, we can simply discount everything that Lambert, et. al. have to say in defense of the Hockey Team.

Update: Looks like Tim Lambert also failed his statistics qualifying exam. According to Tim Lambert,

1. On the one hand we can't make probabilistic statements about confidence intervals.
2. On the other hand we can make probabilistics statements about confidence intervals.

You know, this is really great. This is the kind of thing that Lambert will focus on in his attacks on guys like John Lott. Sweet, sweet irony.
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¹Also according to Tim Lambert we can assume that Brad DeLong failed his qualifying exam in econometrics.

Markets Factor in Acts of Terror

One of the things about the 9/11 attacks that many people talked about was how it was in part planned to have an impact on Wall Street and Capitalism (i.e. market based economies). And I don't think many can argue with this. However, one thing about markets is that they do not "stand still". Markets change and adapt...evolve if you will (which I always find fascinating with regards to some conservatives who seem to like this in markets, but not in nature...but that is another topic altogether). Now Tobias Levkovich has argued that markets have adapted and now acts of terror are likely to have far less impact than they did on 9/11.

FOR Tobias Levkovich, chief US portfolio strategist at Citigroup Smith Barney in New York, the timing of last Thursday's terrorist attacks in London could hardly have been better.

That's not to suggest for even a moment that Levkovich welcomed the attacks. Only the sick of mind and the morally bankrupt cheer when senseless murder occurs and no one would put the respected Levkovich in such twisted company.

But last week Levkovich produced a report, Cataclysm, Terrorism & Stocks, which was posted on the brokerage firm's website in the early hours of Thursday morning, just a couple of hours before the London bombs exploded.

In his report, Levkovich told investors that the US market had adjusted to the post-September 11 world and that stock prices now reflected the geopolitical risks and realities of the terrorist age.

As a result, Wall Street was now well-placed to withstand the bearish pressures of any future terrorist attack on home soil.

As soon as I read that I thought of James' post suggesting that the London attack may have been a mistake. In particular this part,

While al-Qaida showed their customary ruthless skill in planning the London bombings, their choice of target may become a major strategic mistake. London was not the only victim of the spate of bombings of the trains and buses of the London transit system. The bombers clearly meant to sow not only panic but financial disruption, hitting stations in the heart of the city of London, where most of world's daily $1.5 trillion trades in currency are made. That plan failed. The London markets -- which did not close -- quickly sank 3 percent, but then recovered.--link

[snip]

Written between the lines was the message that any terrorist strike that causes a dip in the market represents an excellent buy opportunity because the markets will quickly return to normal.

Levkovich isn't the first to reach conclusions of this nature. Baron Nathan Rothschild was once famously quoted saying the best time to invest was when the streets were flowing with blood.

And last year Don Luskin, writing in Capitalist Magazine, said: "Don't be afraid to hold stocks just because you know there will be another attack - a lot of the risk is already priced in. And when there is another attack, buy the market."

Boy I bet that last one will annoy Brad DeLong (for those of you who don't get this reference, DeLong is sort of a Krugman defender against Luskin's many attacks on Krugman often disparaging Luskin as an idiot in general). Anyhow, keep this in mind the next time you see a terrorist attack, it might very well be a good time to buy stocks as the weak-kneed and foolish sell off valuable stocks driving their price temporarily down.

One thing the article does not touch upon that I'm curious about is the magnitude of terrorist acts and things like WMDs. Would these conclusions hold if the terrorists got their hands on some really good chemical or biological weapons? And what if terrorists manage to find their "Holy Grail" a nuclear weapon that is small and portable? I can't help but wonder if the theory breaks down at that point...or maybe not.

Markos Zuniga's Post Hoc Ergo Propter Hoc

And inconsistent data, and his ability to look at it all and decide that Bush is the cause for increased terrorism! Frankly, in looking at that post I'm at a bit of a loss as to where to begin. I think the easist place to start is with the post hoc ergo propter hoc. First up is this graph with Markos for some reasons tells us that Bush is responsible for the increase in terrorism.

Based on this graph Kos writes,

Notice when attacks where going down, and when they started going up. Since it's hard to read, the black bar, at the low point of international terrorist attacks, is the year 2000. Bush took power in2001, and the rest is history.

In other words, because George W. Bush was elected President of the U.S. (and apparently no other reason) the number of interanational terrorism incidents world wide increased. A boring old logic oriented person would call this post hoc ergo propter hoc. Bush was elected, then terrorist incidents went up...so naturally the reason was the election of George W. Bush. Oh woe is us, if we'd only elected Albert Gore, Jr.

Also, note that Markos' opening thesis is as follows,

The wingers, stung from the spectacular failure of their "flypaper" theory, are now desperately trying to rebut the obvious fact that Iraq has fueled terrorist attacks.

"What about 9-11?" they shriek. "That happened before Iraq!"

"What about the WTC bombing", they add. "That also happened before Iraq."

Yeah, sure. Of course those happened before Iraq. No one is claiming that terrorism was created by Iraq. We're arguing that letting Al Qaida off the hook in Afghanistan and pursuing an unecessary war in Iraq has fueled terrorism.

And by all objective measures, that's been the case.

However, the data in this graph does not sit well with the opening thesis. According the graph that Markos thinks is so damning there was a slight increase from 2003 to 2004, but not triple! But hey, lets not let something like inconsistent data get in the way of arriving at a preferred conclusion. One could try to dodge with noting that the graph lists incidents and the MSN article talks about major incidents. One itty bitty problem though: the graph lists total incidents at about 300...the MSN article says 650 major incidents. Not like I expect Markos to, you know, actually try to figure out why there is a difference. If one goes to the site that generates the graph, and whips up his or her own graph they will note that there is footnote talking about how after 1998 the data is for both domestic and international terrorism. Still no love there for Markos either since, the graph is created by omitting domestic terrorism and the cut off date is 1998, not 2001.

Further this raises a question about Markos' "objective measures" that he notes in first half of his post. Seems to me based on what we've seen of the information Markos has provided measuring terrorists incidents isn't quite so objective.

Also, if we go further back one would see that right when Markos' patron saint Clinton plopped his hind end into the big chair in the Oval Office there was a mild spike in terrorism. Hmmm...maybe if we had elected George H. W. Bush to a second term we could have avoided that spike. Using Markos Zuniga logic of course.

If we go even further back and look at the data from 1975 on (the last 30 years) we'd see that the last several years of terrorism incidents on average are about average. Given that we are fighting a war on terrorism that is not all that bad (and it isn't all that great either). If we go back even further, we can see that terrorism ramped up, then down and is now ramping back up. Are we to believe that this is all due which Presidents are elected...if so, then we need to go take back the Nobel Peace Prize from Jimmy Carter.

And last but not least lets remove the Middle East from the data. Here is the new picture.

In short, this data source that Markos seems to put such store in actually indicates that perhaps there is a flypaper effect. If we define things even more selfishly and restrict things to Europe and North America the results are even more stark with the high point being 1995.

Now none of this answers the question of why the count has tripled from 2003 to 2004 by the U.S. count. But, the idea that Markos is trying to float that the flypaper strategy is not working doesn't appear to be supported by the data he has pointed too. Not that I'm shocked by this.

Update: I am also unimpressed with Markos' claim that we have let Al Qaeda off the hook in Afghanistan. There was helicopter that was shot down in that region prior to the 4th of July weekend. That helicopter had a SEAL Team and other special forces operators on board. Their job was most likely to hunt for Taliban and Al Qaeda fighters and kill/capture them.

Clinton Comparisons

John Rogers over at Kung Fu Monkey has an interesting comment (if the link is bloggered scroll down to the entry titled 'elllllloooo Clinton!). John takes exception to the frequent response to a statement about the Bush Administration that starts with "But Clinton....".

Oh, and the one thing that really, REALLY drives me nut-shudderingly mad is when I point out something about the current administration's policies, and I hear: "But Clinton --"

Now I've done this myself and the purpose of this comparison is not to justify anything the Bush Administration has done, but to try and hold up a mirror in front of people like John and say, "Look at yourself." Many of Bush's policies are portrayed as the acts of the Anti-Christ by many on the Left. The problem is that Bush and Clinton aren't that far apart on many policies and to pretend that they are makes those who think they are so far apart appear to be either unhinged or deliberately dishonest.

For example, in this post I reference Clinton in regards to cutting the social safety net. The big reason for such a reference is not to justify what President Bush has done or proposing doing, but to highlight that the complaint of cutting the social safety net is by itself facile and stupid. In fact, the typical response to that by the Liberals/Lefties has been marked by either insanity or deliberately dishonesty. Look at the update in that post. The poster (pseudononymous poster Richard Cranium) claims that people can't save anymore? Is this at all in line with the facts? I don't know for sure but I seriously doubt it. Today we all sorts of retirement saving vehicls such as IRAs, 401ks, etc.

Fine, you want to ban references to Clinton, then how about rejoining the real world for a few minutes.

Does Poverty Cause Terrorism?

Shockingly, no. The big driver according the Alberto Abadie is the amount of political freedom in a country (working paper abstract). Further, Abadie's results appear to be robust between both international and domestic terrorism¹.

Unlike other recent studies on the causes of terrorism, Abadie's work explores not only transnational instances of terrorism but also domestic ones. This difference is telling: In 2003, the MIPT Terrorism Knowledge Base reported only 240 cases of transnational terrorism compared to 1,536 instances of domestic terrorism. Furthermore, Abadie suggests that the determinants of transnational and domestic terrorism may differ. "Much of modern-day transnational terrorism seems to generate from grievances against rich countries," he writes. "In addition, in some cases terrorist groups may decide to attack property or nationals of rich countries in order to gain international publicity. As a result, transnational terrorism may predominantly affect rich countries. The same is not necessarily true for domestic terrorism."

This is a pretty substantial criticism against the view that it is poverty that plays a role in terrorism. Further, the risk of terrorism is not a simply function of political freedom in that both countries with a great deal of political freedom and highly authoritarian countries are least likely to have problems with terrorism.
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¹It is hard to say that it is indeed robust since I'm going off just the NBER Digest article and the working paper.

Tax Cuts and Single Mothers

In the July 2005 NBER Digest Online is an interesting article on the effects of tax cuts for single mothers. While the tax cuts reduce government revenues, they also increase the incentive to leave welfare and seek paid employment. The researchers, Nada Eissa, Henrik Kleven, and Claus Kreiner, look at the tax acts of 1986, 1990, 1993, and 2001. The conclusion is that each tax act lowered the tax burden on single mothers and improved their welfare substantially. One of the interesting findings that is new to me is that the research found that while women did leave welfare and look for work in all four cases, the number of hours worked did not change. The digest article suggests that this effect might be due to the fixed costs associated with paid employment (commuting costs, child care costs, and preparing and recovering from work).

Another interesting finding was that the 1986 tax act had the largest impact on single mothers.

Looking at the 1986 tax reform, the tax burden on poor working mothers fell 7.94 percent. But for every tax dollar lost or every extra dollar in EITC payments, the welfare real income gain (dollars added to the pockets of these mothers and thus to the nation's income) amounted to $9.38. In the later tax episodes, the welfare gain per dollar spent was somewhat smaller, mainly because the 1986 reform had already reduced the inefficiencies substantially. Nonetheless, the tax cut of 2001 under the Bush Administration did provide a substantial welfare gain of $1.69 for every tax dollar spent. For all four reforms considered, most of the welfare gains were created by more women going to work, rather than by changes in the number of hours worked by those already working.

Kind of amusing when one recalls all the rhetoric about how bad these tax cuts were for people at the bottom of the income distribution. Additionally the research has this to say about the current tax burden of single mothers.

By 2004, a worker filing a head-of-household tax return faced a federal income tax schedule with six tax brackets, and rates ranging from 10 to 35 percent. Earnings would be shielded from taxes by the standard deduction ($7,150) and by the personal exemption ($3,100 per person). If the taxpayer had two children, she would pay 10 percent in federal income taxes only on earnings above $16,450. She would face either no state income tax (in Florida or Texas) or as much as a 5 percent state income tax (in Massachusetts or Oregon).

Additionally, the authors note, this taxpayer would have paid payroll taxes for Social Security and Medicare of 7.65 percent on her first dollar of earnings. And, she would be eligible for the EITC. To be eligible, her Adjusted Gross Income (AGI) must fall below some limit ($33,692 if she has more than one child). The size of the credit depends on the amount of earned income and the number of qualifying children. The credit is refundable if the head of household has no federal tax liability.

Twenty years earlier, the authors add, this taxpayer would have faced a very different tax scheme, with a much smaller EITC and 15 income tax brackets - ranging from zero to 50 percent. For female household heads, the changes in the EITC since then have played a central role in improving their income position.

Those evil Republicans...what are they up to passing tax laws that are beneficial to single mothers? Clearly their goal is to undermine the institution of marriage. Yeah, that is how to spin this.

Cross posted at Outside The Beltway.