William M. Briggs

Watch out Sam Harris, Gordon Hodson and Michael A. Busseri of Brock University are giving you competition for the worst use of statistics in an original paper.

Their “Bright Minds and Dark Attitudes: Lower Cognitive Ability Predicts Greater Prejudice Through Right-Wing Ideology and Low Intergroup Contact” published in Psychological Science¹—headlined in the press as Low IQ & Conservative Beliefs Linked to Prejudice—is a textbook example of confused data, unrecognized bias, and ignorance of statistics.

Hodson and Busseri on are track to beat out Harris’s magnificent effort, and they might also triumph over the paper which “proved” brief exposure to the American flag turns one into a Republican and the peer-reviewed work “proving” exposure to 4th of July parade turns one into a Republican.

Let’s see how they did it.

The authors intimate that “individuals with lower cognitive abilities may gravitate toward more socially conservative right-wing ideologies that maintain the status quo and provide psychological stability and a sense of order”. They say that this “is consistent with findings that less intelligent children come to endorse more socially conservative ideologies as adults”.

How did they prove that idiots and conservatives are racists? They gathered two large data sets from the UK, one started in 1958 (NCDS), the other in 1970 (BCS); about 16,000 individuals in total, roughly equal numbers of males and females. The quizzed the groups when they reached 11 and 10 years old on their “intelligence”; they then came back to these individuals when they were 33 and 30 and asked them about their “socially conservative ideology and racism.”

The authors do not say how many people they used in their analysis; how many individuals were lost in the 20 years between surveys is not noted in their paper. My read of the NCDS website (pdf) makes the loss about 30%. That leaves about 11,000.

Intelligence was defined in one database as scoring well on matching the similarity between 40 pairs of words, and on matching the similarity of between 40 pairs of shapes and symbols. On the other database, this changed to drawing 28 missing shapes, recalling digits from 34 number series, identifying the definitions of 37 words, and “generating words that are semantically consistent with presented words” 42 times.

Thus the two samples measure similar but different abilities. The NCDS (pdf) also had available the Peabody Individual Achievement Test Math and Reading sub-scales which were not used as intelligence measures. Why?

When the kids became 33 and 30 year olds, they were asked whether they agreed with 13 or 16 questions like, “Schools should teach children to obey authority”, “Family life suffers if mum is working full-time.”

Another was, “People who break the law should be rehabilitated.” Just kidding! It’s actually, “People who break the law should be given stiffer sentences.” The bias in the question wording is ignored.

Another question was, “None of the political parties would do anything to benefit me.” Is agreeing or disagreeing with that a “conservative” position? What would the Occupy people say? Another, “Being single provides more time to experience life and find out about yourself.” Conservative or liberal?

According to the NCDS (pdf), there were about 50 questions, of which only 13 were used. A “conservative”, then, is whatever Hodson and Busseri say it is. The same thing goes for what a “racist” is.

For these questions “reliabilities ranged from .63 to .68.” This means the questions are imprecise and imperfect, so that if you use the raw results in subsequent analysis, you must “carry forward” the uncertainty in reliability. Did Hodson and Busseri do this? No.

One would have guessed from the title, that the authors looked at how the scores on the intelligence questions correlated with the scores on the attitude and racism questions, taking into account the uncertainty in the reliability. You would be wrong.

They first modeled the intelligence questions to create one “latent” (unobserved) measure, called “g”. The uncertainty in creating “g” is then ignored in all subsequent analysis. They did the same for the attitude questions, creating a “latent” (actually unobserved) variable called “conservative ideology.” Uncertainty in its creation is also ignored. Then the individuals’ education and socioeconomic status and separately their parent’s socioeconomic status (which again were the results of models) were put into a model with “g” and “conservative ideology” to predict “racism” (the uncertainty of which, as was already said, was ignored). The picture below summarizes their findings.

Lo, they found small p-values. The authors appear unaware that samples of this size are practically guaranteed to spit out small p-values.

What makes the study ludicrous, even ignoring the biases, manipulations, and qualifications just outlined, by the authors’ own admission the direct effect size for “g” on “racism” is only -0.01 for men and 0.02 for women. Utterly trivial; close enough to no effect to be no effect, and statistically “significant” only because of the massive sample size.

The effect size for “conservative ideology” directly predicting “racism” is higher (0.69 and 0.51). But all that means is that the questions the authors picked for these two attitudes are roughly correlated with one another. In other words, “None of the political parties would do anything to benefit me” is crudely correlated with “I
wouldn’t mind working with people from other races” and so forth.

Yet the authors have the temerity to conclude, “These results from large, nationally representative data sets
provide converging evidence that lower g in childhood predicts greater prejudice in adulthood and, furthermore, that socially conservative ideology mediates much of this effect.”

Truly, statistics can “prove” anything.

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¹doi:10.1177/0956797611421206

Thanks to reader Jonathan Woolley who suggested this study.

Update This post of such importance, that it remains on top today. See below for more comments.

Presented for your satisfaction, a way to cheat either yourself or others using time series. The patter below is only a suggestion.

Presentation

Just look at these anomalies, which are related to rampant, deadly climate change. Higher anomalies are worse for all of mankind in every imaginable way.

The anomalies are presented as monthly measures, over roughly a 10-year time period. A regression was fit to them and is plotted. The per-decade increase in this not-good anomaly is 0.87 per decade. Why, after 20 years, the anomaly will be almost twice as large as it is now!

The 95% confidence interval for the decadal change is 0.44 to 1.3. That means that the anomalies are surely heading up!

But ignore these sorrowful facts, because there is good news to be had. Here are some more anomalies.

These anomalies are on the way down, thus our spirits should be on the rise. In fact, the anomalies will drop at 0.57 over the next 10 years. And after 20 years, they’ll be down more than one full point!

The 95% confidence interval for the decadal change is -1.1 to 0. That means that the anomalies are surely heading down!

How the trick works

The pictures are the same!

Even if you flash up both pictures, the audience members will never notice that they are seeing the same anomalies. Yes, it’s true. You’ll worry that somebody will catch on, but they won’t! I have seen this done many times and nobody ever notices that the pictures are identical—except, of course, for those colorful straight lines. And the starting date.

Now take a look at these anomalies, which are the same as above, and see if you can spot the difference.

Instead of one regression line, there are 24. The first one is drawn using the entire time series. The second one is drawn using the entire time series except for time point number 1. The third removes time points 1 and 2, and so on. There are 24 lines in total, showing anything from a large increase to a large decrease, and each drawn by choosing a new starting point.

Do you get it? This is the whole trick! Nobody ever asks why you chose a particular starting point. You can tell any story you like and people will never think to ask what would happen if you were to use a slightly different data set.

Of course, very clever magicians will manipulate both starting and end points, but it’s best not to meddle with the end points until you become a master. People will (or should) naturally ask why you haven’t included the most up-to-date data, but they will absolutely never ask why you only used some of the history and not all of it.

Statistics

The time series above was generated by the R armia.sim() function, using a mean 0, standard deviation 1, AR(0.64,0,0,0,0,0,0,0,0,0,0,0.35) process, which mimics many different real-world monthly time series. But try your own model. It works for models of any kind. And it’s fun!

Next thing is to show how reliable this trick is. The true answer—given our evidence E that the model is mean 0, etc.—is that the anomalies neither increase nor decrease over a decade. The slope of any regression line, in other words, should be 0. Or the confidence intervals of any line drawn should include 0. Of course the actual results will vary.

It’s your confidence intervals which are the real convincers in the trick. Did you notice that both confidence intervals (for the first two figures) confirm the hypothesis that things are getting better and things are getting worse? Isn’t that great!

To show the reliability of this, suppose your funding depends on things getting worse: you need the anomalies to increase. Therefore, you’ll pick a starting date which gives you the best evidence. Not every time series that is truly unchanging (as our E says it is) will cooperate such that you can definitely show an increase. But you can limit the damage against yourself by showing the smallest possible decrease.

I simulated 1000 different time series, each time picking the best starting point (to show the largest possible increase). Remember: if no cheating occurred, the mean of these samples should be 0. It isn’t. It’s much higher at 0.21—with a 95% confidence interval of -0.88 to 1.31.

Notice how much wider this (better) interval is. It’s better because it takes into account cheating.

What if you don’t want to cheat? Well, your interval will still be wider than if you just ran the regression on the data at hand. Except if the data at hand is all the data that will every occur (and if it is, there is no real to run a time series), the arbitrariness of the starting (and ending_ point must be accounted for. If it isn’t, then you will go away too confident of yourself.

The lesson is, of course, that straight lines should not be fit to time series.

Update More comments.

Question: why fit a straight (or any shaped) line to a time series like this? There are three reasons: (1) to discover whether there was a trend, (2) to predict the future, and (3) to use the analysis as part of a larger analysis.

(2) is a respectable goal, and should be encouraged. Most who fit lines to time series have this goal in mind, at least tacitly; that is, they at least imply that the line they have fitted will “continue” into the future. Therein lies a problem. For that line is an all-too-sure guess of what the future will be.

Notice that we stated specifics of the line in terms of the “trend”, i.e. the unobservable parameter of the model. The confidence interval was also for this parameter. It most certainly was not a confidence interval on the actual anomalies we expect to see.

If we use the confidence interval to supply a guess of the certainty in future values, we will be about 5 to 10 times too sure of ourselves. That is, the actual, real, should-be-used confidence interval should be the interval on the anomalies themselves, not the parameter.

In statistical parlance, we say that the parameter(s) should be “integrated out.” So when you see a line fit to time series, and words about the confidence interval, the results will be too certain. This is an inescapable fact.

(1) is also a goal, but a shady one. If we want to know if there has been a change from the start to the end dates, all we have to do is look! I’m tempted to add a dozen more exclamation points to that sentence, it is that important. We do not have to model what we can see. No statistical test is needed to say whether the data has changed. We can just look.

I have to stop, lest I become exasperated. We statisticians have pointed out this fact until we have all, one by one, turned blue in the face and passed out, the next statistician in line taking the place of his fallen comrade.

It is true that you can look at the data and ponder a “null hypothesis” of “no change” and then fit a model to kill off this straw man. But why? If the model you fit is any good, it will be able to skillfully predict new data (see point (1)). And if it’s a bad model, why clutter up the picture with spurious, misleading lines?

Why should you trust any statistical model (by “any” I mean “any”) unless it can skillfully predict new data?

Again, if you want to claim that the data has gone up, down, did a swirl, or any other damn thing, just look at it!

(3) If you fit a line and then use the parameter estimates of that line as input into other analysis (as was done in our sample paper, referenced below), your results will be too certain. We all know the dangers of smoothing time series. If you’ve forgotten, I, II, III.

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This post was inspired by an actual paper—where I do not accuse the authors of cheating; but they do use time series with different starting and ending dates and then combine those time series to make a conclusion. We can see now that they will be too sure of themselves.

The paper is “Observed changes in top-of-the-atmosphere radiation and upper-ocean heating consistent within uncertainty” by Norman Loeb and others in the journal Nature Geoscience. I’m pressed for time, so for background on this paper, surf to Roger Pielke Senior’s place.

From the abstract:

We combine satellite data with ocean measurements to depths of 1,800 m, and show that between January 2001 and December 2010, Earth has been steadily accumulating energy at a rate of 0.50 +/- 0.43 Wm^-2 (uncertainties at the 90% confidence level). We conclude that energy storage is continuing to increase in the sub-surface ocean.

Most curiously, the authors choose the “90% confidence interval” instead of the usual 95%. Why? Skip the discussion of the meaninglessness of confidence intervals and interpret this interval in its Bayesian sense. Then this means that the coefficient of the regression associated with time is estimated at 0.5 W^-2 with a 90% chance of being anywhere in the interval 0.07 to 0.93 Wm^-2.

This is an unobservable coefficient in a model, mind. It is not an amount of “energy.” To get to the actual energy, we’d have to integrate out the uncertainty we have in the coefficients.

Anyway, the change from the usual certainty level also means—I’m estimating here—that the coefficient of the regression associated with time is estimated at 0.5 W^-2 with a 95% chance of being anywhere in the interval -(some-number) to one-point-something Wm^-2. In other words, the intervals have to be widened, and probably such that the lower portion of the interval is negative: it is almost certainly near 0. Like I say, I’m guessing, but with enough gusto to be willing to bet on this. Any takers?

We have to know about the regression. Details? The authors put details in tiny print and in a supplement. Here’s the small print (bolding mine):

Global annual mean net TOA fluxes for each calendar year from 2001 through 2010 are computed from CERES monthly regional mean values. In CERES_EBAF – TOA_Ed2.6r, the global annual mean values are adjusted such that the July 2005–June 2010 mean net TOA flux is 0.58 +/- 0.38 Wm^-2 (uncertainties at the 90% confidence level). The uptake of heat by the Earth for this period is estimated from the sum of: (1) 0.47 +/- 0.38 Wm^-2 from the slope of weighted linear least square fit to OHCA to a depth of 1,800 m analysed following ref. 26; (2) 0.07 +/- 0.05 Wm^-2 from ocean heat storage at depths below 2,000 m using data from 1981 to 2010 (ref. 22), and (3) 0.04 +/- 0.02 Wm^-2 from ice warming and melt, and atmospheric and lithospheric warming1,27 . After applying this adjustment, Earth’s energy imbalance for the period from January 2001 to December 2010 is 0.50 +/- 0.43 Wm^-2 . The +/-0.43 Wm^-2 uncertainty is determined by adding in quadrature each of the uncertainties listed above and a +/-0.2 Wm^-2 contribution corresponding to the standard error (at the 90% confidence level) in the mean CERES net TOA flux for January 2001–December 2010. The one standard deviation uncertainty in CERES net TOA flux for individual years (Fig. 3) is 0.31 Wm^-2 , determined by adding in quadrature the mean net TOA flux uncertainty and a random component from the root-mean-square difference between CERES Terra and CERES Aqua global annual mean net TOA flux values.

The same 90% intervals are used, notice. The weights mentioned are hidden in another paper (ref. 26; I didn’t track this down). There is no word on whether the authors (or the others they cite) recognized the correlation in time and thus realize that the estimates of the coefficients, especially their confidence limits, will be suboptimal (too certain). In other words, a straight line regression is not the best model—but it is a model (no probability leakage, anyway! under the evidence that these indexes have no natural boundary). The final uncertainty is estimated by “determined by adding in quadrature” some other numbers.

What a complex procedure! The supplementary paper is little help in reproducing the exact steps taken. That is, it is doubtful that anybody could read this paper and use it as a recipe to reproduce the results (joyfully, the authors do make the data available).

But from a scan of the procedure, and given my comments thus far, it would appear the interval is too narrow. Adding all those different sources together and properly taking into account the uncertainty in each individual procedure is enough to boost the overall uncertainty by an appreciable amount. How much is “appreciable” is unknown. The amount one would have to add to the overall uncertainty is greater than 0. This implies that the final estimate of the coefficient of the regression associated with time should be about 0.5 W^-2 with a 95% chance of being anywhere in the interval minus-something to just-over-one Wm^-2. Consistent with uncertainty indeed.

There is still another source of uncertainty not noticed by Loeb, or indeed by nearly all authors who use time-series regression: the arbitrariness of the starting and ending points. I am sure Loeb did not purposely do this, but it is possible to shift the start or stop point in a time-series regression to get any result you want. For example, in their main paper Loeb et al. show plots from 2001 until 2010. But in the supplement, the data is from mid-2002 through all of 2010. Changing dates like this can booger you up. I’ll prove this in another post.

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Thanks to reader Dan Hughes for helping me find the papers.

Update Be sure to see this post on how to cheat with time series.

Today, a technical interregnum, a necessary pause for proof of that claim that each of us must come equipped with knowledge that cannot be learned. Stuff that is only known to be true only through introspection, via what we call intuition or, sometimes, faith; philosophers usually settle on the technical term a priori (or on phrases more technical still).

Here is one (of many) proofs given by David Stove in his The Rationality of Induction¹ He made this argument in the support of a priori knowledge in his larger work showing induction is reasonable². All you need know about Bolzano (named below), is that he disputed the idea that we all of us come with built-in knowledge. The formula numbers are as they appeared in the book.

Reading this passage, as with reading any proof, requires some sophistication. This cannot be avoided. But if you are comfortable with the idea of built-in knowledge, then you can skip this and start after the quote. Careful readers will recognize that Stove’s simple argument is also a proof that empiricism—the belief that all knowledge comes from observation—is false.

First, as to our knowledge of validity. Bolzano says that the validity of barbara, or rather, that the barbara schema always preserves truth, is a hypothesis reasonably believed by us, just because of the extensive experience we have had of never finding a counter-example to it. That is, our grounds for believing (149), or rather, for believing

     (166) For all x, all F, all G, either ‘x is F and all F are G is false’, or ‘x is G‘ is true,

consist just of observations we have made, such as

     (151) Abe is black and Abe is a person now in this room and all persons now in this room are black.

That is putting it starkly; still it is, in essence, what Bolzano believes. We learn deductive logic by inductive inference.

But now, this is tacitly to concede, to certain propositions of non-deductive logic, precisely the intuitive status which Bolzano expressly denies to any proposition of deductive logic. Our putative logic learner is supposed to be devoid of all intuitive logical knowledge. Yet Bolzano is evidently crediting him with knowing, straight off, at least this much: that

     (167): (151) confirms (149).

Of course, he need not be supposed to know that he knows (167); still, he is evidently being supposed to know it. But to know (167) is to have some logical knowledge, even is only non-deductive logical knowledge.

And Bolzano must suppose that (167) is known by our logic learner intuitively. Otherwise he would have to have learnt it, as he is supposed to be learning (166), by experience. And how would he accomplish this?

It must at any rate be from some observation-statements. I do not know what kind of observation-statements Bolzano would regard as confirming (167): let us just call these observation-statements

     (167) O₁.

But even if our logic learner has found by experience that O₁ he will be no further advanced. To learn (167), he needs to know, not only that O₁, but that

     (169): (168) confirms (167).

But this is a proposition of logic too. If he does not know (169) intuitively, as by hypothesis he does not, then he will have to learn it, too, from experience. No doubt from some observations

     (170) O₂.

But that is not enough. He will also need to know that

     (171): (170) confirms (169);

and so on.

Obviously, he is never going to make it. Experience is not enough.

Especially careful readers—especially those convinced by this proof, as I am—will recognize that in order to interpret this proof, to assimilate it and follow it, requires precisely the kind of built-in knowledge of which the proof speaks. We must have a priori knowledge.

We are finally ready to tackle the notion that some propositions are “just true.” Propositions of this sort are usually the kinds of truths spoken of above, but there are also moral or ethical truths, too (of these, another day). The claim that there exist “universal truths”—propositions which say are just plain true—is consistent with claim that all truth is conditional, because whatever a priori knowledge we have is conditional on our intuition, or is taken “on faith.” To speak of these truths (in a technical sense) we must first affix the condition, “Given my faith or intuition, this proposition is just true.”

This seems to open the way to relativism because, as direct experience tells us, different people will claim a certain proposition true or false just because their intuitions or faith direct them oppositely. Many times, of course, these differences are mistakes in reasoning, or there are other pieces of evidence that are assumed (as in French speaking chickens) that the speaker is not aware of or does not acknowledge. Skip these cases and focus on just those claims where nothing is assumed except intuition or faith.

The claim is that we must have some shared beliefs about what is true, and it is these beliefs which we speak of when we say “there are truths.” One is that “Other minds exist.” Or, more properly, “Given our intuitions or faith, other minds exist.” The only possible escape from believing this shared truth is solipsism, which is “Given my intuition, only I exist.” Deep, or metaphysical solipsists stop here. But other solipsists (there are varieties of the breed) might allow that “Given my intuition, it is true I exist; but I believe it is logically possible that others might exist.”

But to acknowledge “logical possibility” is to acknowledge at least the truth that logical propositions are true. So that if those other minds exist, they must have this knowledge, too because that is what a mind is. And then if one really is a metaphysical solipsist (good luck finding one) it is still true that “all” share beliefs about what is true—it just the case that “all” is one person. (How many solipsists will jump in and tell me, “Briggs, you don’t exist! Stop claiming you do!”)

In practice, of course, we all really do admit that truth (given our intuitions) that others minds exist. This, then, if just one of many truths that exist. The logical knowledge spoken of by Stove are others. Our task is now clear: since truths, via shared faith or intuition exist, we must identify what they are; we must also identify falsehoods, and that which is only probable. More on this later.

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¹p. 162-163. This book, especially the second half, is a treasure that all statisticians, probabilists, and logicians should read.

²Yes, some people think it isn’t. Bolazno was not one of these: he thought all (as in all) knowledge was known empirically.

We often of a proposition use the term self evident or a similar variant. But we often use it incorrectly. We say for instance, “It is obvious that Mr Obama is a bad president,” when what we really mean is, “Given a certain collection of evidence which I am holding and which I assume you also hold, we infer that Mr Obama is a bad president.”

Error creeps in when the inference doesn’t lead to a certain conclusion, and it is instead only probable, or when the two parties do not agree on the same set of probative evidence. Hence arises politics, with which I assume the reader is familiar with many examples.

Sometimes we use the phrase correctly, as in “We hold these truths to be self evident, that all men are created equal (where men means all human beings)…” Or in the proposition, “For all natural numbers x and y, if x = y, then y = x.” Or in, “Obviously I exist.” Or in a host of other propositions which we say as true.

But notice the difference in these two phrases: “It is true I exist” and “It is self evident that I exist.” It is strictly a mistake to use the first phrase, but fine to use the second. The difference is subtle here and the mistake passes unnoticed, but only because of the example: you will agree that it is true that you exist, so the two phrases might seem equal, but they are not.

The first fails because it is like Alice coming up to you and announcing, “It is true that some chickens are creatures understanding French.” It most certainly is not true—unless we first accept the conditions Alice heard but we did not, but which would have allowed us to deduce this proposition as true.

The reason “It is self evident that I exist” works is because it carries with it the evidence we need to judge the truth of the proposition “I exist.” The evidence, which all arguments need, comes from saying “It is self evident”, which is just a translation of “Given my intuition” or “Given my most fundamental thoughts.”

Thus the second phrase is equivalent to “Given my intuition, it is true that I exist.” This allows us to recast our mathematical axiom “Given my intuition, it is true that for all natural numbers x and y, if x = y, then y = x.”

Now, we often as a harmless shorthand skip the qualifier and just say that “It is true that for all natural numbers x and y, if x = y, then y = x.” We get away without the qualifier in cases like this because the truth is so obvious and non-controversial. And it would be tedious to bring along qualifiers for everything item which we assert is true. For example, “It is true the car is in the garage.” Well, to be perfectly clear, you must say at least, “Given my observation and assuming my senses have not failed me and nothing has left the garage since last I’ve seen it, it is true the car is in the garage.” What a chore! Much easier to state, “The car is in the garage.”

No harm is done in the vast majority of these cases because the evidence which is required to make these propositions true is in fact shared by speaker and listener. But, as in the cases of politics, ethics, and metaphysics, it can be a positive menace to fail to specify the conditioning evidence.

Hold on a minute, did we make a slip? Yes, and a big one. Can you spot it?

Re-examine the “full” garage example. To make the proposition “The car is in the garage” true we had to carry a lot of baggage. Perhaps our slip arises because we did not fully specify the exact conditions which make the proposition true? If so, we can fix it up; after all, we use statements like these all the time without running into difficulties. This isn’t our mistake, so let’s just assume we do have the right evidence.

Our slip is something deeper, more fundamental. Ready? We had to know that, “given the evidence, the proposition is deduced.” We had to have built-in knowledge that lets us take statements of evidence and tie them to propositions. In other words, the operation of going from the evidence to the conclusion is a logical step, the validity of which we must take for granted. We don’t just need the evidence and conclusion, we need the logical glue that binds them.

This “glue” is also in our intuition. Alice needed it, and so do we, for every inference we make.

Next time: a proof of this last claim. It will also be the case that relativism is false.

Statistics show—and as we know by now, these cannot be wrong—that Saturday is the least likely day that people like to ponder statistics. Traffic on this blog is at its lowest ebb on this day. (Wednesday is the most likely.)

Still, a millennium of poor souls stop by to say hi. For this most loyal group, I offer what we used to call the “Polish Cheer” (our town had a high proportion of people, like your author in fractional part, of Polish descent):

Ooh-sa-sa-sa
Ooh-sa-sa-sa
Hit ‘em in the head
With a big kielbasa
M
I
L
K
Milk ‘em!
Mooooooooo.

This was sung by members of the high school pep band when the instruments went silent. I was in this band. (“What? This math-computer-logic geek was in the band? Who would have guessed?”)

You might laugh at this seemingly fangless cheer, but I put it to you: how would you like to be milked? Our cry surely struck fear into the hearts of our rivals.

I’d give the stats on wins, and thus the effectiveness of our fight song, but I was so busy blowing my bass sax and giggling at witty ditties like these, that I can’t remember whether our team won any games.

What I can recall with clarity is that we had to wear uniforms, just as the sporting fellows did. This separated us and announced our purpose. Uniforms were merely an extension of the school’s policy on dress. No jeans, shirts had to have collars, etc. Those in violation were sent home, parents were informed. As a consequence, the hallways, classrooms, and bleachers were pleasant, cheerful places to be.

The Wall Street Journal reports that the latest (de)trend in fashion is for teens to wear pajamas everywhere.

Greedy businesses, like Abercrombie & Fitch, giving no thought to their responsibilities, are encouraging this behavior by manufacturing sloppy-on-purpose clothes (for sale at high price). “A wide neck is key, says Jennifer Foyle, chief merchandising officer, because ‘girls are wanting to show their bra straps.’” Charming, charming. How nice to see what fathers have become. Yes, fathers.

The paper continues:

As with a lot of teen behavior, some adults are annoyed. In Louisiana’s Caddo Parish, which encompasses Shreveport, Commissioner Michael Williams is getting national attention for taking a stand. He plans to propose an ordinance outlawing the wearing of pajamas in public.

“The moral fiber in America is dwindling away,” Mr. Williams says. “It’s pajamas today; what is it going to be tomorrow? Walking around in your underwear?”

“Stay off my lawn! What an old man. Parents have been saying this kind of thing about their kids forever. Even the Sumerians complained about teens slacking off.” (I have a quote from an ancient Sumerian text that I can’t lay my hands on at the moment, which proves that, yes, this ancient society did in fact complain about their offspring. Once I find it, I’ll revisit this topic.)

To which I answer: where are these Sumerians? Where are the Romans? O tempora o mores! Where are all the other civilizations who complained about encroaching decadence? They are no more, that’s where. They each and every one of them failed to heed their prophets. They succumbed.

It is no counter-argument to say that because your parents complained, just as their parents before them complained, that the state of dress¹ is therefore not declining. It could be, and obviously is, growing worse with every generation.

There are some areas of this fine country where all that is worn is shorts, a sloppy t-shirt, and some sort of plastic, funky foot gear. Men augment this ensemble with baseball caps. Women, depending on state of their journey toward senescence, either cinch the t-shirt tight or let it billow like a tent. The only reason this style has not triumphed completely is that winter forces extra layers.

Meaning jeans. Here is the truth of jeans. After eighteen, you look ugly in them. You might assume you cut a certain je nais se quois. You do not. You look sloppy. Even the most expensive jeans—which are designed to look as close to non-jeans as jeans can be, so why bother—-do not look nearly as good on you as you think they do.

People have forgotten one purpose of dress: to please others. You are not wearing clothing solely for yourself. You are doing it for your neighbor. Be kind to your neighbor and wear something nice.

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¹Or music, architecture, art, literature, movies, public discourse, etc., etc., etc.

“All cats are creatures understanding French,” said Alice’s father. “And some chickens are cats.”

“Wait, I know!” said Alice, chirruping. “That means that some chickens are creatures understanding French.”¹

“What you said is true, my dear,” said Alice’s father, his voice full of pride.

What Alice said was true. As true as any another truth, too. True as true can be. But it would still be a mistake for Alice, even if she ventured through the looking glass, to announce triumphantly that “It is true that some chickens are creatures understanding French!” That would be to say what is false, or rather it would be to say a nonsensical thing.

Which was Alice’s father’s specialty. Nonsense of a special sort, that is. For if you haven’t guessed, Alice’s father is Charles Dodgson, a.k.a. Lewis Carroll. Dodgson published several—

—Wait! Hold on. Skip the biography. Didn’t I just say that what Alice first said was true? How can it be that her second phrase, identical to the first, is not true?

Well, her second phrase was not identical, was it? The first time Alice spoke she said, “That means…” and it is that “that” that makes all the difference. The second time she skipped this all-important phrase. One simple word separated truth from falsity. Let’s see why.

Dodgson’s example came from his Symbolic Logic (p. 57). He said that his propositions were

so related that, if the first two were true, the third would be true. (The first two are, as it happens, not strictly true in our planet. But there is nothing to hinder them from being true in some other planet, say Mars or Jupiter—in which case the third would also be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singular contingent privilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)

This distinction is crucial, so I will repeat it. What Alice said the first time was true but only because she accepted the first two statements, the things her father said. She brought those first two phrases along with her when she said, “That means…” She left them out in the second instance, where her audience could not be expected to know that all cats, etc.

The first statement was true because of the provisos she accepted. The second statement was nonsensical, because it was not anchored, it was left floating. The audience could not say why “some chickens are creatures understanding French.” without some kind of evidence.

Those in the audience were free to supply their own evidence, of course. One person might have said to himself, “I know of no chickens who can understand French, but I’ll allow the possibility.” Given that, this person would not say Alice’s statement was exactly true, but he would also not claim that it was exactly false. A second person might have said, “Chickens don’t have lips, which are needed to speak French,” and, given that, he would say Alice was speaking a falsehood.

We have learnt two things from this example that we should never forget. We can’t speak of truth or falsity without reference to evidence, and logic is not the study of propositions but the study of connections between propositions.

A careful reader will have paused over this last sentence and say to himself, “If we can’t speak of truth or falsity without reference to evidence, does that apply to the claim that ‘we can’t speak of truth or falsity without reference to evidence’?” I am, after all, claiming it is true that “we can’t speak of truth or falsity without reference to evidence.” What evidence do I offer?

Well, in order not to go too far afield and not burden us in technicalities, I will let you yourself supply that evidence as homework. Can you think of any claim of truth (or falsity) which does not refer to evidence? If you can, then you have refuted my claim. If you cannot, my claim is not necessarily proved, of course, because just because you can’t think of a counter example doesn’t mean a counter example doesn’t exist. Nevertheless, I do make the claim.

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¹I was reminded of this example by reader Scott Bury.

EPA Agent Stephen Segal protects the wetlands

The EPA came to Idaho and said in a booming voice Stop! to Mike and Chantell Sackett, who were building a home on a plot zoned for residences. The EPA shouted because it had determined that a small portion of the Sackett’s half-acre was—are you ready?—a wetland. And therefore sacred.

Not only did the Sackett’s have to cease building, if they didn’t return the property to the EPA’s vision of purity, they would be fined $37,500 per day. This was stated in the EPA’s compliance order.

And, oh yes, they would be fined an additional $37,500 per day for violating the Clean Water Act, an Act which gave EPA power to issue fines for both violating the Act and for violating compliance orders written under the authority of the Act. Got it?

For our less mathematically gifted readers, that’s seventy-five big ones. A day. How many days? Ah, here is where the story gains interest. Forever: that’s how many days.

Evidently, some slick, unelected, unaccountable, uncharitable and foolish bureaucrat thought that this double-dosing of fines would be a good solution to eliminate our deficit. But the joke’s on them, because even if the EPA gets away with cheating the Sacketts out of their money, it would still take over five-hundred-thousand years of daily fines to pay off the deficit—made large in part by paying the salaries of the windy minds who run the EPA.

The story grows in hilarity when you learn that the Sacketts have no recourse. No one to turn to. They cannot ask their mayor, they cannot appeal to Congress. The police won’t help them. They may not even ask a judge for relief, because the EPA has decided that its compliance orders are subject to judicial review only when the EPA says they are.

They can’t even ask the EPA! That is, they can and did ask, but the EPA did not deign to answer. And it is not required to. Pay the fine, sucker.

Well, this was too much for the Sacketts, who decided to sue anyway. Not just about themselves, but about the EPA’s ability to Lord it over all people. The Sacketts found themselves at the Ninth Circuit court. Which decided that, since the EPA would scarcely make a mistake because they are a branch of the government, told the Sacketts to go packing. The Ninth Circuit, if you haven’t guessed from the evidence, is based in California.

The Sacketts would not settle and pressed their case even unto the Supreme Court. Which heard oral arguments on Monday, 16 January 2012.

Said Malcolm Stewart, attorney for the EPA, regarding the greedy double-fine:

The compliance order is intended to specify the violation that EPA believes to have occurred and the measures that EPA believes are necessary in order to achieve prospective compliance. And the statute does provide separately for penalties for violating the statute and penalties for violating the compliance order.

As an exercise of our duty of candor to the Court, we acknowledged in our brief that the government reads the statute to allow the legal possibility of double penalties, that is up to $37,500 per day for violating the statute, up to 37,500 per day for violating the compliance order. I think that’s really a theoretical rather than a practical -­–

He was interrupted by Justice Breyer who had to point out that distinctions of these kinds were irrelevant.

Following so far? Because it’s about to turn strange.

It turns out (how is another question) that the Army Corps of Engineers could travel to the Sackett’s would-be homestead and, if the Corps decides that their land is a wetland, could grant the Sacketts a permit to fill in the wetness so they can build. But if the Corps says that their wee chunk of land was not a wetland, they would not issue a permit. Even if they got the permit, the EPA might not honor it and still fine the Sacketts.

The EPA can issue compliance orders whenever it likes and does not need probable cause. Further, the Sacketts—or you, dear reader—always stand in danger of the EPA swooping down even if your house is already built. There is no statute of limitations under the EPA’s theory of “continuing violation.” If the EPA says it’s a wetland, by golly, it’s a wetland, or was, and pay or please the EPA you must.

The Sackett’s lawyer is only asking for the Supreme Court to allow the EPA’s actions to be subjected to judicial review just as all other actions by the government are. Such a meager request, a pittance! Yet the EPA is fighting hard so that it may remain arbitrary and aloof.

Even if this is granted Justice Scalia made the valid point that “the factual questions that go to whether these are wetlands or not are going to be decided giving substantial deference to the agency’s determination of the facts.”

How can two small people of limited funds battle an array of bureaucrats who have the full majesty and purse of the government behind them? Answer: they cannot.

More to come. This post inspired by HotAir.

Submitted for your approval, yet another paper. On Probability Leakage, posted at Arxiv. Once you, my beloved readers, have had a go with it, I’ll incorporate your comments in an updated version and send it to a peer-reviewed journal, because there is no better guarantor of truth than peer review.

Only a sketch of the paper is given here. Please read it before commenting: it’ll probably save me re-typing what’s already there.

Abstract

The probability leakage of model M with respect to evidence E is defined. Probability leakage is a kind of model error. It occurs when M implies that events, which are impossible given E, have positive probability. Leakage does not imply model falsification. Models with probability leakage cannot be calibrated empirically. Regression models, which are ubiquitous in statistical practice, often evince probability leakage.

Definition

The exact definition is this: we model observables y via some model M, possibly conditional on explanatory data x and indexed on (unobservable) parameters. We can derive a posterior distribution on the parameters given the old data (call it z) and assuming the truth of M. Ordinary (and inordinate) interest settles on the posterior of the parameters.

The posterior, or functions of it, are not observable. We can never know whether the posterior says something useful or nonsensical about the world. But because we assume M and we have seen z, it logically follows that the posterior predictive distribution p(y|x,z,M) exists (this “integrates out” the parameters and says something about data not yet seen). This distribution makes observable statements which can be used to verify M.

Now suppose we know, via some evidence E, that y cannot take values outside some set or interval, such as (y_a,y_b). This evidence implies Pr(y < y_a | E) = Pr(y > y_b | E) = 0. But if for some value of x (or none is x is null), that Pr(y < y_a | x, z, M) > 0 or that Pr(y > y_b | x, z, M) > 0, then we have a probability leakage; at least, with respect to E.

The probabilities from the predictive posterior are still true, but with respect to M, z, and x. They are not true with respect to E if there is leakage. This probability leakage is error, but only if we accept E as true. Leakage is a number between 0 (the ideal) and 1 (the model has no overlap with the reality described by E).

Model Falsification

The term falsified is often tossed about, but in a strange and loose way. A rigorous definition is this: if a M says that a certain event cannot happen, then if that event happens the model M is falsified. That is, to be falsified M must say that an event is impossible: not unlikely, but impossible. If M implies that some event is merely unlikely, no matter how small this probability, if this event obtains M is not falsified. If M implies that the probability of some event is ε > 0 then if this event happens, M is not falsified period.

Probability leakage does not necessarily falsify M. If there is incomplete probability leakage, M says certain events have probabilities greater than 0, events which E has says are impossible (have probabilities equal to 0). If E is true, as we assume it is, then the events M said are possible cannot happen. But to have falsification of M, we need the opposite: M had to say that events which obtained were impossible.

Box gave us an aphorism which has been corrupted to (in the oral tradition), “All models are wrong.” We can see that this is false: all models are not wrong, i.e. not all are falsified.

Calibration

Calibration has three components: probability, exceedance, marginality. All three are proved impossible if there is probability leakage. If M is to be evaluated by a strictly proper scoring rule, the lack of calibration guarantees that better models than M exist.

Example

Statistics as she is practiced—not as we picture her in theoretical perfection—is rife with models exhibiting substantial probability leakage.

Regression is ubiquitous. The regression model M assumes that y is continuous and that uncertainty in y, given some explanatory variables x, is quantified by a normal distribution the parameters of which are represented, usually tacitly, by “flat” (improper) priors. This M has the advantage of mimicking standard frequentist results.

…The logical implication of M is that, for these values of x, there is about a 38% chance for values of y less than 0 (impossible values) at Location A. Even for Location B, there is still a small chance 2% for values of y less than 0 (impossible values).

Conclusion

An objection to the predictive approach is that interest is solely in the posterior; in whether, say, the hypothesis (H) that absentees had an effect on abandonment. But notice that the posterior does not say with what probability absentees had an effect: it instead says if M is true and given z, the probability that the parameter associated with absentees is this-and-such. If M is not true (it is falsified), then the posterior has no bearing on H. In any case, the posterior does not give us Pr(H|z), it gives Pr(H|M,z). We cannot answer whether H is likely without referencing M, and M implies the predictive posterior.

The area to the left of the vertical line represents probability leakage. The “normal model” says our uncertainty in y is characterized by a normal distribution. The “Location A” and “Location B” is from a larger regression model where one regressors is a categorical variable based on one of two locations.

The following question was posed at the (inaptly named) magazine Scientific American, “Is there a difference between the brain of an atheist and the brain of a religious person?” It was asked of one Andrew Newberg, “director of research at the Myrna Brind Center of Integrative Medicine at Thomas Jefferson University and Hospital in Philadelphia.”

Newberg said yes.

But he admitted that the “the neural picture is not yet complete.”

Several studies have revealed that people who practice meditation or have prayed for many years exhibit increased activity and have more brain tissue in their frontal lobes, regions associated with attention and reward, as compared with people who do not meditate or pray. A more recent study revealed that people who have had “born again” experiences have a smaller hippocampus, a part of the brain involved in emotions and memory, than atheists do. These findings, however, are difficult to interpret because they do not clarify whether having larger frontal lobes or a smaller hippocampus causes a person to become more religious or whether being pious triggers changes in these brain regions.

Newberg left out the most important point of this “recent study”: it does not confirm that people who have been “born again” have a smaller hippocampi than do atheists. Newberg’s understanding of statistical evidence is poor. Three points about studies of this kind (for more, look at this critique of another Newberg study):

1. The sample sizes can usually be counted on the fingers of one or two researchers, making the possibility of over-certainty high;
2. The kind of people chosen to study came from an extremely narrow demographic, in time, place, age, socio-economic, and genetic background. It is a stretch past the snapping point to suggest, as Newberg does, that these few folks represent all humanity;
3. In this study (and all similar) it is not the case that all theists had smaller hippocampi than atheists. What happened was that a few more theists than atheists had smaller hippocampi; it is even so that some theists had larger hippocampi than some atheists.

It is also the case that nobody knows whether smaller or larger hippocampi are better or worse (for their possessors) in any tangible sense. It makes a pleasing sound, of course, to suggest that larger is better, but this supposition is unwarranted—at best: at worst it is foolish.

Various experiments have also tried to elucidate whether believing in God causes similar brain changes as believing in something else. The results, so far, show that thinking about God may activate the same parts of the brain as thinking about an airplane, a friend or a lamppost. For instance, one study showed that when religious people prayed to God, they used some of the same areas of the brain as when they talked to an average Joe. In other words, in the religious person’s brain, God is just as real as any object or person.

One begins to wonder if Newberg can remember what he says from one moment to the next, for this second set of “evidence” is not consonant with the first. If God is as real as “any” object or person—and just what exactly, measurably, unambiguously does “any object or person” mean? any?—then the only thing that is different between the brains of theists and atheists must be so minuscule that we have little hope of finding it.

If the brain acts that the same way for God as it does for pencil erasers, pebbles, grains of sand, air, water, aspirins, electrons, Uncle Mike, and on and on forever, then all that can be different is one small “belief switch”, turned “on” for theists and “off” for atheists.

But even this is nonsense. Talk of “areas of the brain” is ridiculously loose, and is a bad, unshakable habit of people like Newberg (see this egregious study of Sam Harris).

Research also suggests that a ­religious brain exhibits higher levels of dopamine, a hormone associated with increased attention and motivation. A study showed that believers were much more likely than skeptics to see words and faces on a screen when there were none, whereas skeptics often did not see words and faces that were actually there. Yet when skeptics were given the drug L-dopa, which increases the amount of dopamine in the brain, they were just as likely to interpret scrambled patterns as words and faces as were the religious individuals.

The “research” that “suggests” these findings is subject to the identical criticism given above. E.g. it is not the case in this wee sample that every theist had more dopamine than every theist, etc.

Notice how Newberg exposes his bias. Let it be so that theists “tend” to see patterns in patternless data more than do atheists. Sounds bad, almost as if the atheists have not evolved into a higher state. But if theists are quicker to see patterns, then they ought to be better scientists, more acute detectives, quicker to find true signals hidden in the noise. We do find, do we not?, among those of lasting genius (Newton, Galileo, Bach, Shakespeare, etc.) vastly more theists than atheists.

This means the mutation that leads to atheism is harmful. Good thing, then, that we find that atheists breed less than theists. Their faulty, harmful-to-intelligence genes have a good chance of passing out of the population.

Article originally linked on Hot Air.