William M. Briggs

Statistician to the Stars!

Page 144 of 547

The Sexual Habits Of The Tunisian Male Quantified At Last

Baseball is not available to think of in Tunis

You know that we have begun at the peak and that it’s all downhill when a scientific paper starts with a sentence like this:

La sexualité occupe une place très importante dans la vie de l’être humain.

Every caution, flag, and signal should be waving on high; each bell, alarm, and klaxon should be blaring in your ears. A sign with letters three feet high reading Theory Ahead should flash before your mind’s eye. Only an academic, or a middle schooler writing his first essay, could put those words onto paper and think they were worth reading.

The paper is “La Sexualite des Hommes Tunisiens” by Fakhreddine Haffani, the Chef de service de psychiatrie adulte à l’hopital Razi à la Manouba.

We can forgive Haffani, at least a little, knowing that academics are forced to write badly and on subjects which are not of the least interest to anybody because if they do not, out the door they go, cast into the outer darkness, there to live the life of a peripatetic, sans paycheck. Why, if they did not publish bad papers all that would be left to them is blogging.

Thus we cringe only slightly when we read of Haffani’s excuse: “En Tunisie, le comportement sexuel masculin est un continent non encore exploré.” Very well, let’s exploré. Haffani performed his science by asking 300 men, aged 20 to 69, a bunch of questions on their sexuality.

The statistics come hard and fast. Like this one: “L’âge moyen de notre population est de 38,26 ans.” Now that’s 38.26 and not 38.25 years, an important distinction; yes, to the tune of about three-and-a-half days difference. Science is nothing if not precise.

There came the curious heading “Le but de l’activité sexuelle” (The purpose of sexual activity; all translations performed by yours truly as in a high-wire act without a net). Apparently, “16% pensent accomplir une obligation” (think it’s a duty), a number which seems high given the age of the sample. But then we haven’t seen many Tunisian women. Or perhaps the answer comes in noticing that 26.4% (and not 26.3%) of married men find their relations an obligation. This result awaits a theory.

Perhaps this one: “L’amour n’est pas nécessaire pour avoir des rapports sexuels, mais les rapports sexuels sont nécessaires dans une relation amoureuse” (you don’t need love for sex, but it still takes two to tango). A full 73.3% (and not just 73.2%) of Tunisian men agree with this theory. One wonders what the other 26.7% were doing at the time the question was asked.

What about homosexuality? Well, “Un seul homme (0,33%) de notre population était exclusivement homosexuel” (just one guy). Eighty to 90% of the other fellows think activities d’homme ` homme are a non-non. This troubles le bon docteur, who said “Ceci prouve que la mutation sociale a peu ou pas d’influence sur notre conception de l’homosexualité, puisque l’homophobie, qui tend à disparaître dans l’Occident, reste bien ancrée dans la mentalité des jeunes de notre société” (Why, oh why!, can’t we be like other enlightened societies?).

It isn’t all bad news.

La grande majorité des hommes (90 %) se dit satisfaite de la taille de leur pénis et ce sont généralement les plus âgés (moyenne d’âge de 39 ans), alors qu’une minorité ne le l’est pas (10 %) et ce sont les plus jeunes (moyenne d’âge de 30,8 ans)

I leave this untranslated, except to note that nine out of ten Tunisians walk about with a satisfied smile.

Lastly, we come to what can only be described as the oddest statistic. Under the heading “La durée du coït” the reader is pleased to discover “Cette durée moyenne est proportionnelle au niveau d’instruction” (education has benefits).

En effet, elle est de 1 min et 5 sec pour les analphabétes, 1 min et 9 sec pour le niveau primaire, 1 min et 15 sec pour le niveau secondaire et 1 min et 40 sec pour le niveau supérieur.

Which is to say, a full 1 minute and 5 seconds for the illiterate, which is less time than it takes to make toast, all the way to an astonishing extra 35 seconds for those men in the possession of a college degree. (How these numbers were arrived at, we do not ask.)

Le bon docteur did not mention that this is more than sufficient time, for it is to be noted that the population of Tunis has more than doubled since 1960, and is now nearly 11 million.

Readers are cautioned that this site’s spam filter is set to sensitive, and that words found in the locker room won’t be found here.


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Merci beaucoup to Claude Boisson who alerted us to this topic.

People Growing Dumber Or Research Growing Slacker?

Hipp Chronoscope

If there was ever a paper which was in danger of refuting itself by its title, this is it. “Were the Victorians cleverer than us? The decline in general intelligence estimated from a meta-analysis of the slowing of simple reaction time” in the peer-reviewed journal Intelligence says that human beings have been growing steadily stupider since the late Nineteenth Century.

If so, one wonders if that is true of the press who uncritically reported the work as well.

We might guess college teachers Michael Woodley, Jan te Nijenhuis, and Raegan Murphy, the authors of this work, had grown suspicious that the decline in student intelligence was part of a larger effect. “Is everybody,” they must have wondered, “surer of their own abilities with less and less cause? If so, how can we prove it?”

How about unambiguously measuring IQ of a fair sample (not neglecting the dark corners of the earth) of similarly aged human beings each year for many years and then showing how the distribution of scores changes through time?

Too tough, that. Better to measure how long it takes people to swat a paddle after they hear a bell.1

The idea is reaction time is correlated to intelligence, or so people claim. Quicker times are not always associated with bigger brains—think of poor Admiral Nelson!—but it kinda, sorta is.

One thing we don’t want to do is just look at a numerical average of scores. Masks far too much information. Think about it. It could be that, in a certain time and place, a lot of folks test slow-stupid and a similar amount score fast-smart; a sort of U-shape in the distribution of scores. Its numerical mean would be identical to the scores of a different group of folks who all scored about the same.

Better to look at the distribution—the full range, the plus and minuses—than just the numerical mean.

Our authors looked at the numerical mean of reaction times. But don’t hold it against them. The mistake they made is so ubiquitous that it is not even known to be one.

Anyway, the authors gathered their data from a study of another man, one Irwin Silverman2, who searched through the archives and found papers which experimented on reaction times. The average times from most of these papers (our authors tossed some out) were used in the current study.

A picture of the machine used back in Francis Galton’s day, the Hipp chronoscope, is pictured above. Modern-day chronoscopes do not look the same. A show of hands, please: how many think that measuring the same man first on the Hipp chronoscope and then again on a computer will result in identical scores? They’d have to be, else people (to whom electricity was new and freaky) who hit Morse-code paddles in 1888 would not be comparable to people (who grew up with video games and cell phones) clicking a mouse in 2010.

How did our authors adjust for these differences? Shhhhh.

Figure 1

Figure 1

The plot shows where the magic happened. Each dot represents the average reaction time for a reaction-time study (y-axis) in the year the study was conducted (x-axis). Small dots are studies with fewer than 40 folks, larger open circles are those studies with more than 40.

See the way the line, made by a fancy statistical model, slopes upward? That line says people are growing stupider. Never mind you have to squint to see it. Never mind reaction time isn’t IQ. Never mind the enormous gaps in time between the old and new studies. And never mind that if you extrapolate this model it proves that Eighteenth Century denizens would have all bested Einstein at Chess and that those fifty years from now will listen exclusively to NPR. Concentrate instead that a wee p-value has been produced, therefore the the authors’ hypothesis is true.

Our authors apply a generous coating of theory (“dysgenic model”) to explain this crisis. Silverman disagrees with “dysgenics” and says it’s because of “the buildup of neurotoxins in the environment and by the increasing numbers of people in less than robust health who have survived into adulthood”.

My theory is that instead of IQs shrinking, people are increasingly able to find patterns in collections of meaningless dots.


Update Occurred after chatting with Stijn de Vos (‏@StijnDvos) that if this research were true, we should hang out by the Whack-A-Mole to discover future Nobel Prize winners.

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Thanks to John Kelleher for alerting us to this topic. See comments here, too.

1I have in mind the “sobriety test” taken by Dr Johnny Fever, WKRP; a clip of which I could have showed except for the massive greed of the recording industry; but never mind, never mind.

2“Simple reaction time: it is not what it used to be”, American Journal of Psychology. 123.1 (Spring 2010): p39.

What’s The Official Name For Governance By Thuggery?

“I knew nothing! Noooothing!”

To summarize what we know, but only in brief and leaving out many smaller scandals:

  1. The White House lied about the cause of Benghazi. They knowingly, willingly, and falsely claimed the attacks arose because of “spontaneous” anger about an unknown video. The purpose of this ruse was to deflect potential criticism of the president arising shortly before the election. The handling of Benghazi itself was due to either ignorance or incompetence. This is less troublesome than the deception.
  2. The IRS systematically and over a long period of time, and with the eventual knowledge of or by the explicit direction of the White House, illegally and immorally targeted individuals and groups because these people were in political opposition to the Democrat party. It will interesting to discover just how many fines and taxes were incorrectly assessed and how many applications for tax-exempt status were wrongly denied.
  3. The administration knowingly, willingly, and falsely accused a reporter of treason, a crime punishable by death. They lied to trick, or to allow cover for, a judge to issue a warrant which let the White House rifle through the reporter’s personal communications (reportedly even his parents’ phone calls). They did this because the reporter and the company which employs him was not friendly to the Democrat party. It will be interesting to discover what the White House made of the purloined communications.
  4. The administration used similar excuses (i.e. “National security”) to bug the AP, and even Gallup after that company had the temerity to release a poll which showed the President trailing his opponent.
  5. The administration dispatched the HHS secretary to shake down health care businesses, requesting they voluntarily fund a project that was denied funds by Congress. The HHS is writing the regulations which govern these businesses, regulations which necessarily must increase under Obamacare.
  6. The HHS mandated that employers, just because they are employers and for no other reason, must pay for their employees conception prevention medication and devices and for abortifacients in case these should fail. That this violates the religious practices of Muslims, Christians, (some) Jews, and others was ignored because it was discovered that the responsibility of employers to pay trumps the Constitution.
  7. The President routinely and systematically uses language which does not imply but directly accuses any who disagree with him of being not just wrong but evil. He has, more than once, urged citizens to report (rat out) neighbors who are not friendly to the Democrat party. He used government funds (i.e. your money) to set up an apparatus to collect this information.
  8. The administration knowingly and willingly released weapons such that they would end up in the hands of citizens of another country, hoping these weapons would be used for crimes and for murder. A clear case of entrapment, to say the least. One such weapon was used to kill an American citizen. The official who ordered this Fast and Furious campaign was held in contempt by Congress. The official ignored this charge. It will interesting to see why Mexico never declared war on the USA.
  9. The administration, at least the EPA, systematically and willfully covered up information, denying access to Congress and to the press, using the subterfuge of fictitious names and false email accounts.
  10. The administration, via the Department of Homeland Security, systematically tracks, stores, and analyzes communications of this country’s citizens. Its 2009 report warned of “rightwing” “chatter” about the economy.

An interesting test of an individual’s Ideology Quotient is the number and intensity of excuses offered in mitigation of these activities. The more the person espouses that the (utopian) ends justify the (brutal) means, the higher the IQ.

Here is a tell-tale. It is not an argument, in an answer to any of these crimes or behaviors, to say that another also committed them. That is, it is a fallacy to say that because the President lied about the causes of Benghazi that other presidents lied, therefore there is no lie.

Another fallacy: the President didn’t know, therefore there is no crime or immoral behavior. Accepting that the President did not know about his employee’s behavior is not an excuse for that behavior. If no one is looking, a crime is still a crime. This fallacy also backfires, because if we do accept the President’s self-proclaimed ignorance of his subordinates’ activities, then he is guilt of incompetence.

David Axelrod funnily anticipated this last argument and sought to head it off by claiming the government is now so “massive” that no president can oversee it.


Subjective Versus Objective Bayes (Versus Frequentism): Part Final: Parameters!

ET Jaynes, Chance Master

(All the stuff in this series is, in a fuller form, in my new upcoming book, which is tentatively called Logical Probability and Statistics—but I’ve only changed the title 342 times, so don’t count on this one sticking. Incidentally, I’m looking for a publisher, so if you have a definite contact, please email me.)

Read Part IV.

From where do parameters emerge? And what is the difference between treating them objectively and subjectively? A lot of controversy here. The difficulty usually begins because the examples people set themselves to discuss these subjects so far advanced that they have too many (hidden) assumptions built in such that it makes understanding impossible. From our previous examples, we have seen it is better to start small and build slowly. We can’t avoid confusion this way, but we can lessen it.

If you haven’t reminded yourself of the last two posts on the statistical syllogism, do so now, for it is assumed here.

Premise: “There are N marbles in this bag into which I cannot see, from which I will pull out one and only one. The marbles may be all black, all white, or a mixture of these.” Proposition: “I pull out a white marble.”

What is the probability this proposition is true given this and only this premise? Well, what do we know? Black marbles are a possibility, and so are white. Green ones are not, nor are any other colors. We also know that the number of white marbles may be 0, 1, 2, …, N. And the same with black, but with a rigid symmetry: if there are j white marbles then there must be N – j black ones. (There may be other things in the bag, but if so, we are ignorant of them.)

This is not like the example where all we knew was that “Q was a possible outcome”, and where we assigned (0, 1] to the probability “A Q shows”, because there the number of different kinds of possibilities were unknown, and could have been infinite. Here there are two known possibilities. And N is finite.

Suppose N = 1. What can we say? “There are 2 states which could emerge from this process, just one of which is called white, and just one must emerge.” The phrasing is ugly, and doesn’t explicitly show all the information we have, but it is written this way to show the continuity between this premise and the one from last time (i.e. “There are n states which could emerge from this process, just one of which is called Q, and just one must emerge” and “A Q emerges”).

The probability of “I pull out a white marble” is, via the statistical syllogism, 1/2—again, when N = 1. This accords with intuition and with the definite, positive knowledge of the color and number of marbles we might find.

Now suppose N = 2. The number of possibilities has surely increased: there could be 0 whites, just 1 white, or 2 whites. But we’re still interested in what happens when we pull out just one and not more than one. That is, our premise is “There are 2 marbles in this bag into which I cannot see, from which I will pull out one and only one. The marbles may be all black, all white, or a mixture.” Proposition: “I pull out a white marble.”

If there were 0 whites, the probability of pulling out a white is clearly 0. If there were 1 white, we have direct recourse to the statistical syllogism and conclude the probability of pulling out a white is 1/2. If both marbles were white, the probability of pulling out a white is 1. Agreed?

But what, given our premise, is the probability that both marbles aren’t white? That just 1 is? That both are? Easy! We just use the statistical syllogism again. As: “There are 3 states which could emerge from this process, and only one of these three must emerge, which are 0 whites, 1 white, 2 whites” and “The state is 0 whites” or “1 white” or “2 whites”, and the “process” is whatever it was that put the marbles in the bag. Evidently, the probability is, via the statistical syllogism, 1/3 for each state.

It now becomes a simple matter to multiply our probabilities together, like this (skipping the notation):

     1/3 * 0 + 1/3 * 1/2 + 1/3 * 1 = 1/6 + 2/6 = 1/2,

where this is the probability of each state of the number of white marbles times the probability of pulling a white marble given this state, summed across all the possibilities (this is allowed because of easily derived rules of mathematical probability).

With N = 2, the answer for “I pull out a white marble” is again probability 1/2. The same is true (try it) for N = 3. And for N = 4, etc. Yes, for any N the probability is 1/2.

This is the same answer Laplace got, but he made the mistake of using the chance the sun would rise tomorrow for his example. Poor Pierre! Not a thing wrong with his philosophy, but every time somebody heard the example he turned into a rogue subjectivist and would not let Laplace’s fixed premises be. Readers could not, despite all pleas, stop themselves from saying, “I know more about the sun than this bare premise!” That is a true statement, but it is cheating to make it, as all subjectivists substitutions are.

Emphasis: we must always take the argument as it is specified. Adding or subtracting from it is to act subjectively, that is, it changes the argument.

We don’t have to use marbles in bags. We can say “white marbles” are “successes” in some process, and “black marbles” failures. As in it is a success if the sun rises tomorrow, or that a patient survives past 30 days, or more than X units of this product will be sold, and on and on. If we know nothing about some process except that only successes and failures are possibilities, and that there may be none of either (but must be some one one), and that we have N “trials” ahead of us, and N is finite, then the chance the first one we observe is a success is 1/2.

A “finite” N can be very, very, most very large, incidentally, so there is no practical limitation here. Later we’ll let it pass to the limit and see what happens.

This is the objectivist answer, which takes the argument as given and adds nothing about physics, biology, psychology, whatever. Of course, in many actual situations we have this kind of knowledge, and there is nothing in the world wrong with using it, provided it is laid out unambiguously in the premises and agreed to. To repeat: but if all we had was only the simple premise (as we have been using it), then the answer is 1/2 regardless of N.

The emergence of parameters and models

Since we have the full objectivist answer to the argument, it’s time to change the argument above to something more useful. Keep the same premise but change the conclusion/proposition to “Out of n < N observations, j will be successes/white,” with j = 0,1,2,…,n. In other words, we’re going to guess how many successes we see from the first n (out of N). Did we say same premise? Emphasis: same premise, no outside information. Resist temptation!

If n = 1, we already have the answer (the probability is 1/2 for j = 0, and j = 1). If n = 2, then j can be 0, 1, or 2. The statistical syllogism comes to our aid as before and in the same way. Before we’ve seen any “data”, i.e. taken any observations, i.e. have any knowledge about how many successes and failures lie ahead of us, etc., the probability of j equaling any number is 1/3 because there are 3 possibilities: no successes, just one, both. The result also works for any n: the probability is 1 / (n+1), and this works with n all the way up to N.

We’re done again. So now we must pose new arguments. Keep the same premise but augment it by the knowledge that “We have already seen n1 out of the n observations” where, of course, n1 is some number between 0 and n. We need a new conclusion/proposition. How about “The next observation will be a success/white marble”, where “next” means the “n+1″th.

In plainer words, we have some process where we’re expecting success and failures—N of them where N may be very exceptionally big—and we have already seen n1 out of n successes, and now we want to know the probability that the next observation will be a success. Make sense?

Notice once more we are taking the objectivist stance. There are no words, and no evidence, about any physicality, nor any about biology, nor anything. Just some logical “process” which produces “successes” and “failures.” We must never add or subtract from the given argument! (Has that been mentioned yet?)

Proving the answer now involves more math than is economical to write here. But it comes by recognizing that we have taken n marbles out of the bag, leaving N – n. Of these first n, some might be successes, some failures, with n1 + n0 = n. We’ll use these observations as new evidence, that is new premises, for ascertaining the probability that next (as yet unknown) observation is a success.

The probability of seeing a new success (given our premise) is (n1 + 1) / (n + 2). If we only pulled n = 1 out and it was a success, then the probability the next observation is a success is 2/3; if the first observation was a failure then probability next observation is a success is 1/3. Again, this is independent of N! But it does assume N is finite (though there is no practical restriction yet).

This answer is also the same as Laplace got. Where he went wrong was to put in an actual value for n: what he thought was the number of days that sun had already come up successfully, and then used the calculation to find the probability (given our premise) the sun would come up tomorrow. Well, this was some number not nearly equal to 1, because Laplace didn’t think the world that old. Since the probability was not close to 1, and people thought it should, logical probability took a big hit from which it is only now recovering. His critics did not understand they were changing his argument and acting like subjectivists.

Let’s play with our result some more. It’s starting to get good. Suppose n is big and all the observations n1 are successes, then (n1 + 1) / (n + 2) approaches 1. If we let the “process” go to the limit, then the probability becomes 1, as expected. The opposite is true is we never see any successes: at the limit, the probability becomes 0. Or, say, if we saw n1 = n/2, i.e. our sample was always half successes, half failures, then at the limit, the probability goes to 1/2.

It is from this limiting process that parameters emerge. Parameters are those little Greek letters written in probability distributions. But before we start that, let’s push our argument just a little further and ask, given our premise (which includes the n observations) not just what is the next observation will be, but what the next m observations will be. Obviously, m = 0 or 1 or 2 etc. all the way up to m.

The probability m takes any of those values turns out to be what is called a beta-binomial distribution, whose “parameters” are m, n1 + 1, and n0 + 1. Parameters is in quotes because here we know their exact values; there is no uncertainty in them; they have been deduced via the original premise in the presence of our observations. Notice that this result is independent of N, too (except for the knowledge that n and n + m are less than or equal to N).

The beta-binomial here is called a “predictive” distribution, meaning it gives the probability of seeing new “data” given what we saw in the old data.

In other words, we don’t need subjective (or objective) priors when we can envision a process which is analogous to a bag with a fixed number of successes and failures in it (with total N), and where we take observations n (or m) at a time. That’s it. This is the answer. We have deduced both a model in which are not needed unobservable parameters. No Greek letters here. The answer is totally, completely objective Bayes.

There is no subjectivity, no talk of “flat” or “uninformative” priors, no need for frequentist notions, no hypotheses, nothing but one bare premise, some rules of mathematical probability accepted by everybody (because they are true), and we have an answer to a question of real interest. Ain’t that slick!

Where It Starts To Go Wrong

Everything worked because N was finite. We lived in a discrete, finite world, where everything could be labeled and counted. Happily, this world is just like our real world, a hint that difficulties arise when we leave our real world and venture to the abstract world of infinites.

We could let N go to the limit immediately (before taking observations). Now no real thing is infinite. There never has been, nor will there ever be, an infinite number of anything, hence there cannot be an infinite number of observations of anything. In other words, there is just is no real need for notions of infinity. Not for any problem met by man. Which means we’ll go to infinity anyway, but only to please the mathematicians who must turn probability into math.

If N goes to the limit immediately, the number of successes and failures, or rather the ratio of the number of successes (in our imaginary bag) to N becomes an unobservable quantity, which we can call θ. If N passes to the limit, θ can no longer ever equal 0 or 1, as it could in real life, but will take some value in between.

We now have a brand-new problem in front of us. If we want to calculate the probability that the first observation is a success, we have to specify a value for θ by fiat, i.e. arbitrarily and completely subjectively, or we have to let it take each possible value it can take and then multiple the probability it takes this value by the probability of seeing a success given this θ. And then add up the results of each multiplication for each value of θ.

That’s easy to do with calculus, of course. But it leaves us the problem of specifying the probability θ takes each of its values (the number of which is now infinite). There is no other recourse but do to this subjectively. We could say (subjectively) that “Each value of θ is equally likely” or maybe “Let’s assume a flat prior for θ” which is the same thing, or we could say, “Let’s pick a non-informative prior for θ” which is also the same thing, but which involves invoking other probability distributions to describe the already unknown one on θ.

Is there any wonder that frequentists marvel at this sort of behavior? The “prior” appears as it is: arbitrary and subject to whim. (Of course, the frequentist after getting this argument right falsely assumes that frequentism must therefore be true; the fallacy of a false dichotomy.)

It turns out, in this happy case, that if a “flat” prior is assumed, the final answer is the same as above; i.e. we end with a beta-binomial predictive distribution. We start with a “binomial” and get a beta “posterior” on θ, but since we can never observe θ, it is another wonder that anybody ever cares about it. People do, though. Care about θ, to an extent that borders on mania. But that people do is a holdover from frequentist days, where “estimation” of the unobservable is the be-all and end-all of statistics. Another reason to cease teaching frequentism.

But the happenstance between the finite-objectivist answer with the mathematical-subjectivist one in this case is why subjectivism seems to work: it does, but only when the parameters are themselves constrained. Here θ can “live” only in some demarcated interval, (0,1). But if the parameter is itself without limit, as in the case of e.g. “normal” distributions, then we can quickly run into deep kimchee.

The problem comes from going to infinity too quickly in the process, a dangerous maneuver. Jaynes highlighted this to devastating effect in his book, destroying a “paradox” caused by beginning at infinity instead of ending with it. (See his Chapter 15 on the “marginalization paradox.”) If there is any blemish in Jaynes’s work, it is that he failed to apply this insight to every problem. In his derivation of the normal, for example, he began with the slightly ambiguous premise that measurement error can be in “any” direction. Well, if “any” means any of a set of measurable (by human instrument), then we are in business. But if it meant any in an infinite number of directions, then we begin where we should end.

All probability and statistics problems can be done as above: starting with simple premise, staying finite until the end, after which passing to the limit for the sake of ease in calculation may be done, and proceeding objectively, asking questions only about “observable” quantities. That is the nature of objective, or logical, probability.

Questions?

If there are a lot of questions, I might add an addendum post to this series. But please to read all the words above (at least once) before asking. On the other hand, I may just wait for the book.


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