William M. Briggs

Statistician to the Stars!

Author: Briggs (page 151 of 413)

Teaching Journal: Day 1

Truth exists. Therefore, so does falsity. That truth exists is one of the many things we know to be true based on no external evidence. Naturally there are a very large number of things we know to be true given external evidence: most of what we believe to be true is this way.

Incidentally, though I can’t lay my hand on the book, and therefore must paraphrase, a statement by Roger Scruton in his Modern Philosophy: An Introduction and Survey is useful to us. Scruton cogently argues that relativists, in saying that it is certainly true that there are no truths, invite us to disbelieve them. Consequently, I shall not here argue against this (ever) seductive but obviously false philosophy.

If evidence for truth isn’t external, it must be internal. For example, we know via introspection, in our heart of hearts, or, even better, by accepting it on faith, that the following proposition is true: A = “For all numbers x,y = 1, 2, 3, …, if x = y then y = x.” Another way to put it: Given our faith, A is true. Or given our belief, A is true.

We know A is true relative to our faith, our belief. In this way the truth of proposition A is relative, but it’s a special kind of relativity, as we shall see. Contrast A with the proposition G = “George wears a hat” (yes, an old example, well known to regular readers, but still a good one). We have no sense that G is true or false, while A appears true after even momentary reflection. Why? Well, it turns out we are equipped—somehow, never mind here how (but it cannot be an empirical process which gives us this knowledge)—with the knowledge that A is so. But none of us comes pre-made with knowledge of G.

Suppose I give you this evidence, E1 = “All Martians wear hats and George is a Martian.” Now given E1 G is true. Another way to say it: Conditional on E1, G is true. Another: The probability G is true given E1 is 1, or 100%. Another: Accepting for the sake of argument E1, then G.

The truth of G is, just as A was, relative. But we had to supply external evidence which made G true, while we all of us come equipped-from-the-factory with the evidence that makes A true. This is made even clearer by supposing we supply evidence E2 = “Most Martians wear hats and George is a Martian.” Given E2, G is not true, but neither is it false. It is somewhere in between. That somewhere in between falsity and truth is where probability lies.

G then can be true, false, or merely probable depending on the evidence with which it is assessed. It is important to understand that for most everyday propositions it is we who supply the evidence. For example, your enemy may insist that G is false because he holds that E3 = “No Martian wears a hat and George is a Martian.” The argument then (ideally, anyway) moves away from G and to which of E1, E2, or E3 is true.

But how do we know that any of these are true? Well, they aren’t any of them obviously true, as A was, so we need to supply further evidence with which to gauge each E. What if we take F = “There are no Martians”? What happens to G for each E? What if F = “Observations suggest there are no Martians”? How about F = “I have no idea if there are any Martians”? And F = “There are either Martians or there aren’t”?

That’s it for the post; much more in the book, which you must read to keep up (Chapters 1 and 2 today). But there’s also material in this post which isn’t in the book. Which means I have to finish re-writing the book. Time…

Homework

Read then do the questions in Chapter 1, paying especial attention to the challenge to name other propositions which you “just know” are true. Are they really true, or are there hidden or tacit assumptions which you are using? You have to be careful because it is easy to fool yourself. However, this is not necessarily a bad thing.

For example, a person may hold that B is true “just because”, or by faith, or relative to introspection, or whichever synonymous phraseology you prefer. But it might turn out we can prove that B is true only because W, X, Y, … and so on are true; and they true only because some fundamental axiom C is true. But none of this makes B false. Indeed, suppose B = “Four divided by two equals two.” Well, it’s just true! It isn’t, but if you were to believe it, you would not be making an error, at least about the truth of B.

Here is an exercise to prove that probability is not subjective. The United Nations Conference on Sustainable Development, or Rio+20 Earth Summit begins this week (guess where: no, really, guess: why is your guess right or wrong?). Canada’s Environment Minister Peter Kent said “We just aren’t seeing people arriving in the frame of mind to make significant progress towards significant commitments. And we clearly need that.”

What is the probability that H = “The Rio+20 Leaders make significant progress towards significant commitments”? Supply the exact evidence and chain of argument you use to specify this probability. Try not to be facetious and you will learn something. This is a real assignment, incidentally.

The moral of today’s lesson is: know what you’re arguing about.

Teaching Journal: Day 0

This begins my two week tarriance at Cornell, teaching ILR-ST somenumberorother, a course in the Masters of Professional Studies, which is not unlike, though not entirely the same as, an MBA. Students are mostly working professionals who have not seen the inside of an algebraic equation in many a year. Though to learn these things they’ve come to the right guy, for

I’m very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I’m teeming with a lot o’ news —
With many cheerful facts about the square of the hypotenuse.

And from these theorems it’s a straight line to the, well, to the straight line, the second of only two equations with which I re-familiarize students. The other is Bayes’s theorem, so named after the Most Reverend Thomas Bayes. Nothing but simple multiplication and division, really. Even you, dear reader, can do it.

I invite you to play along with us this year. You’ll need a protractor, a compass, a new box of crayons, or whatever writing implement you prefer, and my class notes. The latest version of the book/class notes, free for the asking, can be found at my book page. The paper copy is passable, but (somehow) my enemies have surreptitiously ferreted into it an array of embarrassing typos.

This blog will thus function sort of, kind of, a little like, but not really, an on-line class. But with no registrations or promises. And with most of the onus on you to keep up.

Like I mentioned a few weeks back, I run the class with a lot of back and forth between me and the students. Mostly questions and answers and more questions and more answers. Trouble is, I don’t have a list of preprepared questions, just ideas. Everything changes depending on what students already have in their heads.

Speaking of that: the easiest students to teach, incidentally, are those who haven’t had a statistics course before. Everything is strange to them, but new and congenial. The hardest to influence are those who anchor their minds to a fixed point: for them, everything is interpreted in the context of the old; the new is never new. Actually, of course, I don’t run into the latter type in this classroom.

Class officially starts tomorrow, 9 am sharp. When I lectured undergraduates, I’ve been known to begin speaking at the strike of the bell even into an empty room. This is an old, and now unbreakable habit inculcated into me by my Uncle Sam, who taught me that “on time” meant 15 minutes early, that “late” meant on time, and that there is no allowable third category. But you, dear reader, may show up whenever it is convenient, or even not at all.

Homework today is to mediate on the reasons, motivations, and other character flaws that would lead somebody to want to take a class in statistics in the middle of summer vacation, in the (forecasted) ninety-degree heat.

Today is my day to struggle to recall what it is I want to teach. I usually review my notes, knowing that for the first day I never go much beyond Chapter 1.

Oh yes. The first reminder. I require all students to collect their own data. It should look something like the appendicitis.csv file on the books page. You can read all about the structure of this file in the book (can’t recall the chapter).

This should be data of interest to you, to answer a question you want answered. My students generally pick something work-related, but many choose hobbies or other interests. I’ve had weather in golf tournaments, Top Chef (who’s favored, New York- or LA-based chefs), boxing (who’s favored, righties or south-paws), wine tasting, and on and on.

It’s up to you to figure this out. But if you want to play, start thinking about this now. Two weeks isn’t enough time to procrastinate.

I’ll try and answer whatever questions you have, but be warned that I probably won’t be able to answer everything. I’ll just refer you to old blog posts or relevant sections in the book.

The only bad part is that since you’re doing this remotely, you won’t be able to join us for the wine tour and tasting next Saturday. I’ll post pictures.

Get a good night’s sleep. See you in the morning.

What Do You Think Of The New Look? (Back To Old: Temporarily)

Everybody gets to vote. Though, like in any socialist paradise, there is only one candidate. This one. This new design, I mean.

The old one was growing stale, it was hard to read in some contexts, and much of the site wasn’t used. I’ve eliminated all but three pages: the main-post one, which you are seeing now and which of course must stay; the old Start Here, which is a list of fundamental posts; and a modified About, which combines contact information with my CV and so on.

This style focuses on the posts, or rather, wordy posts, which is all we really do here. Larger font, wider separations between paragraphs, lighter colors, easier on the eye. No having to click on a post to read it, either, as before.

The links to other blogs and sites and other functionality are now at the bottom of the main-post page. These items were scarcely used by anybody, but they are still helpful to have, so they stay but are tucked away.

Comments should be easier, at least in the sense that there is a guide showing which HTML commands are allowed. I still do not allow nested comments. I find them ugly; and they are too apt to become squished.

I’m particularly interested in hearing from anybody who reads the site on a mobile device or tablet computer. It looks great on a/my laptop, anyway. There are some tweaks of the code I know of, but I’m sure there are others you will suggest. What do you think of the within-posts linking? Little dots as underline. No color.

The theme is a (minorly) modified version Esquire theme by Matthew Buchanan. This explains the colors, which for now I’m leaving default. Still some problems with the blockquotes. My mug is covering the search drop down. I’m working on it.

Update Thanks for the comments everybody. I’ve temporarily switched back to the original format to give me time to work on the new. There are lots more problems with this old version, but we’re all used to them and three’s something to be said about familiarity.

The “extra” whitespace on the test theme wasn’t extra. You can’t have text running from left end of the screen to the right. All the text has to exist in a box to be readable. You don’t notice it in the old theme because it’s not white but blue gray space.

Also, this old theme is even worse for mobile viewers…

Personality Predicted By Pedal Extremity Wrappings?

Fats Waller knew which end was up. The mark of mate-ability, he infallibly sang, was to be had by looking down. Shakespeare agreed: he wrote, “Farewell Love and all thy laws for ever, Thy Birkenstocks shall tangle me no more.”

Virgil said, “A fault is fostered by concealment in poor footwear.” And Molière observed, “All the ills of mankind, all the tragic misfortunes that fill the history books, all the political blunders, all the failures of the great leaders have arisen merely from a lack of taste in shoes.” Even St Paul himself said, “Love is patient, love is kind, love is not envious or boastful or arrogant or rude. But love insists that one never wear Crocs.”

With these timeless insights in mind, look at the picture in the upper-right corner of this post. What kind of person would you say wears this shoe? The answer is below, but don’t cheat. Test your powers of observation first.

The answer:

The wearer is a man, obviously. A man’s man, at that. Somebody bold, brash; yet wise, kind to strangers. Discriminating. A fellow who reeks of good taste. A success.

Your list was doubtless filled with these and similar words. And that’s because it’s easy to tell (nearly) everything you need to know about a fellow by fixing an eye on his footwear. As seen above, this was common wisdom even in the Bard’s day. And it is truer today because the number of ways to err increaseth daily. Even yesterday I spied a business-suited man who legs disappeared into black-cotton espadrilles—and this was far from any sanitarium!

Now while it might have always been known that a man’s shoes reveal more about him than the contents of his bookshelves, we did not until yesterday know that we knew this with a certain level of statistical significance. That remiss has been remedied by Omri Gillath and his four-minus-one fast friends in their peer-reviewed paper “Shoes as a source of first impressions” in the Journal of Research in Personality.

Our scientific quartet “investigated people’s precision in judging characteristics of an unknown person, based solely on the shoes he or she wears most often.” They discovered, or rather rediscovered, that folks “accurately judged the age, gender, income, and attachment anxiety of shoe owners based solely on the pictures.”

If you agree with the sentiment “I want to get close to my partner, but I keep pulling back” or “I am nervous when partners get too close to me” or even “I try to avoid getting too close to my partner” then you have the psychological affliction known as “attachment anxiety” which you wear on your feet like a martyr wears his heart on his sleeve.

“Shoes,” the authors inform us, “serve a practical purpose,” a finding which shows why we instinctively trust scientists. Yet shoes also, our researchers insist, “serve as nonverbal cues with symbolic messages.” They are even guides to the “dogmatism and creativity” of their owners. For example, “People who are extraverted…tend to wear more colorful shoes” while, somewhat surprisingly, the rich opt for “high-end brands”.

The experimenters gathered “208 undergraduate students enrolled in an introductory Psychology course” and had them fill out questionnaires and take pictures of their shoes. They then showed these pictures to “63 [new] undergraduate students”. The pics were used to rate the wearers’ “personality, attachment style, political ideology, and demographic dimensions.”

Do you know what happened? The raters’ judgments of the shoe owners’ self-assessed personality traits were barely to crudely correlated. But the important thing is that many of these small correlations were attached to wee, publishable p-values. Which closes the case and officially provides the proof so desperately needed for the commonsense wisdom that shoes make the man.

Not that shoes give the entire game away. For instance, it was found “people could accurately detect attachment anxiety,” from glancing at footwear, “but not attachment avoidance.” This follows from the theoretical considerations:

People with avoidant attachment…are aloof and repressive in regulating their emotions and relationships with others. Given that they generally do not care about how others perceive them, it is less likely that their shoes would reveal something about who they are (Banai, Mikulincer, & Shaver, 2005).

Nevertheless, “unless a shoe owner purposefully generates a deceptive image, shoes can be a reliable source of information.” The authors leave us with this caution: “Do people buy and wear shoes strategically to portray an image, and can observers detect the ‘acquired image?’ These are fundamental questions in personality and social psychology, and they play out in many domains—shoes are merely one attractive alternative to research.”

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Thanks to Eric Anderson who suggested this topic.

Drug Companies Tweaking Results To Produce Publishable P-values?

A person who Nature is calling a “whistle blower” has written a brief confessional in the British Medical Journal admitting to what might be termed statistical fiddling at a “major” drug company. (The just-over-one-page article requires a subscription to view.)

It is always difficult to trust fully any gossip which is reported anonymously, for if a man can hide behind a letter he may say anything, even that which is not so, without fear of reprisal. Too, the person or organization who publishes the gossip has no way of verifying it.

Now Mr X, if we may call him that, speaks of a company’s post-FDA-approval observational studies on drugs, studies whose primary purpose are to tout the drugs. “Patented Xlimicorconaphil is better than the generic! Ask your doctor if Xlimicorconaphil is right for you. Hint: it is.”

Mr X criticizes the statistical methods of these observational studies. He claims, “the truth is that these studies had more marketing than science behind them.” Worse:

Since marketing claims needed to be backed-up scientifically, we occasionally resorted to “playing” with the data that had originally failed to show the expected result. This was done by altering the statistical method until any statistical significance was found. Such a result might not have supported the marketing claim, but it was always worth giving it a go to see what results you could produce. And it was possible because the protocols of post-marketing studies were lax, and it was not a requirement to specify any statistical methodology in detail. On the other hand, the studies were hypothesis testing (such as cohort studies, case-control studies) rather than hypothesis generating (such as case reports or adverse events reports), so playing with the data felt uncomfortable.

The dreadful, should-be-banned term “statistical significance” means a publishable p-value, i.e. one less than the magic, never-to-be-questioned number, a number given to us (rumor has it) by Merlin himself. The number is sacrosanct, it is written into the law. Studies which cannot produce the required number are shunned. Those that find wee p-values are glorified.

Now especially in observations studies, this desirable creature, the wee p-value, can always be found, as long as one is willing to rummage around the data for a sufficient length of time. Mr X claims that is what his drug company has done. He appears to think this practice unusual and a bit shifty. Shifty it may be, but unusual it is not. It is not confined to observational studies, but appears even in designed experiments. And this is to be expected when success is defined in terms of p-values.

Statistics in this way is like a machine into which is fed data, a crank is turned, and out pops a rotten egg or one made of gold. Turning the crank longer increases the chance of gold. Success is trivially identified, but so is failure. The process requires no thinking (except by the nameless mechanics who keep the machine running).

Mr X also claims:

Other practices to ensure the marketing message was clear in the final publication included omission of negative results, usually in secondary outcome measures that had not been specified in the protocol, or inflating the importance of secondary outcome measures if they were positive when the primary measure was not.

Which sounds like standard politics. But I wonder. How often do drug companies try to hide negative results? Truly negative, I mean. Like discovering that widows who eat Xlimicorconaphil stroke out at rates exceeding the general population? What happens when this aberration finally outs? Smells like jail time.

Instead it’s more likely that the kind of “negative” results Mr X means are slight increases of slightly higher blood pressures in some subset of a subset of the population of those who take especially high doses of Xlimicorconaphil. Not a good thing, but not as awful as death or disfigurement.

Anyway, those negative findings are just that: findings. Produced using the same questionable statistical procedures as the positive findings which Mr X isn’t so keen about. How robust are they then? Probably not very.

The truth for most drugs is usually something like this: Xlimicorconaphil was found, via the usual FDA process, to be marginally better than the generic in some subset of the population. Xlimicorconaphil produces slightly different side effects, or of different intensity or frequency. The drug company, having to recoup its investment, takes this information, dresses it up, and sells the pill as New and Improved!

Nothing shady about this, especially in our all-marketing-all-the-time culture where such behavior is expected of everyone. The real worry is if doctors cease being skeptical gatekeepers.

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Thanks to Brad Tittle who suggested this topic.

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