William M. Briggs

Statistician to the Stars!

Category: Statistics (page 1 of 178)

The general theory, methods, and philosophy of the Science of Guessing What Is.

Discrimination Found By Statistics!


Was it Justice Ginsberg who popularized the fallacy that statistics could prove discrimination? Somebody check me on that. Busy day here.

The fallacy is that statistical models which have “statistically significant” findings identify causes. Which they sometimes can, in an informal way, but probably don’t. Anyway, that’s for another day. Point now is that discovering “disparities” and “gaps” and “discrimination” via statistics is silly.

Headline from the New York Post: Goldman Sach’s differentiating stats dominate sex suit.

Standard story. Couple of dissatisfied women accused Goldman Sach’s of being boy friendly. “They are suing the financial powerhouse, alleging a pattern of underpaying women and promoting men over them.” The ladies’ lawyer and Sach’s each hired their own statistician. The fallacy is already in place, ready to be called upon.

If there was real discrimination against women because they were women what should happen is that it should be proved. How? Interviews with employees, managers, ex-employees, examination of emails, memos, that sort of thing. Hard work, which, given the nature of human interactions, may ultimately be ambiguous, useless to prove anything.

Statistics certainly can’t prove discrimination, because statistics don’t identify causes. And it’s what causes the alleged discrimination that is the point in question. Since statistics can’t answer that question—which everybody should know—why would anybody ever use it?

Laziness, for one. Who wants to do all that other work? For two, it’s easy to get people to accept “discrimination” happened because math. Lawyers working on commission therefore love statistics.

The Post said, “The bank’s expert, Michael Ward of Welch Consulting, said the pay disparities between men and women are statistically insignificant and said Farber [the ladies' expert] was overly broad in his analysis.” “Significance” and “insignificance” are model and test dependent, so it’s easy for one expert to say “insignificant” and another “significant.” The data can “prove” both conclusions.

But something else is going on here, I think. Note that according to Farber, “Female vice presidents at Goldman made an average of 24 percent less than their male counterparts”. Ah, means. An easily abused statistic.

There more. Here’s the final two paragraphs (ellipsis original). See if you can spot the probable error. Hint: the mistake, if there is one, appears to be Farber’s. Of course, there might be no error at all. We’re just guessing.

Farber looked at divisions across the bank, rather than at smaller business units, which, according to Ward, muddied the statistical data.

“Breaking it up into these little pieces means you just won’t find these pay gaps,” Farber said. “It’s always a trade-off in this kind of analysis in getting lost in the trees…or saying something about the forest.”

Get it? Take a moment and think before reading further. It’s more fun for you to figure it out than for me to tell you.

This sentence is to fill space so you don’t easily see the answer.

So is this one.

This might be a case of Simpson’s (so-called) paradox. This happens when data looked at in the aggregate, such as mean pay for men and women at the Division level, shows (say) men with higher means, but when the same data is examined at finer levels like business units, it can show women with higher or the same means as men in each unit (or a mix).

The reason this happens is that the percent of men and women isn’t be the same inside each of the finer levels, and the mean pay differ by levels (no surprise). This link shows some easy examples. It’s more common than you’d think.

Farber looked at aggregates and Ward (properly) examined smaller units, a practice which Farber calls “muddying” the data. Well, it’s a strategy. The name-calling, I mean. Judges looking for an excuse appreciate it.

Even though it’s still looking at statistics, and to be discouraged, it’s better to look at the entire pay distributions, not just means, and at even finer levels, say business units and various years of experience in the same job title. But it’s chasing fairies. It can never prove anything.

And even if the data show a difference everywhere it could be that women in each unit are paid less, but not because they are women, but because women in negotiating their “packages” might do so inefficiently compared to men. Who knows?

Statistics is no substitute for hard work.

Paper Claims Surprisingly Strong Link Between Climate Change And Violence. Nonsense.

The official numbers

The official numbers

When does more crime happen, in winter or summer? Why? Too easy. How about this one: according to the FBI, what was the violent crime rate over time? No need to guess. It’s pictured above. The per capita all violent crime percent from 1960 to 2012 (the last year available). Looks to be coming down some since 1991, wouldn’t you say? (The plots for other crime types, including gun crimes, all have the same general shape.)

Say, isn’t the time range of this plot the period where the our-of-control global warming “climate catastrophe” began in earnest? Let’s look at what NOAA’s GISS says:

The official numbers

I’m not in the least interested in arguing about this data; for the sake of argument, let’s just accept it as it is. Look, however, only at the black dots, which are the actual data. The red line is a smoother, i.e. a model, and is not what happened. The model is not the data! Don’t smooth your time series data! (Look here and here for why.)

Let’s tie it all together. Does it look to you like climate change is “correlated” with the violent crime rate? If you’re Chris Mooney or an academic hot for a sensational paper or a member of the media anxious to signal your cooperation with government, you must say yes. Us ordinary folk, not addled by ideology, will say no.

The Washington Post put up yet another fantasy of Mooney’s entitled “There’s a surprisingly strong link between climate change and violence“. I don’t mean to be snarky, I really don’t. But this guy routinely provokes me beyond my ability to resist. May the Lord forgive me.

Mooney cites some new meta analysis, a study I’ll dissect in due course, “of the existing research examining the relationship between climate change and violence and conflict.” Here’s the meat:

Climate variables considered in these papers included temperature increases as well as drought and rainfall changes. Conflict was analyzed in terms of clashes between individuals (like fistfights) and fights between groups (like wars). After taking it all in, the authors found compelling evidence of a link between changes in temperature and increases in conflict, noting that “deviations from moderate temperatures and precipitation patterns systematically increase the risk of conflict, often substantially, with average effects that are highly statistically significant.” Bottom line: In an ever warming world, expect more wars, civil unrest, and strife, and also more violent crime in general.

Yes, that makes sense. A statistical model which analyzes simultaneously fist fights and wars. Almost as sensible as measuring how eight-year-olds spend their allowance and the machinations of the World Bank. Hey! It’s science!

The lesson is: never ever not ever never never believe a meta analysis at its face value. It is one of the most abused statistical techniques. Smoothing time series data is another. Never mind.

Mooney gets one thing partly right when he asks, “Why do hotter temperatures produce more violence?” The obvious answer—as long as we factor out all modern wars, many of which inconveniently occur in winter; in olden days, winter made it difficult to fight; who could have guessed?—is the one we started this post with. People are out in the summer’s long warm days, and inside in the winter’s short cold days. Easy.

Yet not so easy for Mooney and for academics for whom the obvious is never good enough.

Now I would have ignored the article, putting it down as yet another attempt to prove our lying eyes aren’t seeing what they’re seeing (the two graphs above). But Mooney had to go and mention baseball. (I’m a Tigers fan. I don’t want to talk about it.) Mooney thinks a paper he uncovered is terrific proof that climate change makes us more violent.

He quotes from the awful peer-reviewed paper “Temper, Temperature, and Temptation: Heat-Related Retaliation in Baseball” in Psychological Science (2011; 22(4) 423­–428) by Richard P. Larrick and some others. Larrick checked whether increasing temperatures were associated with more beanballs. The authors admitted they were not.

So, their theory busted but still desiring a paper, the authors had to try something else. How about retaliation? Do increasing temperatures cause more? Mooney shows a graph from their paper which is so silly that I refuse to picture it. He presents this graph, as do the authors, as if it were data. Which it is not. It is the output from a preposterously complex regression model (they “control” for 13 things!).

Baseball fans: when do more beanballs, and hence more retaliations take place, in chilly April when the season has just begun and all are of good cheer, or late in hot August when tempers are up and when games start to feel a lot more crucial? Is the observed discrepancy therefore caused by climate change?

Good grief, what a rotten paper, what a rotten theory.

The Problem Of Grue Isn’t; Or, A Gruesome Non-Paradox About Induction

This emerald does not appear to be green, nor grue. Maybe Goodman was right!

This emerald does not appear to be green, nor grue. Maybe Goodman was right!

Skepticism about induction happens only among academic philosophers, and only in print. Tell an induction skeptic to take a long walk off a short dock or hint that his health insurance will be cancelled and you will find an immediate and angry convert to Realism.

Some philosophers come to their skepticism about induction from puzzles which they are unable to solve and reason that, since they cannot solve the puzzles, it’s a good bet to side with skepticism. Well, in some ways this is natural.

A classic puzzle is Nelson Goodman’s “grue”. Goes like this. Grue is a predicate, like green or blue, but with a built-in ad hoc time component. Objects are grue if they are green and observed before 21 October 1978 or blue and observed after that date. A green grape observed 20 October 1978 and a blue bonnet observed 22 October 1978 are grue. But if you saw the green grape yesterday, or remember the blue bonnet from 1976, then neither are grue. The definition changes with the arbitrary date.

So imagine it’s before the Date and you’ve seen or heard of only green emeralds. Induction says future, or rather all unobserved, emeralds will also be green. But since it’s before the Date, these emeralds are also grue, thus induction also says all unobserved emeralds will be grue. Finally comes yesterday—and lo!—a green and not a blue emerald appears, thus not a grue emerald. Induction, which told us it should be grue, is broken!

There have been several exposures of the grue fallacy before, and up until the other day (another date!) I had thought David Stove’s in his Rationality of Induction was best. But I now cast my vote for Louis Groarke’s in his An Aristotelian Account of Induction. He calls belief in Goodman’s fallacy “an adamant will to doubt rather than an evidence-based example of a deep problem with induction” and likens it to the fallacy of the false question (e.g. “Have you stopped cheating on your taxes yet?”).

Groarke says (p. 65):

The proposition, “emeralds are grue,” [if true] can be unpacked into three separate claims: emeralds are green before time t (proposition1); emeralds are blue after time t (proposition2); and emeralds turn from green to blue at time t (proposition3). Goodman illegitimately translates support for proposition1 into support for proposition2 and proposition3. But the fact that we have evidence in support of proposition1 does not give us any evidence in support of all three propositions taken together.

What does the arbitrary time have to do with the essential composition of an emerald? Not much; or rather, nothing. The reason we expect (via induction) unobserved emeralds to be green is we expect that whatever is causing emeralds to be green will remain the same. That is, the essence of what it is to be an emerald is unchanging, and that is what induction is: the understanding of this essence, and awareness of cause.

Groarke emphasizes that the time we observe something is not a fact about the object, but a fact about us. And what is part of us is not part of the object. Plus, the only evidence anybody has, at this point in time, is that all observed emeralds have been green. We even have a chemical explanation for why this is so, which paradox enthusiasts must ignore. Thus “there is absolutely no evidence that any emeralds are blue if observed after time t.”

Two things Groake doesn’t mention. First is that, in real life, the arbitrary time t is ever receding into the future. I picked an obviously absurd date above; it’s absurd because we have all seen green emeralds but no blue ones up to today, which is well past 1978. The ad hoc date highlights the manufactured quality of the so-called paradox. When, exactly, should we use a grue-like predicate for anything?

Secondly, nobody not in search of reasons to be skeptical would have ever thought to apply a predicate like grue to anything. It is entirely artificial. If you doubt that, consider that you can substitute any other predicate after the arbitrary date. It doesn’t have to be blue. Try salty, hot, tall, or fast. An emerald that is green up until t then fast? That’s ridiculous! Yes, it is.

After showing the paradox isn’t, Groake goes on to explain the possible reasons why the paradox has been so eagerly embraced. Cartesian corrosion. That bottomless skepticism which dear old Descartes introduced in the hope of finding a bedrock of certainty. There isn’t space here to prove that, but anybody who has read deeply in epistemology will understand what that means.

Update A glimpse of how much angst the “problem” of grue has created, try this (or similar) searches. Also note the New & Improved title.

Please Don’t Smooth Your (Social Media) Data!

Friends don't let friends smooth.

Friends don’t let friends smooth.


Don’t smooth your data and then use that smoothed data as input to other analysis. You will fool yourself. You will make over-confident decisions. It is the wrong thing to do. It is a mistake. It is a guarantee of over-certainty. I don’t know how to put it more plainly. Lord knows I have tried. See below for a non-success story.

Smoothing means any kind of modeling, which includes running means, just-plain-means, filtering of any kind, regression, wavelets, Fourier analysis, ARIMA, GARCH; in short, any type of function where actual data comes in and something that is not data comes out.

Do not use the something-that-is-not-data as if it is data. This is a sin.

Don’t believe me. Try it yourself. The picture is from an upcoming paper I and some friends are writing.

It shows two simulated normal noise time series, with successively higher amounts of smoothing applied by a k-rolling mean. From top left clockwise: k = 1, 10, 20, 30; a k = 1 corresponds to no smoothing. The original time series are shown faintly for comparison. The correlation between the two series is indicated in the title.

More smoothing equals higher correlations. Since there are no causes between these series, the correlation should be hovering around 0, which it is in the first panel. And that correlation stays near 0—for the original real not fake un-smoothed data. But if you calculate the correlation between the smoothed series…the sky’s the limit!

Now it is not true that in each and every and all instances that smoothing will increase the correlation between two smoothed series. It might be that (in absolute value), for your one-time smoothing, correlation decreases or stays put. But it usually will increase, and usually by a lot.

Why? Imagine any two straight lines with non-zero slopes. These two straight lines will have perfect Pearson correlation, either +1 or -1. Regression and other measures will also show perfect agreement. The proof of this is trivial, and I leave it as an exercise (don’t be lazy; try it). Smoothing makes time series data look more like straight lines, as the pictures show. Simple as that.

There are all manner of fine points I’m skipping and would make wonderful Masters projects. Just what kind of data and what kind of smoothing and what statistical measures are affected and by what magnitude? All these questions are quantifiable and will make for fun puzzles. My experience with actual data and actual smoothing and typical measures shows that magnitude is large.

It happens

Now, without betraying any confidences, let me tell you of the latest in a long and growing string of bad examples. Two companies, one internationally known for their quantitative prowess, another even better known for its ability to make vast wads of money. Call them A (stats) and B (client). I did not work for either A or B, but know and advised certain parties.

B advertised and wondered how much of an effect this had on its measure of success. A said they could tell, using sophisticated Bayesian models incorporating social media data.

Social media!

Wowzee! Tell people you have busted open the secrets of social media and they will dump buckets of cold cash on you. Hint: everybody who says they have it figured out is either exaggerating to themselves or to their clients. (Say, that’s a pretty bold statement.)

Anyway, smoothing occurred. And correlations greater than 0.95 were boasted of. I’m not kidding about this number. Company A really did brag of enormous “impacts” of its smoothed measures. And Company B believed them—because they wanted to believe. Sophisticated Bayesian models incorporating social media data! How could you go wrong?

The real correlations, using unsmoothed data, were near 0. Just as you’d expect them to be for such noisy data as “social media” predicting a company’s measure of success. Do you really think Twitter streams contain magic?

I told all involved. I explained pictures like those above. I was emphatic and clear. I stood neither to gain nor lose regardless of the decision. Only two people (at B) believed me, neither of whom were in a position to make decisions.

At least I am comforted that Reality is my friend here. The company’s will eventually realize, but probably never admit, that their measures are spurious. Because they will realize but not admit, these measures will be quietly abandoned…

…As soon as the next computer self-programmed big data machine learning artificially intelligent smart-phone-data algorithm comes along and seduces them.

Humanae Vitae & The Synod: Theories And Predictions

Selfish genes theory does not predict these.

This originally ran 14 May 2014, but since this weekend is Number Six’s big show, I thought it well to have another look. The title is New & Improved! See the Update at the bottom.

Old predictions

Theories are useful to explain and to predict. Any theory can explain, but only true or likely true theories can skillfully predict.

For instance, Uncle Bob explains that your car won’t start because of Gremlins. His theory, which he drags out on all State occasions, does explain. But it is, as I hope is obvious, a theory which is useless to make predictions.

Two obvious examples. If you say the sun will peek above the horizon at 6:32:17 AM because gravitational theory says it will, and the sun does its duty, your theory has something going for it, especially if the theory makes lots of accurate predictions. And if you say, and say each year for two decades, that the planet’s global average temperature will soar to “unprecedented” heights, yet the temperature misbehaves and stays put, you’re theory is likely false.

Now how about these predictions, made in 1968, on what would happen were contraception to be embraced (which, of course, it has been). This embracement will:

  1. “[O]pen wide the way for marital infidelity and a general lowering of moral standards”. Nailed it.
  2. Especially in the young, “[A] man who grows accustomed to the use of contraceptive methods may forget the reverence due to a woman”. Nailed it.
  3. He will “reduce her to being a mere instrument for the satisfaction of his own desires”. Hooked-up nailed it.
  4. He will “no longer considering her as his partner whom he should surround with care and affection.” Nailed it.
  5. “Finally, careful consideration should be given to the danger of this power passing into the hands of those public authorities who care little for the precepts of the moral law. Who will blame a government which in its attempt to resolve the problems affecting an entire country resorts to the same measures as are regarded as lawful by married people in the solution of a particular family difficulty?” Oh my, oh my, is that one nailed.
  6. “Who will prevent public authorities from favoring those contraceptive methods which they consider more effective?” None, that’s who: another hit.
  7. “Should [the government] regard this as necessary, they may even impose their use on everyone.” HHS mandate, anybody? Nailed it again.
  8. “It could well happen, therefore, that when people, either individually or in family or social life, experience the inherent difficulties of the divine law and are determined to avoid them, they may give into the hands of public authorities the power to intervene in the most personal and intimate responsibility of husband and wife.” Smack! Pow! Wow! Capital-N ailed it.

Who is this guy, this prescient sage, who, drawing from some theory, foretold the world of 2014 so accurately? Well, his name was Paul, and as he came from a long line of Pauls in the same Institution to which he was appointed Leader, he called himself Number 6. One thing we know, given his batting average, we should accord the theory which created these predictions pretty high weight.


Number 6’s theory also explains the popularity of divorce, out-of-wedlock births, and the rise in the belief of individuals’ “unlimited dominion” over “his specifically sexual faculties.” Number 6 didn’t actually specify these as predictions, though, taking them “as read.”


Now the most fascinating thing about these predictions is how they came to be made. What happened was this. Number 6’s predecessor called on a Commission of experts, who met and deliberated from 1963 to 1966 (Number 6 boosted Commission membership halfway through), giving a report to Number 6 two years before he made his predictions. Nobody was in any rush.

The Commission was loaded with sober academics and had the support of a good portion of the leadership of Number 6’s Institution. Word leaked out, as word always leaks out, about the Commission’s efforts and opinions, and this excited popular and media support. The Commission, not wanting to be on the wrong side of history, favored contraception. After all, this was a different world than that world which came before this world: or something.

After several years of glowing expectations, most expected Number 6 to endorse the Commission’s report. He did not.

Boy!, did tempers flare. To say the free-for-all crowd were displeased is a massive understatement. Number 6 and his Institution were ridiculed in the press and in academia and, indeed, by some leaders in Number 6’s Institution. One academic member of the Commission called Number 6’s predictions “that horrible document.” A prominent leader in Number 6’s Institution publicly charged Number 6 with “an anti-collegial act”. Ouch.

That fellow was far from alone. Many other leaders and groups of leaders castigated Number 6 openly. These dissidents went so far as to tell the common folk to ignore Number 6 and do what they please. And they did. And where they did do as they pleased, it was found that the Institution lost members.

Of course, it is not often remembered, perhaps willfully, that Number 6’s theory made stunningly accurate predictions, whereas his enemies’ theories made inaccurate ones.

The reason for that digression is important because again Number 6’s Institution will meet to discuss matters pertaining to human sexuality and the family. The meeting will go for at least two years. Experts will be confided in. Reports will be written.

As before, the press and a sizable chunk of leadership is on the side of liberalization; they particularly favor giving the nod to divorce but also to so-called same-sex marriage and perhaps even abortion. The world has changed, these people say, and therefore the Institution must also change—to become something that is not the Institution.

The Institution’s new leader Francis is being groomed by the liberalizers as the man with the plan, as somebody who is willing to set aside the old truths for new ones. These new dissidents are in the habit of parsing every public word of Francis’s to find in them support for their new truths. So adept are they at this that almost before Francis is done speaking, a news item or blog post is up saying, “Change we can believe in is coming!”

New predictions

My guess, working within Number 6’s theory, is whichever leader is in charge after the family synod is over will support tradition. The ban on contraception will be upheld. Marriage will be, as it can only be, declared to be between one man and one woman. Sodomy will still be a sin. Divorce will still be forbidden and not supported in the Institution’s activities. Abortion, if mentioned, will still be condemned.

The howling which will greet the announcement that there cannot be new truths, but only Truth, will be wondrous to behold, especially since, as before, liberalizers expect the vote to go with them. New dissidents will arise who, again as before, will tell people to ignore official proclamations and do what they want.

What will happen to rebellious families is obvious: more of what Number 6 said, a decrease in interest in marriage, increased state control over all things sexual, recognition that children belong to the state and not “parents”, and because of the dissolution of the family, an increase in support of euthanasia.

And people, even seeing the accuracy of these predictions, will still largely disbelieve the theory.

Update 14 May. It’s coincidence day at WMBriggs.com: Are Our Relationships Threatening The State?

Update 18 October 2014. Not that I want to brag, but it looks like the Truth Theory is still holding strong. But what a week!

The question is whether, after the conclusion of next year’s synod, Pope Francis will emulate his brother Number 6, or will he seek more worldly pastures?

I predict the former, in the following sense. I say Tradition holds, whether Pope Francis wants it to or not. It won’t matter what he or what anybody else wants, sin will still be sin. Doctrine will remain unchanged. Now, how will this Great Continuance happen? I have no idea. But I am reminded of the tale of Arius, a bishop who led one of the Church’s earliest heresies, a man whose power of convincing other Church fathers waxed and waned, but which never deserted him, and who, on his way to a final crucial meeting where he might have convinced others of his fallacy, had this happen to him:

It was then Saturday, and Arius was expecting to assemble with the church on the day following: but divine retribution overtook his daring criminalities. For going out of the imperial palace, attended by a crowd of Eusebian partisans like guards, he paraded proudly through the midst of the city, attracting the notice of all the people. As he approached the place called Constantine’s Forum, where the column of porphyry is erected, a terror arising from the remorse of conscience seized Arius, and with the terror a violent relaxation of the bowels: he therefore enquired whether there was a convenient place near, and being directed to the back of Constantine’s Forum, he hastened thither. Soon after a faintness came over him, and together with the evacuations his bowels protruded, followed by a copious hemorrhage, and the descent of the smaller intestines: moreover portions of his spleen and liver were brought off in the effusion of blood, so that he almost immediately died. The scene of this catastrophe still is shown at Constantinople, as I have said, behind the shambles in the colonnade: and by persons going by pointing the finger at the place, there is a perpetual remembrance preserved of this extraordinary kind of death.

The answer thus comes via Peter Kreeft, who is fond of quoting a Southern Baptist minister who managed to sum up the lessons of the Bible in four words. “I’m God. You’re not.”

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