Guest Post by Willis Eschenbach
OK, quick gambler’s question. Suppose I flip seven coins in the air at once and they all seven come up heads. Are the coins loaded?
Near as I can tell, statistics was invented by gamblers to answer this type of question. The seven coins are independent events. If they are not loaded the chances of a heads is fifty percent. The odds of seven heads is the product of the individual odds, or one-half to the seventh power. This is 1/128, less than 1%, less than one chance in a hundred that this is just a random result. Possible but not very likely. As a man who is not averse to a wager, I’d say it’s a pretty good bet the coins were loaded.
However, suppose we take the same seven coins, and we flip all seven of them not once, but ten times. Now what are our odds that seven heads show up in one of those ten flips?
Well, without running any numbers we can immediately see that the more seven-coin-flip trials we have, the better the chances are that seven heads will show up. I append the calculations below, but for the present just note that if we do the seven-coin-flip as few as ten times, the odds of finding seven heads by pure chance go up from less than 1% (a statistically significant result at the 99% significance level) to 7.5% (not statistically unusual in the slightest).
So in short, the more places you look, the more likely you are to find rarities, and thus the less significant they become. The practical effect of this is that you need to adjust your significance level for the number of trials. If the significance level is 95%, as is common in climate science, then if you look at 5 trials, to have a demonstrably unusual result you need to find something significant at the 99% level. Here’s a quick table that relates number of trials to significance level, if you are looking for the equivalent of a single-trial significance level of 95%:
Trials, Required Significance Level 1, 95.0% 2, 97.5% 3, 98.3% 4, 98.7% 5, 99.0% 6, 99.1% 7, 99.3% 8, 99.4%
Now, with that as prologue, following my interest in things albedic I went to examine the following study entitled Spring–summer albedo variations of Antarctic sea ice from 1982 to 2009 :
ABSTRACT: This study examined the spring–summer (November, December, January and February) albedo averages and trends using a dataset consisting of 28 years of homogenized satellite data for the entire Antarctic sea ice region and for five longitudinal sectors around Antarctica: the Weddell Sea (WS), the Indian Ocean sector (IO), the Pacific Ocean sector (PO), the Ross Sea (RS) and the Bellingshausen– Amundsen Sea (BS).
Remember, the more places you look, the more likely you are to find rarities … so how many places are they looking?
Well, to start with, they’ve obviously split the dataset into five parts. So that’s five places they’re looking. Already, to claim 95% significance we need to find 99% significance.
However, they are also only looking at a part of the year. How much of the year? Well, most of the ice is north of 70°S, so it will get measurable sun eight months or so out of the year. That means they’re using half the yearly albedo data. The four months they picked are the four when the sun is highest, so it makes sense … but still, they are discarding data, and that affects the number of trials.
In any case, even if we completely set aside the question of how much the year has been subdivided, we know that the map itself is subdivided into five parts. That means that to be significant at 95%, you need to find one of them that is significant at 99%.
However, in fact they did find that the albedo in one of the five ice areas (the Pacific Ocean sector) has a trend that is significant at the 99% level, and another (the Bellingshausen-Amundsen sector) is significant at the 95% level. And these would be interesting and valuable findings … except for another problem. This is the issue of autocorrelation.
“Autocorrelation” is how similar the present is to the past. If the temperature can be -40°C one day and 30°C the next day, that would indicate very little autocorrelation. But if (as is usually the case) a -40°C day is likely to be followed by another very cold day, that would mean a lot of autocorrelation. And climate variables in general tend to be autocorrelated, often highly so.
Now, one oddity of autocorrelated datasets is that they tend to be “trendy”. You are more likely to find a trend in autocorrelated datasets than in perfectly random datasets. In fact there was an article in the journals not long ago entitled Nature’s Style: Naturally Trendy . (I said “not long ago” but when I looked it was 2005 … carpe diem indeed.) It seems many people understood that concept of natural trendiness, the paper was widely discussed at the time.
What seems to have been less well understood is the following corollary:
Since nature is naturally trendy, finding a trend in observational datasets is less significant than it seems.
In this case, I digitized the trends. While I found their two “significant” trends in the Bellingshausen–Amundsen Sea (BS) at 95% and the Pacific Ocean sector (PO) at 99% were as advertised and they matched my calculations, unfortunately I also found that as I suspected, they had indeed ignored autocorrelation.
Part of the reason that the autocorrelation is so important in this particular case is that we’re only starting with 27 annual data points. As a result, we’re starting with large uncertainties due to small sample size. The effect of autocorrelation is to reduce that already inadequate sample size, so the effective N is quite small. The effective N for the Bellingshausen–Amundsen Sea sector (BS) is 19, and the effective N for the Pacific Ocean sector (PO) is only 8. Once autocorrelation is taken into account both of the trends were not statistically significant at all, as both were down around the 90% significance level.
Adding in the effects of autocorrelation with the effect of repeated trials means that in fact, not one of their reported trends in “spring-summer albedo variations” is statistically significant, nor even near to being significant.
Conclusions? Well, I’d have to say that in climate science we’ve got to up our statistical game. I’m no expert statistician, far from it. For that you want someone like Matt Briggs, Statistician to the Stars. In fact, I’ve never taken even one statistics class ever. I’m totally self-taught.
So if I know a bit about the effects of subdividing a dataset on significance levels, and the effects of autocorrelation on trends, how come these guys don’t? Be clear I don’t think they’re doing it on purpose. I think that this was just an honest mistake on their part, they simply didn’t realize the effect of their actions. But dang, seeing climate scientists making these same two mistakes over and over and over is getting boring.
To close on a much more positive note, I read that Science magazine is setting up a panel of statisticians to read the submissions in order to “help avoid honest mistakes and raise the standards for data analysis”.
Can’t say fairer than that.
In any case, the sun has just come out after a foggy, overcast morning. Here’s what my front yard looks like today …
The redwood tree is native here, the nopal cactus not so much … I wish just such sunny skies for you all.
Except those needing rain, of course …
w.
AS ALWAYS: If you disagree with something I or someone else said, please quote their exact words that you disagree with. That way we can all understand the exact nature of what you find objectionable.
REPEATED TRIALS: The actual calculation of how much better the odds are with repeated trials is done by taking advantage of the fact that if the odds of something happening are X, say 1/128 in the case of flipping seven heads, the odds of it NOT happening are 1-X, which is 1 – 1/128, or 127/128. It turns out that the odds of it NOT happening in N trials is
(1-X)N
or (127/128)N. For N = 10 flips of seven coins, this gives the odds of NOT getting seven heads as (127/128)10, or 92.5%. This means that the odds of finding seven heads in ten flips is one minus the odds of it not happening, or about 7.5%.
Similarly. if we are looking for the equivalent of a 95% confidence in repeated trials, the required confidence level in N repeated trials is
0.951/N
AUTOCORRELATION AND TRENDS: I usually use the method of Nychka which utilizes an “effective N”, a reduced number of degrees of freedom for calculating statistical significance.
where n is the number of data points, r is the lag-1 autocorrelation, and neff is the effective n.
However, if it were mission-critical, rather than using Nychka’s heuristic method I’d likely use a Monte Carlo method. I’d generate say 100,000 instances of ARMA model (auto-regressive moving-average model) pseudo-data which matched well with the statistics of the actual data, and I’d investigate the distribution of trends in that dataset.
[UPDATE] I found a better way to calculate effective N. See below.
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The way NOAA has adjusted the temperatures of the 20th century, based on what valid criteria I have no idea, how can anyone have any degree of confidence in what they put forward? And when you go back to the 19th century how does anyone purport to have an accurate record of global temperatures? What percentage of the world were temperatures even record in back then?
The chance of getting heads or tails is a bit less than 50%. Because there is a small chance for the coin to end up on the edge?
And it can happen. We have a small key that opens our letterbox. We are refurbishing our hallway, and currently have a bare concrete floor covered with a membrane. I threw the key onto the floor and didn’t hear the familiar sound of it bouncing. I looked back and it was on its edge! It was so astounding that I quickly called my wife to witness it. What’s even weirder is that the key is very thin, so its edge is about just 1 millimetre. This was the third thing (of a sort) to happen in a few weeks. On two occasions, separated by about four weeks, I threw a dishwasher tablet into the dishwasher and it landed on its edge. The second time it landed on its top edge (even less likely). At least the tablet has a wide edge, so the odds aren’t bad, but a key landing on its edge when it is so thin? Surely the odds are extraordinary?
Perhaps you have a natural magnet under your property?
Or maybe stuff just happens sometimes.
Yep, stuff happens sometimes, that’s all it is!
Check out the classic Twilight Zone episode “A Penney For Your Thoughts” to see what happens if you manage to get a coin to land on its edge.
Who said it is a small chance ?
Sometimes the coin toss before a game or such is done on a mat where the coin won’t bounce.
So what if you do the coin toss on a flat patch of beach sand, where an edge on coin can dig into the sand and stay there.
So now what is the probability of it being less than 45 degree tilt from perfectly edge on ??
“the odds of finding seven heads by pure chance go up from less than 1% (a statistically significant result at the 99% significance level) to 7.5% (not statistically unusual in the slightest)”
What would you consider “statistically unusual in the slightest”? 5.0%? Would 5.00001% then not be “statistically unusual in the slightest”? The difference between the two is insignificant in itself. Certainly 7.5% is statistically unusual in some sense, just not as statistically unusual as the arbitrarily-chosen threshold of 5.0%.
I once read the following comment by a person who taught statistics.
He assigned his students a task of flipping a coin 100 times and recording the sequence of heads and tails. He then took the results and informed the class which students had done the exercise and which had “dry-lab” the results, i.e. made then up. His answers were mostly correct.
His secret (which he conveyed to the students to make the point) was that most people think that the same result occurring three times in a row (e.g. three heads), and especially four or more times in a row, was very, very unlikely. The dry-lab results were those that had no (or very few) heads or tails occurring three or more times.
In reality these multiple occurrences are more common than people think.
(A little OT, but similar): Before announcing that the class topic is on pseudo random number generators, I tell my students to write down 5 digits of their choice. We then plot the frequency distribution of those choices. The frequency of the middle digits are higher than the tails, 0 & 1, and 8 & 9; and the frequency of the odd digits are higher than the even digits. This illustrates that people are biased in their choice of digits, as one would expect a more or less even distribution if the choices were done randomly.
I then have them write down 100 consecutive digits chosen at random, (in lines of 10 digits each where the first digit of a succeeding line follows the last digit of the preceding line.) After they’re done, I have them circle the number of pairs of same digits, and the number of triplets of same digits. Most of the students are concentrating so hard on avoiding such occurrences in their efforts to “be random” that they fail to meet the statistical average of 10 pairs and 1 triplet.
The conclusion is that generating pseudo random sequences is difficult, as humans aren’t randomly inclined when they aren’t trying to be random, and they aren’t randomly inclined when they are trying to be random.
Well, then there is the whole question of “Ought the numbers actual BE random?”. Yes, for a random number generator or a truly fair coin, but real coins are not always evenly balanced and all…
As per the ice data set, this “natural bias” comes from the existence of a natural 60 year ocean / weather cycle, a roughly 8 year Southern Ocean Antarctic Circumpolar Wave
https://en.wikipedia.org/wiki/Antarctic_Circumpolar_Wave
the 18 ish year Saros Cycle of lunar tidal forces
https://en.wikipedia.org/wiki/Saros_%28astronomy%29
and the 1500 year cycle of tides caused by lunar cycles as longer term influences have effect.
So we KNOW there will be various interacting cycles causing observed pseudo-trends in the data as “heads and tails” of those sine waves line up, or not, with the start and end of the data observed…
So how does it make sense to apply a test of non-random to a non-random data set to find ‘trend’?
Yes EM. Is that not the reason for “Autocorrelation”. However the example Willis gave was,
““Autocorrelation” is how similar the present is to the past. If the temperature can be -40°C one day and 30°C the next day, that would indicate very little autocorrelation. But if (as is usually the case) a -40°C day is likely to be followed by another very cold day, that would mean a lot of autocorrelation.”
This is an example of non randomness WITHIN the period of study, but your example is of non randomness outside the period of study. I do not know if autocorrelation corrects for non random trends outside the period of study.
Willis?
I spent a decade in grad school in a visual psychophysics lab . The bias of humans to disbelieve the frequency of long runs was pervasive . To make the point , here’s a couple of sequences of 100 pluses and minuses from a common random number generator which I’m sure will pass any simple chi^2 test or such .
++—–+-+-+-+-++-+-++-++-++–++++–+—+-+++-++-++-+++—–++—++++-+–++++–++-+-+-++—+++-+-+++
——-+—-+++-+–+-+-+++–+++–+—-+++-+—-+++-+-++-+——+-+-++–+++-+++—+—+++++++–++++–
Something that one can see every night on UK television where there is late night roulette.
One frequently sees runs of 3 and even 4 reds (or blacks) in a row. I have not looked for odd/even streaks because one would have to look at the numbers in detail, I merely channel hop over this late night gambling fad. But every night one can see this as they show the last dozen spins of the wheel, and certainly 3 of one of the two colours in a row is a common occurence.
Well I have used statistical mathematics; or some aspects of it, for over 50 years, in my daily work; which often included recording the results of repeated experiments; or “trials” as Willis calls them.
But I don’t flip coins, so I haven’t done what Willis has.
So I am all the time (or have been) computing the “average value” of some data set of numbers, along with things like standard deviations.
But in my case, the numbers in my data sets, had a common property. All of the members of my typical data set, were supposed; in the absence of experimental errors, to be exactly the same number.
So my purpose in averaging, was to tend to reduce the random experimental error in my result. Systematic errors, of course posed additional uncertainties, but absent that, my expectation was that the probable random error in my result would diminish in about the square root of the total number of trials, or experiments.
With the caveat, of excluding systematic errors, my expectation is that this statistical average is the best result I can get from that experiment or measurement.
Now that is quite different from tossing a coin, as in Willis’s trial.
With my luck, if I tossed seven coins, just once, all seven of them would likely land on edge, just to annoy me.
So I don’t toss coins, just in case, that should happen.
But the use of statistics, and averaging, related to climate “science”, seems to be quite a different proposition all together.
People in that field, seem to take single, non-repeatable measurements of different things (Temperatures ?? e.g. ) in quite different places, at quite different times, all of which should yield quite random unrelated results, with no expectation at all, that any of those measurements would be the same.
They then engage in what amounts to numerical Origami, during which they discard, all of these unrelated experimental observations, and replace them with either an entirely new and fictitious set of numbers, often referred to as “smoothing”, or else discard them all completely to be replaced by a single number; “the average.”
Now the algorithms of statistical mathematics are all described in detail, in numerous standard texts on that art form; and it is an art form, trying to make nothing out of something.
So you can take a very useful square of paper, on which you could write a nice poem, and by applying a simple algorithm, you can fold that useful piece of paper, and get an ersatz frog that can even jump; but is now much more difficult to right a poem on.
Well of course, the algorithms of statistical mathematics place very few restrictions on the elements of the data set.
The only requirements is that each of them is a real number. That is “real” in the mathematical sense, so NO imaginary, or complex numbers, and NO variables.
Each element is an exactly known number; although it is not necessarily the exact value of anything physically existing.
So the result of applying the algorithm is always exact, and any practitioner, applying the same algorithm, will get the exact same result from the same data set.
So statistics is an exact discipline, with NO uncertainty in the outcome.
And it always works on ANY set of numbers whatsoever that meet the “real” number condition.
The numbers of the data set, do not have to be related in any way. You could choose all the telephone numbers in your local phone yellow pages. Well you could also include the page numbers, or the street address numbers as well, or any subset of them.
So NO uncertainty surrounds the outcome of the application of ANY statistical mathematics algorithm.
Now if you like uncertainty, my suggestion is to look instead, not at the numbers you get from doing statistics, but at the absurd expectations for what meaning lies in that outcome.
There is NO inherent meaning, whatsoever. The result is just a number.
If you add the integers from 1 to 9 inclusive, you get a sum of 45 (always), and dividing by the number of elements (9) you get the average value of 4.5 (always) and as you can see it isn’t even one of the numbers in the data set.
The average value of all the phone numbers in your yellow pages, is likely to not even be a telephone number at all. Averaging numbers that aren’t even supposed to be the same, simply discards all those numbers in favor of a completely fictitious one.
So our self delusion, is in what we expect the outcome of our statistics to mean.
It means only what we choose it to mean. There is no inherent meaning.
Just my opinion of course. Most people (maybe 97%) would likely disagree with me.
Even some well-known scientists fall in that trap. Several recent posts have quoted papers by Mike Lockwood, whose grip on statistics is not the best. In his famous paper in Nature that the coronal magnetic field has more than doubled in the last hundred years, Lockwood claimed that his finding was significant at the 99.999999999996% level [he ignored or didn’t know about auto-correlation].
Doctor you are the best at getting to the point that I have ever seen! Please live many more years.
42
Which is ASCII for * , the wildcard character on DOS machines… essentially, it means match everything. So, the ultimate answer, is “everything”.
I took exactly 1 course is statistics when I studied engineering. It was an elective at that. Wish I had taken more. Who would think it’d be that useful? About the only thing I remember about it, besides how to calculate lottery odds (never have bought a ticket) is it’s usually easier to find the odds of something not happening and go from there – as you’ve shown above.
ps Willis, Plant some yellow leafed Japanese Maples under the Redwoods. The effect is excellent on those mostly cloudy days you have there.
The lottery is a tax on the mathematically challenged.
When it gets above $100 million, I get mathematically challenged,
Aside from being a highly regressive tax on people with poor math skills, state lotteries have some redeeming qualities. Students learning statistical mechanics can relate to the expectation value of the ticket exceeding its face value for a sufficiently high prize. For quantum mechanics, the ticket represents Schrödinger’s Cat; the wave function collapses at the drawing. This can then be extended to the probability density of electron position in an atom or molecule.
I also slip in that the students are about as likely to be killed in a traffic accident going to buy the ticket as they are to win the top prize. Bad sport I guess.
It’s good having some new teaching models to replace the old ones. I’ve often found myself trying to explain levers and leverage using the playground “see-saw” that’s familiar from my childhood — only to be reminded that youngsters today (in the US, at least) have never seen these. Liability problems, I guess.
The mathematics of statistics lead to some results that many will find nearly impossible to believe, if they rely on “common sense” to decide what is or is not true.
The odds of winning a certain version of the Florida Lotto were at one time about 23 million to one, because there were that many different combinations of the numbers possible.
Many people will take this to mean that if one buys one ticket per drawing, then after 23 million drawings, they will have won. Of course this is not true…some will win many times in that many drawings, and some will not win at all.
But what are the odds if one buys multiple tickets per drawing? Math tells me that if I buy ten tickets (all different of course), then my odds of winning have increased all the way up to one in 2.3 million. But I have had people who were otherwise seemingly intelligent insist that this was BS. No way can spending just ten dollars magnify the chance of winning that much, they said. What about all those other 20.7 million possible combinations, they point out.
Innumeracy takes many forms it seems.
My favorite statistical surprise though, is the more or less well known Birthday Problem: How many people must one gather in a room for the odds to be fifty-fifty that two of them (or more) will have the same Birthday? The answer is quite surprising, if you have never heard the problem before and are not able to instantly figure such things out in your head.
I have spent years hacking the math of state lottos…and you now give it away? Btw, my math was so successful in Illinois that they added two numbers (55, 56) to the system. RATS. Like banning Vegas blackjack card counters.
So you are the one! Odds are now over 50 million to one to win, plus they slashed the prizes for getting 3, 4, or 5 correct.
Anyway, those MIT folks gave the casinos a good go, though, huh?
Seriously though, I thought they added numbers because of those organized efforts to plan how to win: Groups would wait for a long time with no winner, when the jackpot would then become far higher than the odds against winning. The lottery then becomes, from a statistical point of view, a “good bet”.These groups then sent people out to purchase tickets, large numbers of tickets. In fact the goal would be to purchase one of every possible combination, thus ensuring a win. The only risk would be if others also won…or if they ought almost but not quite every combination and lost after spending millions of dollars.
By adding so many combinations, it became logistically far more problematic to ever buy one of each ticket in time.
But it could have been you, sir. I was just guessing as to why the changes.
Although, the steep odds also made giant jackpots more common, at times reaching a quarter or a half of a BILLION dollars!
Imagine…being one of the richest people in the country from a (now) two dollars ticket.
You want more imfo…?
Could you loan me $3.00?
Dr. Istvan,
I would love more info sir. Always appreciate your comments.
And I have zero imfo, so that would be a treat as well…assuming it is not a typo?
🙂
Mr. Robertson,
As soon as I collect my winnings from picking the place horse in race #7 at Churchill Downs, I will be flush and can spot you the $3.
🙂
On the Birthday problem, I don’t recall the precise number, but as I recall it is less than 30, maybe 22? I could, of course, look it up, but that would be cheating.
Our church prints out birthdays of all the members on the back of a directory. There are 192 names. There are 39 shared birthdays. That’s 1 in 5. There are shared birthdays in every month. In the month with the small number of names, 9, there are two sets of shared birthdays. It is a remarkable confirmation of this phenomenon. For the month of November, there are 24 names; there are 6 days in November with shared birthdays, as many as 4 one one day.If I were to go on the basis of this data set, I’d have to conclude that on average, you’ll find a shared birthday in any group much larger than 8 or 9..
If the lottery is pure and fair then there are two factors that determine the odds of winning a big prize:
1. The odds of choosing the winning number, and
2. The odds of other people also choosing the winning number.
Since the odds of the winning number being chosen is the same as the odds for any other number being chosen the focus of ones attention should be applied to number 2, which is the odds of other people choosing the same number as you choose.
Since some numbers are more popular than others one should focus his attention on uncovering the least popular numbers and then bet on these numbers exclusively.
Yeah, but maybe they are the least popular for a reason?
No matter what the stats tell me, I would be surprised as all get out to se the combination of 1-2-3-4-5-6 with a powerball of 7 ever come up.
The again, I was quite surprised when the winners were 8-17-23-25-41-49 w/ PB of 34…so what do I know?
“Yeah, but maybe they are the least popular for a reason?”
The reason probably has more to do with the thinking (or feelings) of human beings rather than the probabilities offered up by mathematics.
Certain numbers in certain cultures are deemed to have different powers than other numbers, meaning that in certain cultures some numbers are considered to have the power of being, say, “lucky” and other numbers are considered to have the power of being “unlucky”. The number 13 is seen by many people in the West as having the power of being unlucky. In Eastern cultures other numbers have similar powers.
Whatever the case, these powers of “lucky” or “unlucky” are powers bestowed upon numbers by humans and not bestowed on numbers by mathematics; A number does not know (nor does it behave as if it knows) whether it is a “lucky” number or an “unlucky” number.
If a person believes that some numbers are lucky and some numbers are unlucky then I believe this tells us more about the person than it does about the number.
Yes, I was kidding about that part. Sorry, thought it was obvious.
You are right of course, some people do have superstitions about numbers.
We had a technician where I work who was going to quit because he was assigned truck number 13.
When I reported this, to my surprise they changed the number. If someone thinks they are jinxed, I suppose that is as bad as actually being jinxed.
For the birthday problem, you multiply together the odds of people having separate birthdays, like this
((364/365) x (363/365) x (362/365) x (361/365), …..etc until the answer drops below 0.5.
As you say, the answer is suprisingly low – about 23 I seem to remember.
Okay, now that I posted, I looked up the answer and it is 23, so my guess of 22 was almost dead on.
There must be a different explanation for the results I gave about the church group. The statistics of the Birthday problem would say that in a group of 192, the odds are about 100 percent that there is at least one shared birthday. It doesn’t project the odds of the average number of shared birthdays. As I noted in my other post, there are 39 shared birthdays in the group of 192. What are the odds of that? Or, put differently, in a group of size N, how many shared birthdays are there likely to be on average?
I have heard about that problem applied to a soccer match, it is more common in Europe to tell it that way, I guess. The problem says: What is de probability that two people on the soccer pitch share a birthday?
There are 2 teams, 11 players per team, and the referee, total 23 people, and the probability is a bit more that 50 percent.
Church Question: Why assume all birthdays are equally probable?
9 months after Christmas is more likely – if only because targeting the child’s birth to make them the most mature in the school year is common. And couples have time off work at Christmas,
As for one shared birthday in each month, it’s only one sample. You’d need to see of that was common in about 30 churches of about 200 members to find if that’s significant.
M Courtney: In the East Neuk of Fife, according to the registrar when I registered my daughter’s birth, it’s 9 months after Hogmanay. 9I hasten to add I’m not a Fifer, I just happened to be stationed there at the time0
My two oldest kids were born in late September, and I once joked with a friend that this was about nine months after Christmas. He replied that his two kids had the same birthday. I, of course, asked if that was related to any particular date nine months before, and with a great big smile, he said “Yeah, my birthday!”
MCourtney,
My younger sister and I were both born on April 8th, two years apart.
And of my other seven siblings, two were on July 3rd.
“According to a public database of births, birthdays in the United States are quite evenly distributed for the most part, but there tend to be more births in September and October.[11] This may be because there is a holiday season nine months before (as nine months is the human gestation period), or from the fact that the longest nights of the year happen in the Northern Hemisphere nine months before as well. However, it appears the holidays have more of an effect on birth rates than the winter weather; New Zealand, a Southern Hemisphere country, has the same September and October peak with no corresponding peak in March and April.[12] The least common birthdays tend to fall around public holidays, such as Christmas, New Years and fixed-date holidays such as July 4 in the US. This is probably due to hospitals and birthing centres not offering labor inductions and elective Caesarean sections on public holidays.
Based on Harvard University research of birth records in the United States between 1973 and 1999, September 16 is the most common birthday in the United States and December 25 the least common birthday (other than February 29, because of leap years).[13] More recently[when?] October 5 and 6 have taken over as the most frequently occurring birthdays.[14]
In New Zealand, the ten most common birthdays all fall between September 22 and October 4. The ten least common birthdays (other than February 29) are December 24–27, January 1–2, February 6, March 22, April 1 and April 25.[12]
According to a study by the Yale School of Public Health, positive and negative associations with culturally significant dates may influence birthrates. The study shows a 5.3 percent decrease in spontaneous births and a 16.9 percent decrease in cesarean births on Halloween, compared to other births occurring within one week before and one week after the October holiday. Whereas, on Valentine’s Day there is a 3.6 percent increase in spontaneous births and a 12.1 percent increase in cesarean births”
https://en.wikipedia.org/wiki/Birthday
Birthdays are definitely autocorrelated. There was a big birth spike in Houston 9 months after Ike. My own kid included.
Also, you have a spike of honeymoon babies in February-March from the May-June wedding season, and as mentioned, the September bulge from Christmas Break.
The one that gets people the best is the Monty Hall Problem. I told it to someone who I know is a maths fan, and he couldn’t handle it. He even said I was wrong, so I had to get him to Google it. Likewise, my brother-in-law (much more clever at maths than I could ever be) said the same – that it can’t be so. The reason is that people forget it isn’t a 50/50 chance. It’s 2/3 if you change.
Yes, very interesting…see below.
Birthday problem, feh. My job includes running the database of researchers at our facility, so I learn a lot of birthdays; I’ve entered 4000+ people since I started, and only one shares my birthday, and not one of the 300 staff in my building does. Also, you can figure out what that day is — scroll through one of those calendars that list “important events on this date in history”, and it’s the day where you reeeally have to stretch your definition of “important”.
🙂
You are citing a very different problem. To get the odds of any given birthday matched to 50% takes a much, much larger population, and as is pointed out below, birthdays are not purely random, so it further depends on what the birthday is.
Taylor
That was one of my favorite openings with my new math classes when I taught high school. Because the breakeven number is in the 20s, it was a perfect case to use in classes, which typically numbered in the low 30s. I’d first make a bet with them that two had the same birthday, they would invariably disagree, and then we’d go around the room – never failed once…
Taylor
At somewhere in the mid thirties, the certainty gets near 99%.
I had a somewhat headed debate in a Chili’s back in grad school. A fellow student was talking about improving the odds of winning the lottery, and he said you could start by eliminating all combinations of even numbers, all combinations of odd numbers, all combinations of sequential numbers, etc, because those sequences rarely happened. He was incapable of understanding that every single combination – assuming no loaded balls – was just as likely to happen as another. I couldn’t believe an engineering grad student would fail to realize something so basic in probability.
Menicholas, by “same birthday” do you mean “born on the same day–say January 5, of specifically 1960;” or do you mean “born on the same calendar day–i.e., January 5 of any year?” If the people are selected from a classroom of students in grades 1 through 12, the two answers (the number of people required to have a fifty-fifty chance that two or more people were born on the same day, and the number of people required to have a fifty-fifty chance that two or more people were born on the same calendar day) won’t differ that much because most students in such classrooms are of a common age. However, if the people are selected from a less age-restricted distribution (say the population of a large city), the two answers will be markedly different.
IMO, the “birthday problem” as commonly stated is an example of the Bertrand Paradox–see https://en.wikipedia.org/wiki/Bertrand_paradox_%28probability%29: “Joseph Bertrand introduced it in his work Calcul des probabilities (1889) as an example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined.” Specifically, as commonly stated the “birthday problem” doesn’t specify how the “people are chosen;” or if as is reasonable to assume they are chosen randomly, doesn’t specify the method that produces the random variable.
Mr. Coray,
The question as I have heard it is specifically about same birthday, not birth date.
As are the stats quoted.
But you raise a valid point.
“Note the distinction between birthday and birthdate: The former occurs each year (e.g. December 18), while the latter is the exact date a person was born (e.g., December 18, 1998).”
https://en.wikipedia.org/wiki/Birthday
And the whole thing apparently assumes a random assemblage of people.
A meeting of the Aries Babies That Are Also Oxes Club will have a different result, I would think.
I only ever met one other person in my adult life who was also one, and she and I had a happy twelve years together.
🙂
I tell people that choosing 1, 2, 3, 4, 5 and 6 has the same chance of winning as any other number, and they don’t believe me.
But to increase your chances of being the ONLY winner of the big jackpot, you should choose numbers above 31, as many people include their birthday in their selection.
BTW, nice opuntia. I sure would love a cutting. Have none with yellow flowers.
Most of the opuntia in the Arizona desert have yellow flowers, along with the chollas.
Mine have red.
Willis, I am not a self taught statistician. Am a Harvard taught PhD level econometrician amongst other fairly useless Harvard taught avocations. What you have just posted is truly beautiful. In two senses. 1. Skip the complicated math, just use the common sense intuitions that underly all math formalisms. 2. Illustrate simply. Five sectors increase significance odds>5. You should become a teacher…belay that, sailor, you already are. Amply evidenced here.
Highest regards on a ‘beautiful’ post.
+1!
One hundred scientists independently decide to try the same experiment. Five of them find significant results, and 95 don’t. Which studies get published?
Great corrollary to Willis’ main theorem.
GREEN JELLY BEANS CAUSE ACNE
http://www.explainxkcd.com/wiki/index.php?title=882:_Significant
\
http://www.explainxkcd.com/wiki/images/3/3f/significant.png
Stats can be so misleading. My fav statistics problem: What is the probability that at least 2 people have their birthdays on the same day in a party of 20 people?
About 40%.
With two people there is one pair, so the probability is 1/365
With three people there are three distinct pairings , so the probability is 3/365
With four people, there six distinct pairings, so the probability is 6/365
With five people there are 10 distinct pairings, so the probability is 10/365
With six people there are 15 distinct pairings, so the probability is 15/365
With seven people there are 21 distinct pairings, so the probability is 21/365
8—>28/365
9—>36/365
10—>45/365
11—>55/365
…..
20 gives (20*19)/(365*2) or 190/365…..about 0.52
and
N gives
“””””””””””””””””””””””””””””””””””””””””””””””””””
Opps
N gives N times N-1 divided by (2) divided by 365
Hmm. I am not a statistician either, but I do know how to look stuff up:
https://en.wikipedia.org/wiki/File:Birthday_Paradox.svg
“20 gives (20*19)/(365*2) or 190/365…..about 0.52”
But:
“in fact the ‘second’ person also has total 22 chances of matching birthday with the others but his/her chance of matching birthday with the ‘first’ person, one chance, has already been counted with the first person’s 22 chances and shall not be duplicated), ”
https://en.wikipedia.org/wiki/Birthday_problem#/media/File:Birthday_Paradox.svg
A friend of mine was born on February 29th. 😉
Again, another population-dependent example. If everyone was born in the same year, then the required number decreases only slightly (366 choices vs. 365). However, if it’s a random age group, then chances are diminished a bit more, because it’s more difficult to get a pairing. Of course, if you’re trying to match that specific person, it is more difficult still (see my comment above). It’s also worth noting that leap year introduces a complexity into the general calculation that is not considered in any of the examples so far, since any solution should include assumptions about the probability of 2/29/XX birthdays in the population. Calculating that should not be attempted by the faint hearted.
Taylor
OK, all you statisticians, who is gonna win the 7th at Churchill Downs ?
I’ll take the # 6.
23 minutes to post.
Is it raining?
Because the 4 horse’s mother was a mudder. And his father was a mudder.
If it is raining, put me down for #4.
But if it is not, I will go with the favorite #2 Adhara at 3:2, on general principle.
Funny, but this ain’t about principles 🙂
It went 5/2/6.
Drats 🙂
Woo hoo!
The general principle is that the favorite usually wins. Paramutual wagering leads to the correct horse being the favorite more often than not.
If it was that easy, there wouldn’t be so many people living under the bridges near the racetracks.
Ya think ??
Easy Nellie !!
Yes, you are right. I was referring to the wisdom of crowds, but this is likely not a valid exercise in such.
My pappy played the ponies…I am a poker man.
Elegant post Willis, I having no statistical education and being a horrid gambler with minimal math skill I struggle with understanding autocorrelation in subdivided datasets and significance in extrapolated trendiness. What interests me is the data sets themselves, and the seemingly recent and intensified data adjustments. I wonder how any trend or statistical significance can be arrived at when the data is adjusted and more disturbing (maybe) data infilling from vast regions of inadequately collected data and further distressing is the measurements taken from poorly sited instrument stations save of course satellite sampling. My intuition screams bloody murder that the data itself and that much of what I read regarding the ever rising global temperatures are adjusted (always) so that the surface and troposphere is warming worse than we thought and always CO2 is the main driver of said warming. I wonder what is the statistical significance of routine data adjustments? If this makes any sense I will appreciate any thoughts !
I am surprised that autocorrelation was not corrected for. I did one introductory class in Econometrics and that is one of the first things you get taught after the basics of OLS, R^2, t-tests etc. It should be noted that there can be both positive autocorrelation and negative. Positive AC is when you get longish trends and negative AC is when the data tends to fluctuate from one period to the next. Both can be tested for with simple statistical tests (e.g. Durbin-Watson). Heteroscedasticity is a fun one BTW – when the variance of the residual (error term) is not constant. Changing variance … only a statistician could dream up such a concept. I must tip my hat to Ron Oaxaca of UofA for being such a tough but fair econometrics teacher.
Bully
+10 for mentioning your teacher alone!
Nice!
Auto
Thanks Auto – I was surprised to see he is still teaching there. I was at UofA about 25 years ago.
The difference between a coin flip and a study of ice in the Antarctic is that you can calculate the probability of the coin before hand. You can’t know what the probability of the ice is except by testing it year on year and even then you can never know which results are random anomalies and which are expected due to the energy received. Therefore, to assign a probability to these events at all is completely meaningless. You can have 10,000 years of data, then suddenly get what seems like a freak result in the present. You won’t know if it really is a freak event or if it signals a change in climate conditions until you have waited for 5, 10 or even 100 years and can look back.
Here is a question on climate statistics.
If you travel to Yamal, cut samples from 34 trees and discard 33 of them, what are your chances of retaining the one sample that proves a theory?
That depends. Is the person doing the selecting, and deciding if it is proved, a warmista?
If so the odds are 97% in favor of a consensus saying it is proven, but later found to be completely wrong.
Depends. Is it to be used right-side-up or upside-down?
Ad where the heck is Yamal?
Are you referring to Siberia?
Brrr…no thankee!
Do they have trees there?
Is it not all permafrost?
I once spent 3 days on the trans-Siberian express. I reckon I have seen about half the world’s trees.
Richard B
yes, the tour companies promote the Siberian tree-watch – and even some of the Canadian 50 mph to another-place-with trees. And whilst I dare say Jasper and its ilk – in each of the biggest countries on the plant – have some fascinating side roads etc. – I know I’d rather do the Caucasus or the Atlantic Provinces – on a pleasure trip.
Pay me to look at trees – fine – everyone has a price.
Auto, who would rather look at oceans and seas
“However, they are also only looking at a part of the year. How much of the year? Well, most of the ice is north of 70°S, so it will get measurable sun eight months or so out of the year. That means they’re using half the yearly albedo data. The four months they picked are the four when the sun is highest, so it makes sense … but still, they are discarding data, and that affects the number of trials.”
I think it makes a difference, whether they decided which months to used before or after the examined the data.
Likewise, it makes a difference into how many geographic sectors they divide the arctic circle, and where they put the divisions.
In the latter case, I’m willing to assume they chose the sectors (based on the five seas) before making observations or calculations, and suspend suspicion. And, for myself, I’m willing to agree that looking at the albedo when the sun is highest, as you say “makes sense”, and again suspend suspicion.
But as you say, auto-correlation is another matter.
It is not a coin that is an example of an event in Mr. Eschenbach’s example but rather is a flip. Before conducting a scientific study of ice in the Antarctic, one would have to identify the functional equivalent of a flip. In designing a study, a climatologist doesn’t usually do that. This has dire consequences for the usefulness of the resulting study. It has dire consequences for us when a politician makes conclusions from such a study the basis for public policy.
Under the generalization of the classical logic that is called the “probabilitic logic,” every event has a probability. Usually, one is uncertain of the value of this probability. This value can be estimated by counting the outcomes of events in repeated trials. These counts are called “frequencies.” With rare examples climatological studies don’t produce them. In fact they can’t produce them because the events that need to be counted were not described by the study’s designer.
It is by counting events that scientists provide us with information about conditional outcomes of the future. By ensuring that this information cannot be provided by a climatological study, climatologists ensure that their works are completely useless.
All statistics are tools for analyzing an existing, or past populations.
Any predictions or forecasts are based on the unstated and overwhelming assumption that the future will behave as it has in the past.
Predictions are made by statisticians, not statistics!
The authors are careful to say don’t use these results out of sample. However as others have said without any prior discussion of hypothesized processes the concept of significance applied to the regression is meaningless. The observations are what they are and the linear trend is one of many calculations able to be derived from them.
To suggest significance the authors need to be arguing they are hypothesizing a linear model for the relationships with certain characteristics.
This is an investigative study and significance testing has no place in it.
A quote from William Briggs – “If we knew what caused the data, we would not need probability models. We would just point to the cause. Probability models are used in the absence of knowledge of cause.”
Which is why I have mixed feelings about “Six Sigma” – it implies a great deal of ignorance about your process. OTOH, used correctly, Six Sigma can be useful in reducing the ignorance.
Slywolfe:
That “any predictions or forecasts are based on the unstated and overwhelming assumption that the future will behave as it has in the past” is the position that was taken by the philosopher David Hume in relation to the so-called “problem of induction.” The process by which one generalizes from specific instances is called “induction.” How to justify the selected process is the “problem of induction.” “If one has observed three swans, all of them white, what can one logically say about the colors of swans in general?” is a question that can be asked in light of this problem.
The problem of induction boils down to the problem of how to select those inferences that are made by a scientific theory. Traditionally the selection is made by one or another intuitive rules of thumb. However, in each instance in which a particular rule selects a particular inference, a different rule selects a different inference. In this way, the law of non-contradiction is violated by this method of selection. In Hume’s day, there was no alternative to this method. Thus, over several centuries of the scientific age the method by which a scientific theory was created was “illogical” in the sense of violating the law of non-contradiction. This scandal lurked at the heart of the scientific method.
Modern information theory provides an alternative that satisfies the law of non-contradiction. There is a generalization of the classical logic that is called the “probabilistic logic.” It is formed by replacement of the classical rule that every proposition has a truth-value by the rule that every proposition has a probability of being true. In the probabilistic logic, an inference has the unique measure that is called its “entropy.” Thus, the problem of induction is solved by an optimization in which the entropy of the selected inference is maximized or minimized under constraints expressing the available information. Among the well known products of this method of selection is thermodynamics. The second law of thermodynamics is an expression of this method.
I wish more people here would take up the study of Information Theory. I have suggested it many times, but no one seems interested. Besides what you have said, it is within Information Theory that one will find the proper treatment of such things as Measurement Uncertainty. Classical Statistics, alone, is not a powerful enough tool to analyze the data that is used in the study of climate.
Well said!
I can tell that what you are saying here is important, but I am not sure I completely understand the point.
I will have to read up on “the law of contradiction”. I do not recall hearing this term used before now.
Can you clarify at all just what this means?
Thanks.
I’ll try to clarify. The classical logic of Aristotle and his followers contains three “laws of thought.” One of these is the law of non-contradiction (LNC). Let ‘A’ designate a proposition. The LNC is the proposition NOT [ A AND (NOT A) ] where ‘NOT’ and ‘AND’ are the logical operators of the same name.
The rules of thumb aka heuristics that are usually used by scientists in selecting the inferences that will be made by their theories violate the LNC, for there is more than one rule of thumb each selecting a different inference. One commonly used rule of thumb selects as the estimator of the value of a probability that estimator which is the “unbiased” estimator. Despite the attractiveness of the term “unbiased” and ugliness of the term “biased” the “unbiased” estimator has the unsavory property of fabricating information. An estimator that fabricates no information is “biased” under the misleading terminology that is currently taught to university students. That they are misled in this way skirts the issue of the identity of that “biased” estimator which fabricates no information. The latter estimator is information theoretically optimal.
Can you provide an example or two of this? I would like to understand.
Menicholas:
Your interest seems to be in how one can construct a model without assuming the future will behave as it has in the past. One can do this by selecting optimization as the method by which the inferences made by a model are selected rather than the method of heuristics. Thermodynamics provides an example that will be familiar to those who have studied engineering or the physical sciences. The induced generalization maximizes the missing information per event (the “entropy”) of our knowledge of the microstate of the thermodynamic system under the constraint of energy conservation. An assumption that the future will behave as it has in the past is not made. This assumption is replaced by optimization.
Mr. Oldberg,
“However, in each instance in which a particular rule selects a particular inference, a different rule selects a different inference. In this way, the law of non-contradiction is violated by this method of selection.”
Can you provide an example or two of this? I would like to understand.
“The induced generalization maximizes the missing information per event (the “entropy”) of our knowledge of the microstate of the thermodynamic system under the constraint of energy conservation.”
Well, if it was that simple, why didn’t you say so to begin with *wrinkles nose and ponders what lies beyond edge of universe*
The message is that simple but is hard to decode if one’s background in statistical physics is weak.
Such bad statistics are why the need for Raising the bar on statistical significance
Valen Johnson proposed: Revised standards for statistical evidence PNAS
Yes, I was taught that 0.05 level of significance barely qualified as a valuable hint that something should be further investigated.
I suggest “Statistics Done Wrong: The Woefully Complete Guide” http://smile.amazon.com/Statistics-Done-Wrong-Woefully-Complete/dp/1593276206 as a good tome on this topic. Basically, a lot of things that are “significant”, just plain are not.
I have no affiliation with the author of this book.
Hmmm. What are the odds that every “innocent statistical error” just happens to fall on the side of confirming “climate catastrophe?”
Thanks, Reality, but I’ll repeat what I said. I don’t think that the authors have any bad intent or any desire to deceive. I think they truly believe their results. I think that we are looking at a combination of woeful statistical ignorance and the usual confirmation bias. This confirmation bias is the most likely explanation for the direction of the errors of mainstream climate science.
Unfortunately, we also see the same thing on the other side of the aisle … hopefully not as often, but we’re all bozos on this bus …
w.
It strikes me that confirmation bias is a form of auto-correlation. Yes? No?
“This confirmation bias is the most likely explanation for the direction of the errors of mainstream climate science.”
Yeah, kind of like believing in Jack and The Beanstalk. Shouldn’t mainstream climate science have grown out of it?
This week’s lesson in semantics…
My use of “innocent” automatically brought up the antonym “guilty” to Mr. Eschenbach – and undoubtedly to several here. I should have used “inadvertent,” “unintentional,” “unconscious” (or several other words) instead.
Correctly, the fact of incorrect statistical analysis does NOT imply fraudulent intent by the researcher(s). (The fact that the majority of such errors support only one possible conclusion does, however, indicate some sort of bias in the process, as noted.)
Of course, the definite finding of fraud committed by one or more researchers DOES strongly indicate that that particular set of researchers, and their conclusions at any time (past or future) are fraudulent.
Willis says, ” we are all bozos on this bus” .I like it. A wise man once said, “Those of us to good for this world, are adorning some other”
Now a quick question regarding autocorrelation. You stated…
““Autocorrelation” is how similar the present is to the past. If the temperature can be -40°C one day and 30°C the next day, that would indicate very little autocorrelation. But if (as is usually the case) a -40°C day is likely to be followed by another very cold day, that would mean a lot of autocorrelation.”
This is an example of non randomness WITHIN the period of study, but E.M Smith above gave an example is of non randomness outside the period of study. (The known 60 year ENSO swings, among others)
Does autocorrelation corrects for non random trends outside the period of study?
Climate Science random number generator in “Dilbert”.
http://web.archive.org/web/20011027002011/http://dilbert.com/comics/dilbert/archive/images/dilbert2001182781025.gif
Tossing coins I suppose
Is the same, more or less,
As the computer modeled predictions
They’re using to guess?
http://rhymeafterrhyme.net/false-predictions/
The models that have been used in making policy on CO2 emissions do not make “predictions.” They make “projections.” For predictions there are events. For projections there are none. To conflate “prediction” with “projection” is common and is the basis for applications of the equivocation fallacy in making arguments about global warming ( http://wmbriggs.com/post/7923/ ). Applications of this fallacy rather than logical argumentation are the basis for global warming alarmism.
Projections become predictions
With alarmists like Al Gore,
Making statements about the future
Is what he’s known for.
Predictions are hard…especially about the future.
-Y. Berra
Had to laugh for sure! Good shootin’ Will!
dawtgtomis:
Will’s shootin’ missed his target (if any). Thus, congratulations to Will on his marksmanship are misplaced.
Terry O., when they said “our grandchildren will not know what snow is” (way back when) was a prediction, was it not?
dawtgtomis:
Properly defined, a “prediction” is an extrapolation from an observed state of nature to an unobserved but observable state of nature. For example, it is an extrapolation from the observed state “cloudy” to the unobserved but observable state “rain in the next 24 hours.”
The claim that “our grandchildren will not know what snow is” does not match this description. What is the observed state of nature? What is the unobserved but observable state of nature? These questions lack answers.
I think it is acceptable to break the Antarctic into five separate regions and test for significance without correction – if one had a reason before hand to expect those regions to behave differently. For example, climate models might predict different trends in different regions. Otherwise, sub-dividing into five parts provides four extra opportunities to find “significance” by chance.
Human clinical trials of new drug candidates provides an excellent opportunity to find “significance by chance”. If your drug didn’t provide conclusive evidence of efficacy (p<0.05) with the total patient population treated, then one can look only at only the males, or the female, or the younger patients, or the sicker patients, or those with marker X, Y, or Z etc – and find "conclusive" evidence of efficacy. Then you devise a "compelling rational" for why the efficacious sub-population responded better than the whole treatment group. Fortunately, the FDA has its own statisticians who are motivated to reject such shenanigans – unlike most peer reviewers of climate science papers. They ask the sponsoring company why they included patients less likely to respond in the clinical trial if they had a compelling rational. Then, they tell the sponsor to run a new clinical trial limited to patients now expected to respond to the drug and prove efficacy in this population is not due to chance.
The government's What Works Clearinghouse reviews published studies (many peer reviewed) on education programs and rejects more than 90% for a variety of methodological and statistical flaws. Their results are online.
compelling rationale
rhymes with morale