Archive for the Bad Statistics Category

GAA Clustering

Posted in Bad Statistics, GAA, The Universe and Stuff with tags , , , , , , on July 25, 2022 by telescoper
The distribution of GAA pitches in Ireland

The above picture was doing the rounds on Twitter yesterday ahead of this year’s All-Ireland Football Final at Croke Park (won by favourites Kerry despite a valiant effort from Galway, who led for much of the game and didn’t play at all like underdogs).

The picture above shows the distribution of Gaelic Athletics Association (GAA) grounds around Ireland. In case you didn’t know, Hurling and Gaelic Football are played on the same pitch with the same goals and markings on the field. First thing you notice is that the grounds are plentiful! Obviously the distribution is clustered around major population centres – Dublin, Cork, Limerick and Galway are particularly clear – but other than that the distribution is quite uniform, though in less populated areas the grounds tend to be less densely packed.

The eye is also drawn to filamentary features, probably related to major arterial roads. People need to be able to get to the grounds, after all. Or am I reading too much into these apparent structures? The eye is notoriously keen to see patterns where none really exist, a point I’ve made repeatedly on this blog in the context of galaxy clustering.

The statistical description of clustered point patterns is a fascinating subject, because it makes contact with the way in which our eyes and brain perceive pattern. I’ve spent a large part of my research career trying to figure out efficient ways of quantifying pattern in an objective way and I can tell you it’s not easy, especially when the data are prone to systematic errors and glitches. I can only touch on the subject here, but to see what I am talking about look at the two patterns below:

You will have to take my word for it that one of these is a realization of a two-dimensional Poisson point process and the other contains correlations between the points. One therefore has a real pattern to it, and one is a realization of a completely unstructured random process.

random or non-random?

I show this example in popular talks and get the audience to vote on which one is the random one. The vast majority usually think that the one on the right that  is random and the one on the left is the one with structure to it. It is not hard to see why. The right-hand pattern is very smooth (what one would naively expect for a constant probability of finding a point at any position in the two-dimensional space) , whereas the left-hand one seems to offer a profusion of linear, filamentary features and densely concentrated clusters.

In fact, it’s the picture on the left that was generated by a Poisson process using a  Monte Carlo random number generator. All the structure that is visually apparent is imposed by our own sensory apparatus, which has evolved to be so good at discerning patterns that it finds them when they’re not even there!

The right-hand process is also generated by a Monte Carlo technique, but the algorithm is more complicated. In this case the presence of a point at some location suppresses the probability of having other points in the vicinity. Each event has a zone of avoidance around it; the points are therefore anticorrelated. The result of this is that the pattern is much smoother than a truly random process should be. In fact, this simulation has nothing to do with galaxy clustering really. The algorithm used to generate it was meant to mimic the behaviour of glow-worms which tend to eat each other if they get  too close. That’s why they spread themselves out in space more uniformly than in the random pattern.

Incidentally, I got both pictures from Stephen Jay Gould’s collection of essays Bully for Brontosaurus and used them, with appropriate credit and copyright permission, in my own book From Cosmos to Chaos.

The tendency to find things that are not there is quite well known to astronomers. The constellations which we all recognize so easily are not physical associations of stars, but are just chance alignments on the sky of things at vastly different distances in space. That is not to say that they are random, but the pattern they form is not caused by direct correlations between the stars. Galaxies form real three-dimensional physical associations through their direct gravitational effect on one another.

People are actually pretty hopeless at understanding what “really” random processes look like, probably because the word random is used so often in very imprecise ways and they don’t know what it means in a specific context like this.  The point about random processes, even simpler ones like repeated tossing of a coin, is that coincidences happen much more frequently than one might suppose.

I suppose there is an evolutionary reason why our brains like to impose order on things in a general way. More specifically scientists often use perceived patterns in order to construct hypotheses. However these hypotheses must be tested objectively and often the initial impressions turn out to be figments of the imagination, like the canals on Mars.

Research Excellence in Physics

Posted in Bad Statistics, Education, Science Politics on May 16, 2022 by telescoper

For no other reason that I was a bit bored watching the FA Cup Final on Saturday I decided to construct an alternative to the Research Excellence Framework rankings for Physics produced by the Times Higher last week.

The table below shows for each Unit of Assessment (UoA):  the Times Higher rank;  the number of Full-Time Equivalent staff submitted;  the overall percentage of the submission rated  4*;  and the number of FTE’s worth of 4* stuff (final column), by which the institutions are sorted. The logic for this – insofar as there is any – is that the amount of money allocated is probably going to be more strongly weighted to 4* (though not perhaps the 100% I am effectively assuming) than the GPA used in the Times Higher.

1. University of Oxford 9= 171.3 57 97.6
2. University of Cambridge 3 148.2 64 94.8
3. Imperial College 18= 130.1 49 63.7
4. University of Edinburgh 13= 118.0 51 60.2
5. University of Manchester 2 87 66 57.4
6. University College London 24= 112.5 42 47.3
7. University of Durham 23 84.2 45 37.9
8. University of Nottingham 7 63.9 59 37.7
9. University of Warwick 20 79.2 47 37.2
10. University of Birmingham 4 55.2 66 36.4
11. University of Bristol 5 54.1 61 33.0
12. University of Glasgow 12 58.2 53 30.8
13. University of York 13= 59.9 51 30.5
14. University of Lancaster 21 56.1 46 25.8
15. University of Strathclyde 13= 46.7 52 24.3
16. Cardiff University 18= 52.2 46 24.0
17. University of Exeter 22 49.4 48 23.7
18. University of Sheffield 1 34.7 65 22.5
19. University of St Andrews 8 40.8 55 22.4
20. University of Liverpool 16 44.4 49 21.7
21. University of Leeds 9= 34 53 18.0
22. University of Sussex 26 42.7 42 17.9
23. The University of Bath 24= 38.8 42 16.3
24. Queen’s University of Belfast 31 49.7 32 15.9
25. Queen Mary University of London 28= 48 33 15.8
26, University of Southampton 27 41.7 38 15.8
27. The Open University 32= 41.8 36 15.0
28. University of Hertfordshire 38 42 32 13.4
29. Liverpool John Moores University 17 25.8 50 12.9
30. Heriot-Watt University 9= 21 55 11.6
31. King’s College London 28= 33.9 34 11.5
32. University of Portsmouth 6 19.8 58 11.5
33. University of Leicester 35= 34.3 28 9.6
34. University of Surrey 35= 30.6 31 9.5
35. Swansea University 32= 25.2 32 8.0
36. Royal Holloway and Bedford New College 35= 19.1 36 6.9
37. University of Central Lancashire 39 19.3 25 4.8
38. Loughborough University 40 19.8 22 4.4
39. University of Keele 32= 9 38 3.4
40. The University of Hull 30 11 28 3.1
41. University of Lincoln 43 15.2 16 2.4
42.The University of Kent 41 19 12 2.3
43. Aberystwyth University 44 18.2 7 1.3
44. University of the West of Scotland 42 8 11 0.9

Using this method to order institutions produces a list which clearly correlates with the Times Higher ordering – the Spearman rank correlation coefficient is + 0.75 – but there are also some big differences. For example, Oxford (=9th in the Times Higher) and Cambridge (3rd) come out 1st and 2nd with Imperial (=18th in the Times Higher) moving up to 3rd place. Edinburgh moves up from =13th to 4th. The top ranked UoA in the Times Higher table is Sheffield, which drops to 18th in this table. Portsmouth (6th in the Times Higher) drops to 32nd in this version. And so on.

Of course you shouldn’t take this seriously at all. The lesson -if there is one – is that the use of the Research Excellence Framework results to produce rankings is a bit arbitrary, to say the least…

Census Day

Posted in Bad Statistics, Biographical, The Universe and Stuff with tags , on April 3, 2022 by telescoper

Today is April 3rd 2022 which means that it’s Census Day here in Ireland; I’ve just finished filling in the form, which is 24 pages long but it turns out lots of the pages are duplicates for use in homes with multiple occupancy, and others don’t apply to me at all, so in fact I only had to complete 8 pages and it didn’t take all that long.

The Census should have taken place last year but was postponed because of the Covid-19 pandemic. Apparently the corresponding 2021 census in the UK went ahead, though I wasn’t at, and couldn’t get to, the property I still own in Wales so couldn’t participate. Although I was initially threatened with a fine, the UK Census people seem to have given up trying to chase me. I blogged about the previous census in Wales in 2011 here.

On the holiday after St Patrick’s Day I was at home when I noticed a card had been pushed through my letterbox while I was still in the house. It was from a ‘Census Enumerator’ who said he had tried to deliver the form but I was out. I wasn’t out and he hadn’t rung the doorbell. More importantly he hadn’t simply put the census form through the letterbox. In the UK the census forms are just sent out in the post. This little episode didn’t inspire me with confidence. Anyway, the bloke came back a week later and gave me the form. He also asked me for some personal information such as my phone number, which I naturally refused to give him. Apparently he has to collect the form in person too, which seems daft to me. Why can’t people just send their census returns back in the post?

On the last page there is a so-called ‘time capsule’ in which to leave information for historians to read 100 years from now. All I could think of to write was any historians reading this in 2122 would probably think that it was absurd to be doing this wasteful paper-based census when the digital age started some time ago, so I just said for the record that I was one of the people who thought that in 2022…

Solar Corona?

Posted in Bad Statistics, Covid-19, mathematics, The Universe and Stuff on December 8, 2021 by telescoper

A colleague pointed out to me yesterday that  evidence is emerging of a four-month periodicity in the number of Covid-19 cases worldwide:

The above graph shows a smoothed version of the data. The raw data also show a clear 7-day periodicity owing to the fact that reporting is reduced at weekends:

I’ll leave it as an exercise for the student to perform a Fourier-transform of the data to demonstrate these effects more convincingly.

Said colleague also pointed out this paper which has the title New indications of the 4-month oscillation in solar activity, atmospheric circulation and Earth’s rotation and the abstract:

The 4-month oscillation, detected earlier by the same authors in geophysical and solar data series, is now confirmed by the analysis of other observations. In the present results the 4-month oscillation is better emphasized than in previous results, and the analysis of the new series confirms that the solar activity contribution to the global atmospheric circulation and consequently to the Earth’s rotation is not negligeable. It is shown that in the effective atmospheric angular momentum and Earth’s rotation, its amplitude is slightly above the amplitude of the oscillation known as the Madden-Julian cycle.

I wonder if these could, by any chance, be related?

P.S. Before I get thrown into social media prison let me make it clear that I am not proposing this as a serious theory!

Citation Metrics and “Judging People’s Careers”

Posted in Bad Statistics, The Universe and Stuff with tags , , , , on October 29, 2021 by telescoper

There’s a paper on the arXiv by John Kormendy entitled Metrics of research impact in astronomy: Predicting later impact from metrics measured 10-15 years after the PhD. The abstract is as follows.

This paper calibrates how metrics derivable from the SAO/NASA Astrophysics Data System can be used to estimate the future impact of astronomy research careers and thereby to inform decisions on resource allocation such as job hires and tenure decisions. Three metrics are used, citations of refereed papers, citations of all publications normalized by the numbers of co-authors, and citations of all first-author papers. Each is individually calibrated as an impact predictor in the book Kormendy (2020), “Metrics of Research Impact in Astronomy” (Publ Astron Soc Pac, San Francisco). How this is done is reviewed in the first half of this paper. Then, I show that averaging results from three metrics produces more accurate predictions. Average prediction machines are constructed for different cohorts of 1990-2007 PhDs and used to postdict 2017 impact from metrics measured 10, 12, and 15 years after the PhD. The time span over which prediction is made ranges from 0 years for 2007 PhDs to 17 years for 1990 PhDs using metrics measured 10 years after the PhD. Calibration is based on perceived 2017 impact as voted by 22 experienced astronomers for 510 faculty members at 17 highly-ranked university astronomy departments world-wide. Prediction machinery reproduces voted impact estimates with an RMS uncertainty of 1/8 of the dynamic range for people in the study sample. The aim of this work is to lend some of the rigor that is normally used in scientific research to the difficult and subjective job of judging people’s careers.

This paper has understandably generated a considerable reaction on social media especially from early career researchers dismayed at how senior astronomers apparently think they should be judged. Presumably “judging people’s careers” means deciding whether or not they should get tenure (or equivalent) although the phrase is not a pleasant one to use.

My own opinion is that while citations and other bibliometric indicators do contain some information, they are extremely difficult to apply in the modern era in which so many high-impact results are generated by large international teams. Note also the extreme selectivity of this exercise: just 22 “experienced astronomers” provide the :calibration” which is for faculty in just 17 “highly-ranked” university astronomy departments. No possibility of any bias there, obviously. Subjectivity doesn’t turn into objectivity just because you make it quantitative.

If you’re interested here are the names of the 22:

Note that the author of the paper is himself on the list. I find that deeply inappropriate.

Anyway, the overall level of statistical gibberish in this paper is such that I am amazed it has been accepted for publication, but then it is in the Proceedings of the National Academy of Sciences, a journal that has form when it comes to dodgy statistics. If I understand correctly, PNAS has a route that allows “senior” authors to publish papers without passing through peer review. That’s the only explanation I can think of for this.

As a rejoinder I’d like to mention this paper by Adler et al. from 12 years ago, which has the following abstract:

This is a report about the use and misuse of citation data in the assessment of scientific research. The idea that research assessment must be done using “simple and objective” methods is increasingly prevalent today. The “simple and objective” methods are broadly interpreted as bibliometrics, that is, citation data and the statistics derived from them. There is a belief that citation statistics are inherently more accurate because they substitute simple numbers for complex judgments, and hence overcome the possible subjectivity of peer review. But this belief is unfounded.

O brave new world that has such metrics in it.

Update: John Kormendy has now withdrawn the paper; you can see his statement here.

Still no Primordial Gravitational Waves…

Posted in Astrohype, Bad Statistics, The Universe and Stuff with tags , , , , , , on October 27, 2021 by telescoper

During March 2014 this blog received the most traffic it has ever had (reaching almost 10,000 hits per day at one point). The reason for that was the announcement of the “discovery” of primordial gravitational waves by the BICEP2 experiment. Despite all the hype at the time I wasn’t convinced. This is what I said in an interview with Physics World:

It seems to me though that there’s a significant possibility of some of the polarization signal in E and B [modes] not being cosmological. This is a very interesting result, but I’d prefer to reserve judgement until it is confirmed by other experiments. If it is genuine, then the spectrum is a bit strange and may indicate something added to the normal inflationary recipe.

I also blogged about this several times, e.g. here. It turns out I was right to be unconvinced as the signal detected by BICEP2 was dominated by polarized foreground emission. The story is summarized by these two news stories just a few months apart:

Anyway, the search for primordial gravitational waves continues. The latest publication on this topic came out earlier this month in Physical Review Letters and you can also find it on the arXiv here. The last sentence of the abstract is:

These are the strongest constraints to date on primordial gravitational waves.

In other words, seven years on from the claimed “discovery” there is still no evidence for anything but polarized dust emission…

A Vaccination Fallacy

Posted in Bad Statistics, Covid-19 with tags , , , , on June 27, 2021 by telescoper

I have been struck by the number of people upset by the latest analysis of SARS-Cov-2 “variants of concern” byPublic Health England. In particular it is in the report that over 40% of those dying from the so-called Delta Variant have had both vaccine jabs. I even saw some comments on social media from people saying that this proves that the vaccines are useless against this variant and as a consequence they weren’t going to bother getting their second jab.

This is dangerous nonsense and I think it stems – as much dangerous nonsense does – from a misunderstanding of basic probability which comes up in a number of situations, including the Prosecutor’s Fallacy. I’ll try to clarify it here with a bit of probability theory. The same logic as the following applies if you specify serious illness or mortality, but I’ll keep it simple by just talking about contracting Covid-19. When I write about probabilities you can think of these as proportions within the population so I’ll use the terms probability and proportion interchangeably in the following.

Denote by P[C|V] the conditional probability that a fully vaccinated person becomes ill from Covid-19. That is considerably smaller than P[C| not V] (by a factor of ten or so given the efficacy of the vaccines). Vaccines do not however deliver perfect immunity so P[C|V]≠0.

Let P[V|C] be the conditional probability of a person with Covid-19 having been fully vaccinated. Or, if you prefer, the proportion of people with Covid-19 who are fully vaccinated..

Now the first thing to point out is that these conditional probability are emphatically not equal. The probability of a female person being pregnant is not the same as the probability of a pregnant person being female.

We can find the relationship between P[C|V] and P[V|C] using the joint probability P[V,C]=P[V,C] of a person having been fully vaccinated and contracting Covid-19. This can be decomposed in two ways: P[V,C]=P[V]P[C|V]=P[C]P[V|C]=P[V,C], where P[V] is the proportion of people fully vaccinated and P[C] is the proportion of people who have contracted Covid-19. This gives P[V|C]=P[V]P[C|V]/P[C].

This result is nothing more than the famous Bayes Theorem.

Now P[C] is difficult to know exactly because of variable testing rates and other selection effects but is presumably quite small. The total number of positive tests since the pandemic began in the UK is about 5M which is less than 10% of the population. The proportion of the population fully vaccinated on the other hand is known to be about 50% in the UK. We can be pretty sure therefore that P[V]»P[C]. This in turn means that P[V|C]»P[C|V].

In words this means that there is nothing to be surprised about in the fact that the proportion of people being infected with Covid-19 is significantly larger than the probability of a vaccinated person catching Covid-19. It is expected that the majority of people catching Covid-19 in the current phase of the pandemic will have been fully vaccinated.

(As a commenter below points out, in the limit when everyone has been vaccinated 100% of the people who catch Covid-19 will have been vaccinated. The point is that the number of people getting ill and dying will be lower than in an unvaccinated population.)

The proportion of those dying of Covid-19 who have been fully vaccinated will also be high, a point also made here.

It’s difficult to be quantitatively accurate here because there are other factors involved in the risk of becoming ill with Covid-19, chiefly age. The reason this poses a problem is that in my countries vaccinations have been given preferentially to those deemed to be at high risk. Younger people are at relatively low risk of serious illness or death from Covid-19 whether or not they are vaccinated compared to older people, but the latter are also more likely to have been vaccinated. To factor this into the calculation above requires an additional piece of conditioning information. We could express this crudely in terms of a binary condition High Risk (H) or Low Risk (L) and construct P(V|L,H) etc but I don’t have the time or information to do this.

So please don’t be taken in by this fallacy. Vaccines do work. Get your second jab (or your first if you haven’t done it yet). It might save your life.

Cosmology and the Born-Again Bayesians!

Posted in Bad Statistics, Biographical, The Universe and Stuff with tags , , , , on May 10, 2021 by telescoper

The other day, via Twitter, I came across an interesting blog post about the relatively recent resurgence of Bayesian reasoning in science. That piece had triggered a discussion about why cosmologists seem to be largely Bayesian in outlook, so I thought I’d share a few thoughts about that. You can find a lot of posts about various aspects of Bayesian reasoning on this blog, e.g. here.

When I was an undergraduate student I didn’t think very much about statistics at all, so when I started my DPhil studies I realized I had a great deal to learn. However, at least to start with, I mainly used frequentist methods. Looking back I think that’s probably because I was working on cosmic microwave background statistics and we didn’t really have any data back in 1985. Or actually we had data, but no firm detections. I was therefore taking models and calculating things in what I would call the forward direction, indicated by the up arrow. What I was trying to do was find statistical descriptors that looked likely to be able to discriminate between different models but I didn’t have the data.

Once measurements started to become available the inverse-reasoning part of the diagram indicated by the downward arrow came to the fore. It was only then that it started to become necessary to make firm statements about which models were favoured by the data and which weren’t. That is what Bayesian methods do best, especially when you have to combine different data sets.

By the early 1990s I was pretty much a confirmed Bayesian – as were quite a few fellow theorists -but I noticed that most observational cosmologists I knew were confirmed frequentists. I put that down to the fact that they preferred to think in “measurement space” rather than “theory space”, the latter requiring the inductive step furnished by Bayesian reasoning indicated by the downward arrow. As cosmology has evolved the separation between theorists and observers in some areas – especially CMB studies – has all but vanished and there’s a huge activity at the interface between theory and measurement.

But my first exposure to Bayesian reasoning came long before that change. I wasn’t aware of its usefulness until 1987, when I returned to Cambridge for a conference called The Post-Recombination Universe organized by Nick Kaiser and Anthony Lasenby. There was an interesting discussion in one session about how to properly state the upper limit on CMB fluctuations arising from a particular experiment, which had been given incorrectly in a paper using a frequentist argument. During the discussion, Nick described Anthony as a “Born-again Bayesian”, a phrase that stuck in my memory though I’m still not sure whether or not it was meant as an insult.

It may be the case for many people that a relatively simple example convinces them of the superiority of a particular method or approach. I had previously found statistical methods – especially frequentist hypothesis-testing – muddled and confusing, but once I’d figured out what Bayesian reasoning was I found it logically compelling. It’s not always easy to do a Bayesian analysis for reasons discussed in the paper to which I linked above, but it least you have a clear idea in your mind what question it is that you are trying to answer!

Anyway, it was only later that I became aware that there were many researchers who had been at Cambridge while I was there as a student who knew all about Bayesian methods: people such as Steve Gull, John Skilling, Mike Hobson, Anthony Lasenby and, of course, one Anthony Garrett. It was only later in my career that I actually got to talk to any of them about any of it!

So I think the resurgence of Bayesian ideas in cosmology owes a very great deal to the Cambridge group which had the expertise necessary to exploit the wave of high quality data that started to come in during the 1990s and the availability of the computing resources needed to handle it.

But looking a bit further back I think there’s an important Cambridge (but not cosmological) figure that preceded them, Sir Harold Jeffreys whose book The Theory of Probability was first published in 1939. I think that book began to turn the tide, and it still makes for interesting reading.

P.S. I have to say I’ve come across more than one scientist who has argued that you can’t apply statistical reasoning in cosmology because there is only one Universe and you can’t use probability theory for unique events. That erroneous point of view has led to many otherwise sensible people embracing the idea of a multiverse, but that’s the subject for another rant.

Reaction to the announcement of a new measurement of (g-2)

Posted in Bad Statistics, The Universe and Stuff on April 7, 2021 by telescoper

Today’s announcement of a new measurement of the anomalous magnetic dipole moment – known to its friends as (g-2) of the muon – has been greeted with excitement by the scientific community, as it seems to provide evidence of a departure from the standard model of particle physics (by 4.2σ in frequentist parlance).

My own view is that the measurement of g-2, which seems to be a bit higher than theorists expected, can be straightforwardly reconciled with the predictions of the standard model of particle physics by simply adopting a slightly lower value for 2 in the theoretical calculations.

P.S. According to my own (unpublished) calculations, the value of g-2 ≈ 7.81 m s-2.

 

US Election Night and Day

Posted in Bad Statistics, Biographical, Politics on November 4, 2020 by telescoper

Before you ask, no I didn’t stay up all night for the US presidential election results. I went to bed at 11pm and woke up as usual at 7am when my radio came on. I had a good night’s sleep. It’s not that I was confident of the outcome – I didn’t share the optimism of many of my friends that a Democrat landslide was imminent – it’s just that I’ve learnt not to get stressed by things that are out of my control.

On the other hand, my mood on waking to discover that the election was favouring the incumbent Orange Buffoon is accurately summed up by this image:

Regardless of who wins, I find it shocking that so many are prepared to vote for Trump a second time. There might have been an excuse first time around that they didn’t know quite how bad he was. Now they do, and there are still 65 million people (and counting) willing to vote for him. That’s frightening.

As I write (at 4pm on November 3rd) it still isn’t clear who will be the next President, but the odds have shortened dramatically on Joe Biden (currently around 1/5) having been short on Donald Trump when the early results came in; Trump’s odds have now drifted out between 3/1 and 4/1. Biden is now clearly favourite, but favourites don’t always win.

What has changed dramatically during the course of the day has been the enormous impact of mail-in and early voting results in key states. In Wisconsin these votes turned around a losing count for Biden into an almost certain victory by being >70% in his favour. A similar thing looks likely to happen in Michigan too. Assuming he wins Wisconsin, Joe Biden needs just two of Michigan, Nevada, Pennsylvania and Georgia to reach the minimum of 270 electoral college votes needed to win the election. He is ahead in two – Michigan and Nevada.

This is by no means certain – the vote in each of these states is very close and they could even all go to Trump. What does seem likely is that Biden will win the popular vote quite comfortably and may even get over 50%. That raises the issue again of why America doesn’t just count the votes and decide on the basis of a simple majority, rather than on the silly electoral college system, but that’s been an open question for years. Trump won on a minority vote last time, against Hillary Clinton, as did Bush in 2000.

It’s also notable that this election has once again seeing the pollsters confounded. Most were predicted a comfortable Biden victory. Part of the problem is the national polls lack sufficient numbers in the swing states to be useful, but even the overall voting tally seems set to be much closer than the ~8% margin in many polls.

Obviously there is a systematic problem of some sort. Perhaps it’s to do with sample selection. Perhaps it’s because Trump supporters are less likely to answer opinion poll questions honestly. Perhaps its due to systematic suppression of the vote in pro-Democrat areas. There are potentially many more explanations, but the main point is that when polls have a systematic bias like this, you can’t treat the polling error statistically as a quantity that varies from positive to negative independently from one state to another, as some of the pundits do, because it is replicated across all States.

As I mentioned in a post last week, I placed a comfort bet on Trump of €50 at 9/5. He might still win but if he doesn’t this is one occasion on which I’d be happy to lose money.

P.S. The US elections often make me think about how many of the States I have actually visited. The answer is (mostly not for very long): Kansas, South Dakota, Colorado, Iowa, Missouri, Arkansas, Louisiana, California, Arizona, New York, New Jersey, Maryland, Massachusetts, New Hampshire, Maine, and Pennsylvania. That’s way less than a majority. I’ve also been to Washington DC but that’s not a State..