Archive for the Bad Statistics Category

The Neyman-Scott ‘Paradox’

Posted in Bad Statistics, Cute Problems with tags , , , , on November 25, 2016 by telescoper

I just came across this interesting little problem recently and thought I’d share it here. It’s usually called the ‘Neyman-Scott’ paradox. Before going on it’s worth mentioning that Elizabeth Scott (the second half of Neyman-Scott) was an astronomer by background. Her co-author was Jerzy Neyman. As has been the case for many astronomers, she contributed greatly to the development of the field of statistics. Anyway, I think this example provides another good illustration of the superiority of Bayesian methods for estimating parameters, but I’ll let you make your own mind up about what’s going on.

The problem is fairly technical so I’ve done done a quick version in latex that you can download

here, but I’ve also copied into this post so you can read it below:

 

neyman-scott1

neyman-scott2

I look forward to receiving Frequentist Flak or Bayesian Benevolence through the comments box below!

The Worthless University Rankings

Posted in Bad Statistics, Education with tags , , , on September 23, 2016 by telescoper

The Times Higher World University Rankings, which were released this weekk. The main table can be found here and the methodology used to concoct them here.

Here I wish to reiterate the objection I made last year to the way these tables are manipulated year on year to create an artificial “churn” that renders them unreliable and impossible to interpret in an objective way. In other words, they’re worthless. This year, editor Phil Baty has written an article entitled Standing still is not an option in which he makes a statement that “the overall rankings methodology is the same as last year”. Actually it isn’t. In the page on methodology you will find this:

In 2015-16, we excluded papers with more than 1,000 authors because they were having a disproportionate impact on the citation scores of a small number of universities. This year, we have designed a method for reincorporating these papers. Working with Elsevier, we have developed a new fractional counting approach that ensures that all universities where academics are authors of these papers will receive at least 5 per cent of the value of the paper, and where those that provide the most contributors to the paper receive a proportionately larger contribution.

So the methodology just isn’t “the same as last year”. In fact every year that I’ve seen these rankings there’s been some change in methodology. The change above at least attempts to improve on the absurd decision taken last year to eliminate from the citation count any papers arising from large collaborations. In my view, membership of large world-wide collaborations is in itself an indicator of international research excellence, and such papers should if anything be given greater not lesser weight. But whether you agree with the motivation for the change or not is beside the point.

The real question is how can we be sure that any change in league table position for an institution from year to year are is caused by methodological tweaks rather than changes in “performance”, i.e. not by changes in the metrics but by changes in the way they are combined? Would you trust the outcome of a medical trial in which the response of two groups of patients (e.g. one given medication and the other placebo) were assessed with two different measurement techniques?

There is an obvious and easy way to test for the size of this effect, which is to construct a parallel set of league tables, with this year’s input data but last year’s methodology, which would make it easy to isolate changes in methodology from changes in the performance indicators. The Times Higher – along with other purveyors of similar statistical twaddle – refuses to do this. No scientifically literate person would accept the result of this kind of study unless the systematic effects can be shown to be under control. There is a very easy way for the Times Higher to address this question: all they need to do is publish a set of league tables using, say, the 2015/16 methodology and the 2016/17 data, for comparison with those constructed using this year’s methodology on the 2016/17 data. Any differences between these two tables will give a clear indication of the reliability (or otherwise) of the rankings.

I challenged the Times Higher to do this last year, and they refused. You can draw your own conclusions about why.

Bayes Factors via Savage-Dickey Supermodels [IMA]

Posted in Bad Statistics, The Universe and Stuff on September 12, 2016 by telescoper

How could I possibly resist reblogging an arXiver post about “Savage-Dickey Supermodels”?

arXiver

http://arxiv.org/abs/1609.02186

We outline a new method to compute the Bayes Factor for model selection which bypasses the Bayesian Evidence. Our method combines multiple models into a single, nested, Supermodel using one or more hyperparameters. Since the models are now nested the Bayes Factors between the models can be efficiently computed using the Savage-Dickey Density Ratio (SDDR). In this way model selection becomes a problem of parameter estimation. We consider two ways of constructing the supermodel in detail: one based on combined models, and a second based on combined likelihoods. We report on these two approaches for a Gaussian linear model for which the Bayesian evidence can be calculated analytically and a toy nonlinear problem. Unlike the combined model approach, where a standard Monte Carlo Markov Chain (MCMC) struggles, the combined-likelihood approach fares much better in providing a reliable estimate of the log-Bayes Factor. This scheme potentially opens the way to…

View original post 53 more words

Rank Nonsense

Posted in Bad Statistics, Education, Politics with tags , , , , , on September 8, 2016 by telescoper

It’s that time of year when international league tables (also known as “World Rankings”)  appear. We’ve already had the QS World University Rankings and the Shanghai (ARWU) World University Rankings. These will soon be joined by the Times Higher World Rankings, due out on 21st September.

A lot of people who should know a lot better give these league tables far too much attention. As far as I’m concerned they are all constructed using extremely suspect methodologies whose main function is to amplify small statistical variations into something that looks significant enough to justify constructing  a narrative about it. The resulting press coverage usually better reflects a preconceived idea in a journalist’s head than any sensible reading of the tables themselves.

A particularly egregious example of this kind of nonsense can be found in this week’s Guardian. The offending article is entitled “UK universities tumble in world rankings amid Brexit concerns”. Now I make no secret of the fact that I voted “Remain” and that I do think BrExit (if it actually happens) will damage UK universities (as well as everything else in the UK). However, linking the changes in the QS rankings to BrExit is evidently ridiculous: all the data were collected before the referendum on 23rd June anyway! In my opinion there are enough good arguments against BrExit without trying to concoct daft ones.

In any case these tables do not come with any estimate of the likely statistical variation from year to year in the metrics used to construct them, which makes changes impossible to interpret. If only the compilers of these tables would put error bars on the results! Interestingly, my former employer, the University of Sussex, has held its place exactly in the QS rankings between 2015 and 2016: it was ranked 187th in the world in both years. However, the actual score corresponding to these two years was 55.6 in 2015 and 48.4 in 2016. Moreover, Cambridge University fell from 3rd to 4th place this year but its score only changed from 98.6 to 97.2. I very much doubt that is significant at all, but it’s mentioned prominently in the subheading of the Guardian piece:

Uncertainty over research funding and immigration rules blamed for decline, as Cambridge slips out of top three for first time.

Actually, looking closer, I find that Cambridge was joint 3rd in 2015 and is 4th this year. Over-interpretation, or what?

To end with, I can’t resist mentioning that the University of Sussex is in the top 150 in the Shanghai Rankings for Natural and Mathematical Sciences this year, having not been in the top 200 last year. This stunning improvement happened while I was Head of School for Mathematical and Physical Sciences so it clearly can not be any kind of statistical fluke but is entirely attributable to excellent leadership. Thank you for your applause.

 

 

The Rising Stars of Sussex Physics

Posted in Bad Statistics, Biographical, Education with tags , , , , on July 28, 2016 by telescoper

This is my penultimate day in the office in the School of Mathematical and Physical Sciences at the University of Sussex, and a bit of news has arrived that seems a nice way to round off my stint as Head of School.

It seems that Physics & Astronomy research at the University of Sussex has been ranked as 13th in western Europe and 7th in the UK by leading academic publishers, Nature Research, and has been profiled as one of its top-25 “rising stars” worldwide.

I was tempted to describe this rise as ‘meteoric’ but in my experience meteors generally fall down rather than rise up.

Anyway, as regular readers of this blog will know, I’m generally very sceptical of the value of league tables and there’s no reason to treat this one as qualitatively any different. Here is an explanation of the (rather curious) methodology from the University of Sussex news item:

The Nature Index 2016 Rising Stars supplement identifies the countries and institutions showing the most significant growth in high-quality research publications, using the Nature Index, which tracks the research of more than 8,000 global institutions – described as “players to watch”.

The top 100 most improved institutions in the index between 2012 and 2015 are ranked by the increase in their contribution to 68 high-quality journals. From this top 100, the supplement profiles 25 rising stars – one of which is Sussex – that are already making their mark, and have the potential to shine in coming decades.

The institutions and countries examined have increased their contribution to a selection of top natural science journals — a metric known as weighted fractional count (WFC) — from 2012 to 2015.

Mainly thanks to a quadrupling of its physical sciences score, Sussex reached 351 in the Global 500 in 2015. That represents an 83.9% rise in its contribution to index papers since 2012 — the biggest jump of any UK research organisation in the top 100 most improved institutions.

It’s certainly a strange choice of metric, as it only involves publications in “high quality” journals, presumably selected by Journal Impact Factor or some other arbitrary statistical abominatio,  then taking the difference in this measure between 2012 and 2015  and expressing the change as a percentage. I noticed one institution in the list has improved by over 4600%, which makes Sussex’s change of 83.9% seem rather insignificant…

But at least this table provides some sort of evidence that the investment made in Physics & Astronomy over the last few years has made a significant (and positive) difference. The number of research faculty in Physics & Astronomy has increased by more than 60%  since 2012 so one would have been surprised not to have seen an increase in publication output over the same period. On the other hand, it seems likely that many of the high-impact papers published since 2012 were written by researchers who arrived well before then because Physics research is often a slow burner. The full impact of the most recent investments has probably not yet been felt. I’m therefore confident that Physics at Sussex has a very exciting future in store as its rising stars look set to rise still further! It’s nice to be going out on a high note!

 

 

The 3.5 keV “Line” that (probably) wasn’t…

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , on July 26, 2016 by telescoper

About a year ago I wrote a blog post about a mysterious “line” in the X-ray spectra of galaxy clusters corresponding to an energy of around 3.5 keV. The primary reference for the claim is a paper by Bulbul et al which is, of course, freely available on the arXiv.

The key graph from that paper is this:

XMMspectrum

The claimed feature – it stretches the imagination considerably to call it a “line” – is shown in red. No, I’m not particularly impressed either, but this is what passes for high-quality data in X-ray astronomy!

Anyway, there has just appeared on the arXiv a paper by the Hitomi Collaboration describing what are basically the only set of science results that the Hitomi satellite managed to obtain before it fell to bits earlier this year. These were observations of the Perseus Cluster.

Here is the abstract:

High-resolution X-ray spectroscopy with Hitomi was expected to resolve the origin of the faint unidentified E=3.5 keV emission line reported in several low-resolution studies of various massive systems, such as galaxies and clusters, including the Perseus cluster. We have analyzed the Hitomi first-light observation of the Perseus cluster. The emission line expected for Perseus based on the XMM-Newton signal from the large cluster sample under the dark matter decay scenario is too faint to be detectable in the Hitomi data. However, the previously reported 3.5 keV flux from Perseus was anomalously high compared to the sample-based prediction. We find no unidentified line at the reported flux level. The high flux derived with XMM MOS for the Perseus region covered by Hitomi is excluded at >3-sigma within the energy confidence interval of the most constraining previous study. If XMM measurement uncertainties for this region are included, the inconsistency with Hitomi is at a 99% significance for a broad dark-matter line and at 99.7% for a narrow line from the gas. We do find a hint of a broad excess near the energies of high-n transitions of Sxvi (E=3.44 keV rest-frame) – a possible signature of charge exchange in the molecular nebula and one of the proposed explanations for the 3.5 keV line. While its energy is consistent with XMM pn detections, it is unlikely to explain the MOS signal. A confirmation of this interesting feature has to wait for a more sensitive observation with a future calorimeter experiment.

And here is the killer plot:

Perseus_Hitomi

The spectrum looks amazingly detailed, which makes the demise of Hitomi all the more tragic, but the 3.5 keV is conspicuous by its absence. So there you are, yet another supposedly significant feature that excited a huge amount of interest turns out to be nothing of the sort. To be fair, as the abstract states, the anomalous line was only seen by stacking spectra of different clusters and might still be there but too faint to be seen in an individual cluster spectrum. Nevertheless I’d say the probability of there being any feature at 3.5 keV has decreased significantly after this observation.

P.S. rumours suggest that the 750 GeV diphoton “excess” found at the Large Hadron Collider may be about to meet a similar fate.

The Distribution of Cauchy

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on April 6, 2016 by telescoper

Back into the swing of teaching after a short break, I have been doing some lectures this week about complex analysis to theoretical physics students. The name of a brilliant French mathematician called Augustin Louis Cauchy (1789-1857) crops up very regularly in this branch of mathematics, e.g. in the Cauchy integral formula and the Cauchy-Riemann conditions, which reminded me of some old jottings aI made about the Cauchy distribution, which I never used in the publication to which they related, so I thought I’d just quickly pop the main idea on here in the hope that some amongst you might find it interesting and/or amusing.

What sparked this off is that the simplest cosmological models (including the particular one we now call the standard model) assume that the primordial density fluctuations we see imprinted in the pattern of temperature fluctuations in the cosmic microwave background and which we think gave rise to the large-scale structure of the Universe through the action of gravitational instability, were distributed according to Gaussian statistics (as predicted by the simplest versions of the inflationary universe theory).  Departures from Gaussianity would therefore, if found, yield important clues about physics beyond the standard model.

Cosmology isn’t the only place where Gaussian (normal) statistics apply. In fact they arise  fairly generically,  in circumstances where variation results from the linear superposition of independent influences, by virtue of the Central Limit Theorem. Thermal noise in experimental detectors is often treated as following Gaussian statistics, for example.

The Gaussian distribution has some nice properties that make it possible to place meaningful bounds on the statistical accuracy of measurements made in the presence of Gaussian fluctuations. For example, we all know that the margin of error of the determination of the mean value of a quantity from a sample of size n independent Gaussian-dsitributed varies as 1/\sqrt{n}; the larger the sample, the more accurately the global mean can be known. In the cosmological context this is basically why mapping a larger volume of space can lead, for instance, to a more accurate determination of the overall mean density of matter in the Universe.

However, although the Gaussian assumption often applies it doesn’t always apply, so if we want to think about non-Gaussian effects we have to think also about how well we can do statistical inference if we don’t have Gaussianity to rely on.

That’s why I was playing around with the peculiarities of the Cauchy distribution. This distribution comes up in a variety of real physics problems so it isn’t an artificially pathological case. Imagine you have two independent variables X and Y each of which has a Gaussian distribution with zero mean and unit variance. The ratio Z=X/Y has a probability density function of the form

p(z)=\frac{1}{\pi(1+z^2)},

which is a Cauchy distribution. There’s nothing at all wrong with this as a distribution – it’s not singular anywhere and integrates to unity as a pdf should. However, it does have a peculiar property that none of its moments is finite, not even the mean value!

Following on from this property is the fact that Cauchy-distributed quantities violate the Central Limit Theorem. If we take n independent Gaussian variables then the distribution of sum X_1+X_2 + \ldots X_n has the normal form, but this is also true (for large enough n) for the sum of n independent variables having any distribution as long as it has finite variance.

The Cauchy distribution has infinite variance so the distribution of the sum of independent Cauchy-distributed quantities Z_1+Z_2 + \ldots Z_n doesn’t tend to a Gaussian. In fact the distribution of the sum of any number of  independent Cauchy variates is itself a Cauchy distribution. Moreover the distribution of the mean of a sample of size n does not depend on n for Cauchy variates. This means that making a larger sample doesn’t reduce the margin of error on the mean value!

This was essentially the point I made in a previous post about the dangers of using standard statistical techniques – which usually involve the Gaussian assumption – to distributions of quantities formed as ratios.

We cosmologists should be grateful that we don’t seem to live in a Universe whose fluctuations are governed by Cauchy, rather than (nearly) Gaussian, statistics. Measuring more of the Universe wouldn’t be any use in determining its global properties as we’d always be dominated by cosmic variance