Who’s #1?

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , on May 2, 2012 by xi'an

First, apologies for this teaser of a title! This post is not about who is #1 in whatever category you can think of, from statisticians to climbs [the Eiger Nordwand, to be sure!], to runners (Gebrselassie?), to books… (My daughter simply said “c’est moi!” when she saw the cover of this book on my desk.) So this is in fact a book review of…a book with this catching title I received a month or so ago!

We decided to forgo purely statistical methodology, which is probably a disappointment to the hardcore statisticians.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 225)

This book may be one of the most boring ones I have had to review so far! The reason for this disgruntled introduction to “Who’s #1? The Science of Rating and Ranking” by Langville and Meyer is that it has very little if any to do with statistics and modelling. (And also that it is mostly about American football, a sport I am not even remotely interested in.) The purpose of the book is to present ways of building rating and ranking within a population, based on pairwise numerical connections between some members of this population. The methods abound, at least eight are covered by the book, but they all suffer from the same drawback that they are connected to no grand truth, to no parameter from an underlying probabilistic model, to no loss function that would measure the impact of a “wrong” rating. (The closer it comes to this is when discussing spread betting in Chapter 9.) It is thus a collection of transformation rules, from matrices to ratings. I find this the more disappointing in that there exists a branch of statistics called ranking and selection that specializes in this kind of problems and that statistics in sports is a quite active branch of our profession, witness the numerous books by Jim Albert. (Not to mention Efron’s analysis of baseball data in the 70’s.)

First suppose that in some absolutely perfect universe there is a perfect rating vector.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 117)

The style of the book is disconcerting at first, and then some, as it sounds written partly from Internet excerpts (at least for most of the pictures) and partly from local student dissertations… The mathematical level is highly varying, in that the authors take the pain to define what a matrix is (page 33), only to jump to Perron-Frobenius theorem a few pages later (page 36). It also mentions Laplace’s succession rule (only justified as a shrinkage towards the center, i.e. away from 0 and 1), the Sinkhorn-Knopp theorem, the traveling salesman problem, Arrow and Condorcet, relaxation and evolutionary optimization, and even Kendall’s and Spearman’s rank tests (Chapter 16), even though no statistical model is involved. (Nothing as terrible as the completely inappropriate use of Spearman’s rho coefficient in one of Belfiglio’s studies…)

Since it is hard to say which ranking is better, our point here is simply that different methods can produce vastly different rankings.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 78)

I also find irritating the association of “science” with “rating”, because the techniques presented in this book are simply tricks to turn pairwise comparison into a general ordering of a population, nothing to do with uncovering ruling principles explaining the difference between the individuals. Since there is no validation for one ordering against another, we can see no rationality in proposing any of those, except to set a convention. The fascination of the authors for the Markov chain approach to the ranking problem is difficult to fathom as the underlying structure is not dynamical (there is not evolving ranking along games in this book) and the Markov transition matrix is just constructed to derive a stationary distribution, inducing a particular “Markov” ranking.

The Elo rating system is the epitome of simple elegance.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 64)

An interesting input of the book is its description of the Elo ranking system used in chess, of which I did not know anything apart from its existence. Once again, there is a high degree of arbitrariness in the construction of the ranking, whose sole goal is to provide a convention upon which most people agree. A convention, mind, not a representation of truth! (This chapter contains a section on the Social Network movie, where a character writes a logistic transform on a window, missing the exponent. This should remind Andrew of someone he often refer to in his blog!)

Perhaps the largest lesson is not to put an undue amount of faith in anyone’s rating.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 125)

In conclusion, I see little point in suggesting reading this book, unless one is interested in matrix optimization problems and/or illustrations in American football… Or unless one wishes to write a statistics book on the topic!

Le Monde rank test (cont’d)

Posted in R, Statistics with tags , , , on April 5, 2010 by xi'an

Following a comment from efrique pointing out that this statistic is called Spearman footrule, I want to clarify the notation in

$\mathfrak{M}_n = \sum_{i=1}^n |r^x_i-r^y_i|\,,$

namely (a) that the ranks of $x_i$ and $y_i$ are considered for the whole sample, i.e.

$\{r^x_1,\ldots,r^x_n,r^y_1,\ldots,r^y_n\} = \{1,\ldots,2n\}$

instead of being computed separately for the $x$‘s and the $y$‘s, and then (b) that the ranks are reordered for each group (meaning that the groups could be of different sizes). This statistics is therefore different from the Spearman footrule studied by Persi Diaconis and R. Graham in a 1977 JRSS paper,

$\mathfrak{D}_ n = \sum_{i=1}^n |\pi(i)-\sigma(i)|\,,$

where $\pi$ and $\sigma$ are permutations from $\mathfrak{S}_n$. The mean of $\mathfrak{D}_ n$ is approximately $n^{2/3}$. I mistakenly referred to Spearman’s ρ rank correlation test in the previous post. It is actually much more related to the Siegel-Tukey test, even though I think there exists a non-parametric test of iid-ness for paired observations… The $x$‘s and the $y$‘s are thus not paired, despite what I wrote previously. This distance must be related to some non-parametric test for checking the equality of location parameters.