16 August 2006
While I'm sure that many readers of this blog saw this article in the August 15 NYT science section, it's worth noting anyway as an insight into the sociology of mathematics and a look at some interesting pure mathematics as well (plus the graphics therein were really cool).
It seems that Perelman solved a 102 year old mathematical question of huge importance but wants to have nothing to do with the affect on the field and the resulting acclaim (including an almost certain Fields Medal) since he's disappeared into the Russian forest. Nonetheless, other mathematicians have taken on the task of writing up his results producing proofs in three books that are now available online. These are fascinating to read, even though much of the discussion is at the highest mathematical level, since some of the principles are familiar to us (Cauchy-Schwartz, gradients, Hessians, etc.) from routine work, but obviously appearing in wildly different contexts.
So here's the related question. Suppose, like mathematics, we could list the "big" unsolved problems in political science. What would this list look like? Personally, I'd love to see such a thing. Of course it is unclear whether we have a David Hilbert-like figure to say "As long as a branch of science offers an abundance of problems, so is it alive" and then to go on and identify the 23 most important unsolved problems (the 1900 "Hilbert Challenge":). In this vein, my list of unsolved problems would start with why does the discipline cling to the bankrupt NHST and continue to worship "stars"?
7 August 2006
I've spent quite a bit of time in the last few weeks - probably too much - thinking about the term 'regression' and its use in statistics, and why I find it so dislikeable. I sincerely doubt any campaign I try to start will have any real effect, so let me lay down the reasons why I feel we as scientists should refer to linear modelling as just such, and not as 'regression'.
One reason is that the word only has a tenuous connection to the actual algorithm - the other is that it far too often implies a causal relationship where none exists.
As the story goes, Francis Galton took a group of tall men and measured the height of their sons, and found that on average, the sons as a group were shorter than their fathers. Drawing on similar work he had done with pea plants, he described this phenomenon as "regression to the mean," recognizing that the sample of fathers was nonrandom. A "regression coefficient" then described the estimated parameter which, when multiplied by the predictor, would produce the mean value.
I can only surmise that "determining regression coefficients through minimizing the least squares difference" was too verbose for Galton and his buddies, and "regression analysis" stuck. Now we have lawyerese terms like "multiple regression analysis," which really should read "multiple parameter regression analysis" since we're only running one algorithm, but we appear stuck with it.
So what's the big deal? Nomenclature isn't an easy business, and two extra syllables in "linear model" might slow things down. But aside from my gripe with using "regress" as a transitive verb (the Latin student in me cringing), even the most generous interpretation of the word's root, and the experiments that revealed it, yield to trouble.
"Regression" literally means "the act of going back." If we accept this definition in this context, we have to have something to which we can return. Clearly, this implies discovering the mean - but chronologically, it can only mean discovering the cause, that which came before.
Linear modelling makes no explicit assumptions about cause and effect, a major source of headache in our discipline, but the word itself, consciously or otherwise, binds us to this fact.
The remedy to this is not simple; after all, I'm talking about trying to break the correlation-is-causation fallacy through words, which is both a difficult task and the sort of behaviour that will keep people from sitting with you at lunch. But we can improve things slowly and subtly in this fashion:
1) If you are confident that your analysis will unveil a causal relationship, say so. Call it "regression-to-cause", or "causal linear model", or something like that.
2) If you're not so sure, call it a (generalized) linear model, or a lin-mod, or a least-squares, or another term that does not necessarily imply causation. Resist the temptation to fall back to the word "regression" until a long time has passed.
This doesn't have to be a completely nerve-wracking exercise; just use a strike-through when necessary, to show that the term
regression'linear model' is better suited to describe what we're trying to build here.
1 August 2006
For those interested in more detail about the Texas Redistricting case, and associated Amici brief, that Drew Thomas wrote about a few entries ago, you might be interested in The Future of Partisan Symmetry as a Judicial Test for Partisan Gerrymandering after LULAC v. Perry, by Bernie Grofman and me, forthcoming in the Election Law Journal. An abstract appears below. Comments welcome!
While the Supreme Court in Bandemer v. Davis found partisan gerrymandering to be justiciable, no challenged redistricting plan in the subsequent 20 years has been held unconstitutional on partisan grounds. Then, in Vieth v. Jubilerer, five justices concluded that some standard might be adopted in a future case, if a manageable rule could be found. When gerrymandering next came before the Court, in LULAC v. Perry, we along with our colleagues filed an Amicus Brief (King et al., 2005), proposing that a test be based in part on the partisan symmetry standard. Although the issue was not resolved, our proposal was discussed and positively evaluated in three of the opinions, including the plurality judgment, and for the first time for any proposal the Court gave a clear indication that a future legal test for partisan gerrymandering will likely include partisan symmetry. A majority of Justices now appear to endorse the view that the measurement of partisan symmetry may be used in partisan gerrymandering claims as “a helpful (though certainly not talismanic) tool” (Justice Stevens, joined by Justice Breyer), provided one recognizes that “asymmetry alone is not a reliable measure of unconstitutional partisanship” and possibly that the standard would be applied only after at least one election has been held under the redistricting plan at issue (Justice Kennedy, joined by Justices Souter and Ginsburg). We use this essay to respond to the request of Justices Souter and Ginsburg that “further attention … be devoted to the administrability of such a criterion at all levels of redistricting and its review.” Building on our previous scholarly work, our Amicus Brief, the observations of these five Justices, and a supporting consensus in the academic literature, we offer here a social science perspective on the conceptualization and measurement of partisan gerrymandering and the development of relevant legal rules based on what is effectively the Supreme Court’s open invitation to lower courts to revisit these issues in the light of LULAC v. Perry.