UPDATED: Should Unprecedented Tour Success Warrant Suspicion?

Last week I considered whether a big improvement in a rider's results is reasonable grounds for suspicion of doping. Bradley Wiggins is the current poster boy for this sort of skepticism, having just finished fourth in the Tour de France. After last year's Tour, the news that Bernhard Kohl and Stefan Schumacher had been caught using CERA was easy to believe due to the sense that their performances had improved to an extent that wasn't natural.

However, before jumping to conclusions about Wiggins we should ask whether dopers really show significantly improved results and, if so, whether such improvements also occur for non-dopers. This can be answered with some fairly straightforward statistical testing to compare a rider's current results with their previous results. I previously defined two parameters to quantify the improvement and significance of a given rider's results during a given year:

• R: The difference between a rider’s mean placings in a given year and mean placings in previous years. Larger numbers mean greater improvement.


• P: The likelihood that this difference is real and not simply a result of random fluctuations (statistical significance). Smaller numbers mean greater significance.

When computing P, I first take the logarithm of all results in order to enhance the value of top placings and minimize differences between mid-pack finishes (e.g. the difference between 2nd and 12th is much more important than the difference between 102th and 112nd). See below for the enthralling technical details.

I computed R and P for 934 riders over the years 2003-2009. For each rider, I only considered years in which 10 or more results were in the Cycling Quotient database. Forty-five rider/year pairings showed statistically significant improvements (see technical note for the definition of "significant"). Here they are, with 2009 cases in red:

Having improved an average of 50 places per race, Wiggins's 2009 is on this list. Columbia's Tony Martin is here as well, and has actually gained more than Wiggins this year. But there are very few convicted dopers here; where are our naughty friends? Adding Di Luca's 2009 to the list I showed on my previous post, the results for recent doping positives look like this:

Although a few of these riders show large gains in average results and fairly low P values, none of these riders appear on the above list of significant cases. So I'd have to say there isn't much support to the idea that a big improvement in results is a sign of doping.


Technical Notes: I defined significance as having a P less than 2e-5. This might sound overly conservative -- it means the chances that the rider's current and previous results are the same is only 0.002%. The problem is that I've done 2400 tests, so a cutoff of 5% would give me over 100 false positives. Dividing 0.05 by the number of tests, I get a P cutoff of 2e-5 and don't need to worry about false positives. Additional technical notes of possible relevance here and here (scroll down to the fine print).

TdF Stage 20

I think it's fair to say that Armstrong kept his word on what he would do to last year's top 5. Will he now demand that Sastre apologize for making him apologize during the first week? Oh the drama.

SVD Analysis of the 2006 Tour

As promised, here is more of this dorky, generally pointless stuff – Part 3 of a series on using singular value decomposition to study grand tour results. In this post, I will look at the 2006 Tour de France, a year (in)famous for the Floyd Landis doping spectacle and Oscar Pereiro being declared winner in part due to gaining 30 minutes in a flat-stage breakaway that was not chased. Despite these odd circumstances, the overall complexity of this Tour was less than most grand tours, containing only 3.1 effective stages.

As I discussed in the post on the 2007 tour, SVD basically rearranges all of the results from all of the stages into a series of composite stages, the modes, that appear in the data with decreasing weights. Here are the modes and weights for the 2006 Tour:

The SVD modes are read as columns in the raster plot, with red and green corresponding to greater and lesser times for each stage. Only riders who finished the race are included in the results, and among those excluded is Floyd Landis since his results have been removed from the CQ record.

The first striking feature of the mode patterns is how uneventful the first nine stages were. None of the large SVD modes has a large signal for these stages, meaning very few time gaps occurred. The major climbing stages were Stages 11, 15, 16, and 17, and sure enough these are the primary contributors to Mode 1. Stages 7 and 19 were individual time trials, the first of which had minor effects on the GC. Mode 2 primarily encodes large time losses on Stage 13 – the stage in which Voigt, Pereiro, Chavanel, Quinziato, and Grivko gained enormous time in the peloton. Mode 3 mostly encodes time gains on the Stage 17 to Morzine, which saw the peloton shatter in the wake of Landis’s shady escapade. Modes 4 and 5 are more corrections to mountain stage placings, and the latter modes are minor time gaps on flat stages and time trials.

Each riders’ individual results can be recomputed by summing up these patterns, with each pattern separately weighted according to the individual rider. Looking at the weights for the first two modes for each rider, we see something quite unique in this Tour. This plot shows the extent to which each rider’s results (dot) exhibited Mode 1 (x-axis) and Mode 2 (y-axis), with final GC placing running from red to blue:

Comparing this with the results from the 2007 Tour we see a major rotation such that the results tend to spread from upper left to lower right rather than simply left to right. This, of course, is due to the Stage 13 breakaway. The four points isolate below are Pereiro, Voigt, Chavanel, and Quinziato (Grivko is not included since he did not finish the Tour). Their special status as breakaway survivors is clearly shown in their isolation on the plot. Note that Pererio is to the upper left of his breakaway companions, meaning that his Tour did have the makings for GC success even without the break. Looking at the top 10, we see how he compares to the others:

What are you doing all the way down there, Oscar? Winning the Tour de France, apparently.

TdF Stage 18: The Problem with Greg Lemond

Alberto Contador scored a mighty impressive win in today's ITT. His post-race press conference was clouded by questions about doping, specifically generated by a newspaper column written by Greg Lemond (link is in French). In short, Lemond demands that Contador prove he is clean, otherwise we can only assume he is a doper. The very idea of computing a rider's physiological parameters from TV images is a bit silly, yet Lemond claims a specific VO2 max value for Contador that he considers impossible. When making arguments like this Lemond implicitly assumes arbitrary limits on the capabilities of cyclists, and those limits tend to align with his own performances.

Personally I think Lemond is fine when he talks about the rise of EPO and its effects in the early 90’s. Clearly the EPO users got fast and there was indeed a peloton at two speeds, at least until everyone started using. When Lemond and others bemoan this I think they’re justified, because what they’re talking about is a change in ability across the entire peloton. The average rider got a lot faster, and while technology, training, and team support also improved I don’t think there’s any question that pharmaceuticals were the primary cause.

However, this is much different than looking at the performance of a single rider and calling them dirty, especially if that rider is the best in the peloton. Like everything else, rider ability is spread across a distribution and you can rarely say anything about the extremes of a distribution even if you can characterize the average. By definition, extreme cases like Contador are unlike anyone else in the population. They are literally in a class by themselves, so even if everyone they’re beating is doping it is impossible to know if they are as well (e.g. Armstrong in his prime). Unless they fail a test, of course. One would think that Greg Lemond, himself one of these extreme cases, would recognize this.

If we really did understand physiology to the extent Lemond thinks he does, then we could move past these statistical arguments. But the fact that statistics-based methods (epidemiology, statistical genetics) are still the source of most biomedical knowledge suggests otherwise.

So when Lemond rants about the general use of drugs in cycling he’s got a point, but when he demands that a specific rider justify their results he’s just being a nut.

SVD Analysis of the 2007 Tour

I previously posted about using singular value decomposition (SVD) to analyze stage race results and estimated that based on the patterns in the results, grand tours could be reduced to about 3 or 4 stages without losing much information. The character of these “composite stages” depends on the results of each race on a case-by-case basis. Here I consider the 2007 Tour de France, which was the “simplest” tour, effectively having only 2.8 stages.

SVD basically rearranges all of the results from all of the stages into a series of composite stages, the modes, that appear in the data with decreasing weights. The first mode is the most prominent pattern in the data, the second mode the second most prominent, etc, and the weight (aka the singular value) of each mode is that composite stage’s importance in the global data set. The rearrangement can be represented as a raster plot. Here are the modes and weights for the 2007 Tour:

The SVD modes are read as columns in the raster plot, with red and green corresponding to greater and lesser times for each stage.

Looking up the first mode, it is not surprising that Stages 8, 14, 15, and 16 were mountain stages in the 2007 tour. In most grand tours, the mountain stages dominate the major modes because the time gaps in the mountains are so much greater than those in other stages. Mode 1 in particular has an much larger weight than any other mode, and it describes 93% of the results in the Tour. It can be though of as the base pattern for any rider’s results. This first mode alone correctly determines the three riders on the final podium (Contador, Evans, Leipheimer), although it interchanges Evans and Leipheimer’s final placings. It primarily encodes the climbing stages, but also includes some information on the ITTs in Stages 13 and 19. The second mode encodes a correction to the first, which is a time lost in the Alps relative to the Pyrenees, presumably representing riders who grow stronger in the last week. Mode 3 is mostly gains on Stage 9 paired with losses on Stage 15. Modes 4-8 contain further corrections to the mountain stages. Modes 9-14 generally account for time gaps in stages where breaks played a role and details of time trials. Modes 15-21 are slight differences in sprint stage and prologue performances – these weights are extremely small since most of the field finished sprint stages with the same time.

Each riders’ individual results can be recomputed by summing up these patterns, with each pattern separately weighted according to the individual rider. A GC rider, therefore, will have a small weight for Mode 1 since they didn’t lose much time on the decisive stages, whereas a consistent member of the autobus will have a large weight for that mode corresponding to their large time losses in the mountains. We can, for instance, look at the first two modes for each rider. This plot shows the extent to which each rider’s results (dot) exhibited Mode 1 (x-axis) and Mode 2 (y-axis), with final GC placing running from red to blue:

GC riders have lower values for Mode 1 since they lost the least time on important stages. The blob of blue on the far right is the autobus riders who consistently lost a log of time. Zooming in on the top 10 we get an idea of how the GC is scattered:

Contador has the smallest Mode 1 component – he lost the least time in the big stages. Along with his teammates Leipheimer and Popovych, he also did relatively well later in the race, as signified by his large Mode 2 component. Although Leipheimer did well overall (Mode 1), his early losses in the Alps (Mode 2) were not great enough to overcome Evans. We could continue to add modes and represent the data in higher dimensions, but I think two is enough for a blog post.

The 2007 Tour is a fairly typical case for SVD. Most Tours show a similar dominant mode that combines climbing and time trial stages, encompassing most of the GC. It usually takes the addition of one or two more modes to work out the precise order of the podium and a couple more to fill in the details of the top 10, but the majority of activity encoded in the smaller modes are mid-GC reassortments due to breaks.

I think this generally means Tour results can be summarized by a few patterns in the data and therefore major changes in riders’ performances in the course of a grand tour are very rare. This is, however, a global and quantitative view of the data and should be taken as such; the difference between fourth and seventh place for an individual rider is still be quite important to them and their supporters! Vinokourov’s miniscule time gains on the Champs Élysées in 2005 do not appear until Mode 19 in a 2005 SVD analysis, but it certainly mattered to him and Levi Leipheimer.

Next I’ll consider the 2006 Tour de France. How does Oscar Pereiro’s unorthodox victory look through SVD glasses?

TdF Stage 17

The GC activity today was all very exciting, but the ride of the day has to go to Thor Hushovd. Said Thor:

"Today everything just felt perfect. I attacked over the first climb, did a good descent and then had an amazing day in the front," Hushovd said. "I think this is the best day I've ever had on the bike."

This is being played up as a smackdown to Cavendish (as if this bike race was lacking in manufactured histrionics), but I think that distracts from what an awesome ride this was. It is great to see my geeky blogging disrupted by news of manly riding!

The Tour de France has 3.4 Stages

With a time trial and mountain stage remaining in this year’s Tour de France, there are no lack of assurances that the Tour is not over. But once the GC has been sorted out by the early stages, how much do the results actually change? Is there any major variance in the results between each type of stage, or can the Tour be reduced to one representative mountain stage, a representative time trial, and a typical sprint stage?

To determine how complex the Tour is, I calculated the singular value decomposition (SVD) of stage finish times. SVD is a linear algebra technique that rearranges data into a series of features, called modes, weighted from the most important to least important. For example, it is used in image compression to reduce a matrix of pixels to an approximate matrix that is reconstructed from the a handful of modes. Furthermore, by looking at how quickly the mode weights decrease, one can estimate the effective number of independent components in the data. A famous example was a study of correlated voting records in the US Supreme Court, which concluded that the nine justices could be approximated well by 4.7 independent justices with uncorrelated voting patterns. Another way of stating this is that the information content of US Supreme Court decisions is the same as a court of 4.7 justices rather than nine.

Looking at Tour results, we can ask the same question. The Tour is usually 21 stages, but we can perform SVD analysis and determine the effective number of stages. I did this for a number of recent grand tours using results for time lost on each stage from Cycling Quotient. Here are the results:

The average Tour effectively has 3.4 stages, the average Giro 3.3, and the average Vuelta 4.3. The most complex tour here is the 2003 edition, which involved multiple long breaks and a closely contested GC. The Vuelta is consistently more complex than the Tour and Giro, perhaps because breaks are allowed freer reign or riders in the lead are more likely to crack after a long racing season.

So does the SVD analysis reduce the race to a single climbing, time trial, and sprinting stage, with perhaps a “half stage” for hilly transition stages? To fully answer this, we need to look at each stage race on a case-by-case basis. I will look at a few races in forthcoming posts. Briefly, since the consistently good climbers also tend to be the better time trialists, time trials and climbing stages are often combined into a single representative “GC mode”. Additional modes encode variations around this main trend, such as a difference for some riders between the Alps and Pyrenees, and breakaway results. It should be kept in mind that this analysis is entirely data driven – the modes correspond to how the results panned out rather than any preconceived notion of which stages are important in a grand tour. Since I do not have a handy source for complete results during the Indurain years I could not do the analysis, however I imagine the balance of time trialing and climbing might have been different then.


Technical notes: SVD is a linear algebra routine that produces a unique solution for a given data matrix. The results matrix was composed of stages x riders, such that each matrix element is the time lost in seconds for a given rider on a given stage. Time bonuses were not factored in because of personal laziness. Riders who did not finish all stages were excluded since SVD chokes on missing data. The effective number of stages is computed as the Shannon entropy for the fractional singular values squared. Missing grand tours are due to incomplete results in the CQ database.

Should Unprecedented Tour Success Warrant Suspicion?

In recent years at the Tour de France we’ve seen breakthrough results from riders like Floyd Landis and Bernhard Kohl nullified by doping convictions. In this light, it is tempting to view the current success of Bradley Wiggins with skepticism (see here for a generally civil discussion). Wiggins, now third on GC, is a multiple gold medalist and world champion on the track, but has never had much success in road or stage racing. He credits his new climbing ability to substantial weight loss paired with a new focus on road racing, but a hardcore skeptic who believes Bernhard Kohl is not going to be so easily convinced.

Of course, at the moment there is no way we can know if Wiggins is on the good juice or not. But we can look at his pattern of improvement and ask how extraordinary it actually is. Similarly, we can look at dopers and determine if their results were entirely unprecedented. To this end, I will consider two quantities:

• R: The difference between a rider’s mean placings in a given year and mean placings in previous years. Larger numbers mean greater improvement.


• P: The likelihood that this difference is real and not simply a result of random fluctuations (statistical significance). Smaller numbers mean greater significance.

Data, as usual, are from the indispensable Cycling Quotient. When computing P, I first take the logarithm of all results in order to enhance the value of top placings and minimize differences between mid-pack finishes (e.g. the difference between 2nd and 12th is much more important than the difference between 102th and 112nd). See below for more exciting technical notes.

Here are the career results for Bradley Wiggins:

Wiggins results so far in 2009 are definitely the best of his road career, with a mean placing of 66 compared to 108 over previous years. This corresponds to R = 42.5. The likelihood of this result in random fluctuations is P = 0.00012, suggesting that this is indeed statistically significance. Here is how Wiggins compares to a few other riders:

Sastre’s results appear to have improved slightly, but the P-value tells us this is not significant. Haussler is having a very good season but not to the extent that Wiggins is. Vande Velde was impressive in 2008, but Wiggins has shown an even greater improvement in 2009.

So how rare is this magic season of Bradley Wiggins? Looking back across the CQ data, I identified other riders who have had a season with an improvement greater than or better than Wiggins, defined as a R greater than 42.5 and P less than 0.00012:

Should we be skeptical of Wiggins based on this company? There are a few suspicious characters there, but overall this list is notably light on convicted dopers. We can compare these riders with some of the recent bad boys:

These improvements are generally more modest. Kashechkin, Landis, and Basso are the only riders in this lot whose gains were in the neighborhood of Wiggins's 2009. Three of these riders actually had a drop in results when doping (negative R), although these were not significant.

The data fairly clearly contradict the idea that significant performance improvements are necessarily, or even likely, the result of doping. As always, we should apply the standard caveat that the absence of a positive test does not always imply clean riding. However, without evidence to the contrary I think we should conclude that it is common for riders to significantly improve their performances without the aid of doping. Likewise, it is common for dopers not to have significantly improved results.


Technical notes: Most race results on Cycling Quotient are partial, often listing only the top 10 or 20 riders. For calculation of R and P, only races with 100 or more listed riders were used. Calculations were therefore based on about 900 races from 2000 to 2009, with the later years having more races listed. Roughly half of the races were stages from grand tours, and the remaining results are mostly the major one-day races and lesser stage races. To avoid small sample sizes, R and P were only computed if the rider at hand had more than 10 race results for the year in question and more than 10 results prior to that year. The likelihood P corresponds to the probability that there is no difference between a rider’s mean results in a given year and their mean results in previous years, and was computed using the Student’s t-test without assuming known or equal variances between the two samples.

TdF Stage 16

Well, wasn't that a blast from the past. Nice to see the old man throw down and bridge on a climb like that. He might just hold that second place yet.

TdF Stage 15

Perhaps it's a little early to call it, but the Tour GC tends to be decided by mountaintop finishes and we've probably seen enough. I think the major remaining questions are if Armstrong and Wiggins can maintain their podium spots on Ventoux.

Armstrong got a lot of grief for a comment regarding the quality of the top finishers in last year's Tour, but after today it looks like he might have had a point. Running down the top 5 from '08:

1. Sastre is 11th at 3:52,
2. Evans is 14th at 4:27,
3. Kohl is flipping burgers in Vienna,
4. Menchov is 29th at 11:23,
   
5. Vande Velde is 12th at 3:59.

I am now waiting for Armstrong to "kick their asses". Team time trials do not count.

TdF Stage 14

My goodness, who would have ever thought a win by Sergei Ivanov would get everyone so hot and bothered?

TdF Stage 13

I have read that the other racers call Heinrich Haussler "Barbie" for his attention to fashion and grooming. Crying as he won today's stage probably won't help that image, although a 195 km break ahead of the cold, whimpering peloton should earn him some respect. After he took a painful-to-watch second place at Milano-San Remo and another second at De Ronde, today's result isn't a surprise. Looking at Haussler's pro results prior to the start of the Tour shows how he has been steadily marching down the top ten over the past few years:


[Data from CyclingQuotient]

TdF Stage 12

I've often heard that recovery becomes more difficult with age. If that's true, Lance Armstrong must be pleased with the way the hard stages are spread out in this year's Tour.


I think Nocentini is going to lose his fancy shirt tomorrow.

TdF Stage 11

After years of working to chase down breaks for Robbie McEwen, gangly Belgian Johan Van Summeren got himself off the front today. Great to see him on the other side of that situation for once. Unfortunately for him, Columbia-HTC had plenty of help in bringing the race back together for a bunch sprint. There was a rumor that the finishing incline would be difficult for Cavendish, and I guess it was -- he could only beat Farrar by a wheel.


It must be really fun to be Eisel, Grabsch, or any of the other Columbia-HTC riders right now. There are few things more satisfying in cycling than working your ass off for a teammate that has the talent, smarts, and drive to deliver the win.


In very important news, apparently the UCI leaked to L'Equipe that some unnamed rider thinks Cavendish is an anti-French racist! Cavendish dealt with this matter appropriately:

"For sure I’m going to get arsy at some riders, because, you know, I’m an asshole," said Cavendish. "But it’s irrelevant their nationality, and irrelevant what they look like, or where they come from. Because, like I said, I’m an asshole."

Now Cavendish could have escaped this controversy by pointing out that as a sort-of-British person he is expected to dislike the French, but clearly he has come to the Tour in better form than that. Bravo!

TdF Stage 10

ASO's plans to liven up le Tour by taking everyone's radios away seems to have backfired. In fact, the race was so boring I suspect we saw a subtle protest at the finish -- Hushovd and Farrar agreed not to try coming around Cavendish so that even the sprint would be tedious. Or maybe they were waiting for their DS to point out the finish line in the last 200 meters. In any case the stage made the rest day press conferences look action packed. Maybe next time assign all riders and DS random radios?


In other, equally exciting news it seems that Giro has their new lid on a number of teams (Astana, Garmin, etc). Apparently it's very light and weighs not so many grams, but it looks to me like the helmet Postal was wearing in 1999 (maybe now with a Greek name?). I think this will be a marketing challenge for Giro; will the weight savings overcome the dated styling in the eyes of consumers? If my riding cohort is representative of the market, the answer is no. Lightness is fine, but cycling is all about looking good these days. In white, like a bride.

TdF Stage 9

Egoi Martinez took the mountains jersey today, with Franco Pellizotti 23 points behind. I expected David Moncoutie to make a stronger run for the jersey, but he did enough too keep himself in contention by getting in today's early break and fighting for a few points. Yes, this is a minor matter, but I don't have much more to say about the fairly boring stage today. Whatever happened to that plucky little mountain scamp, Riccardo Ricco? I miss him.


I also missed most of the TV coverage since I was "enjoying" a 40km time trial today. Unfortunately I did just well enough to encourage myself to put some more work into TTs. My current muscle aches provide evidence that Fabian Cancellara's massive glutes have contributed greatly to his success.

TdF Stage 8

I do love a small group sprint, and today Luis Leon Sanchez made it look easy. He waited form Casar to do most of the work to bring Efimkin back, closed the gap himself with a few pedal strokes, then paused and let Casar come around to lead him out. He glanced over his right shoulder as if he was giving Casar permission to go, then came around him easily for the win.


I hung around to watch the Versus interviews this morning. They really need to give George Hincapie a while to recover before doing post-stage interviews. Let him have something to drink, maybe some food to get some glucose back in his brain. Ambushing him while non-functional just isn't fair. Then Thor Hushovd had trouble getting his massive Viking hands through the tiny bike-racer sleeves on that green jersey.


Johan Bruyneel claims confusion at the early attack from Cadel Evans. Don't play coy with us, you Belgian minx! Everyone knows you are the evil genius of le Tour, the puppetmaster who controls all.


Oscar Pereiro apparently gave up today. Is his career Beloking?

TdF Stage 7

Fantastic win by Brice Feillu as France takes first and second on a mountaintop finish. Cue the "Is this what a clean Tour looks like?" speculation. It's another great result for France and a big day for Feillu's title sponsor, le ribcage.


In GC news, so concludes the Great Astana Leadership Drama of 2009. Contador now leads Armstrong and I doubt that is going to change. Armstrong should race for second, donate millions to the UCI anti-doping program, and maybe have a longshot chance for an eight Tour win as we head into fall. Leipheimer looked more comfortable than I expected, and Evans continues to loosen up his style nicely. Andy Schleck was disappointing; I expected him to follow Contador but he couldn't even drop his fat brother. And how about track star Bradley Wiggins climbing with the GC contenders and attacking in the final kilometer? Suspicalicious!

TdF Stage 6

Over 20 riders crashed in today's stage. That looked far more dangerous than the TTT; I wonder who will be the first rider to call for canceling races when it rains. Angry about being left out, René Haselbacher crashed at the Tour of Austria.


The UCI sorta promises next-day turnover for doping results. This could be a problem, since it doesn't allow much time to leak the positives to L'Equipe.


Tomorrow ends with a bit of switchbacky goodness:
















Hard for me to see anyone other than Contador winning.

TdF Stage 5

Well done, Mr Voeckler! I wonder how many other people didn't know he had never won a TdF stage. Maybe he was given an honorary win at some point.


I agree with whiny Rolf Aldag. If you're sending a team to the Tour, it only makes sense to try winning a stage.


And finally, Bradley Wiggins is saying:

[T]here are so many different levels at this Tour: there are 100 blokes who are good and 100 blokes who are not so good, and it's all mixed up.

Is he saying it's a peloton at two speeds, and he's in the fast group? My goodness, that team is suspicious.