The 2013 Tour de France has gotten off to an eventful start, characterized by bus incidents, sprints for yellow jerseys, single-second breakaways claiming yellow, and controversies re timing.

Soon, the Centenary Tour will hit the mountains, and as has become an annual series here on the Science of Sport, we’ll spend some time digging into the rider’s performances on those climbs – the times, the estimates for power outputs and their implications. I can’t stress enough that everything we do is estimated, aimed at providing you with some kind of insight as you watch the race develop. In an ideal world, we would not rely on models that estimate power outputs based on assumptions of bike and rider mass, drafting/wind, and road conditions. We would instead see the actual data from the front of the race.

However, this is unlikely to happen, and so for now we do what is possible, and hope to explain the implicit assumptions as part of the discourse. As always, it is the process of discussing a subject, not the outcome, that is of value.

So, in order to lay some groundwork before the first mountain stages in the Pyrenees starting this Saturday, I thought I’d try to explain the approach and some of the methods we will use over the course of the next few weeks.

## The mountains – times, powers and implications

The Tour de France is won and lost in the mountains, and it’s here that the physiology of the best cyclists in the world is best analyzed. Freed (to some extent, anyway) from the energy savings of drafting on the flat roads, and required to overcome gravity on slopes ranging from 6% to 13%, the power output produced by the cyclist is the key determinant of performance and thus position.

That power output carries with it some important physiological implications, because the whole system, for want of a better term, is ‘closed’ and it is thus possible to estimate, within a reasonable set of assumptions, what kind of physiology drives a given power output. That’s because we know that the ability to ride at a power output X is the result of a given maximal oxygen carrying capacity, a given mechanical efficiency and a sustainable exercise work rate that is relatively uniform within elite athletes (this last variable could be called any number of things – threshold ability, functional power output etc).

Therefore, power output reveals underlying physiology. The problem we have (as outsiders, anyway) is access to data. I have for four years been saying that I think the credibility of the sport (and its presentation to the viewer) would be greatly enhanced if power output was accurately measured and provided openly. People argue against this based on the fact that it would make the racing predictable, that the major contenders would know what their rivals are producing and thus change the racing. I disagree – this is, to me, like saying that the men’s 100m final at the Olympics is boring and predictable because we know, within about 1%, what the winning time is going to be. It doesn’t matter that the best 7 men in the world know that Usain Bolt will run around 9.65s, the spectacle is to see how the result is achieved and who competes to change it.

So, whether cyclists know that winning on Alp d’Huez requires a time of 41 minutes and a corresponding power output of around 6.1 W/kg is immaterial. And finally, I’d point out that the riders and coaches all know this already – it’s not as though Contador will be suddenly enlightened when he discovers that Froome is producing 6 W/kg to ride away from him – he knew it already, it’s just that we didn’t. So in my opinion, the power output data would add to the value of the Tour, and certainly, if understood and explained correctly, enhance its credibility, or flag suspicion, that could be used to do so.

Alas, we are not there yet, and so we look to other performance indicators, and estimates, to provide the insight.

## Time comparison – what is reasonable?

The power output of course produces the time, and so the first point of analysis is to compare times from one year to the next. If a rider in 2013 produces an ascent of Alp d’Huez that is faster than anything Pantani, Ullrich, Virenque and Armstrong were able to produce given the doping of that era, then it should be pause for concern and some suspicion.

This has been an “unpopular” concept because some people assume it to be the equivalent of guilt based on performance – if you are too fast, you must be doping. That’s not entirely true, though it would be almost patronizing to say that there isn’t an element of truth in it. The reality, as far as I’m concerned, is that in the past, doping exerted such a large effect on performance that it pushed performances beyond what is possible with normal physiology. Recent work by Pitsiladis suggests that EPO use improves endurance performance by around 5%, and that’s for relatively short duration endurance. It may be greater in the Tour, and so you have this shift in capacity that moves performances beyond what is normal. So I do subscribe to the belief that what happened in the 90s and 2000s cannot be overcome by normal training, however gifted an athlete may be.

In time, advances in training, technology and preparation may slowly erode that advantage, but within a narrow period of a few generations, the effect of the doping seen in cycling in the 90s and early 2000s was so large that I don’t believe it possible to match doped performances, and so if or when it happens, some very important questions must be asked. Certainly, in the last few years, every Grand Tour has experienced a “slowing down”, and the times in the Giro, Tour and Vuelta have been, with one or two exceptions, slower than they were prior to the biological passport’s introduction. In 2011, I wrote a piece for the New York Times describing this and since then, the performances have remained slower. Not proof of a clean sport, by any means, but an encouraging sign.

Returning to the issue of ‘guilt by outstanding performance’, it’s important to understand that because the analysis of times does not always account for race tactics, environment and other contexts, no rider can ever be declared guilty of doping based on times (and resultant estimates of power output alone). There can be no judgment, only informed questioning. I remember a few years ago, Alberto Contador produced an amazingly fast ascent of Verbier, leading to accusations of doping. As it turned out, they were true…! However, that particular climb was not the ‘proof’ people said it was, because it benefitted from being very short in duration, with a strong wind from behind, and so the performance was inflated and thus wrongly compared to climbs like Alp d’Huez.

The point is, as the next few weeks unfold, **I will report on the climbing times, past and present, and comparing what Froome and Contador and co produce in 2013**. However, these are only indicators, and once the context is understood, then we can begin to understand their implications. I’ll say this – if anyone rides anything close to what the doped generation does, I’ll be among the first to raise those uncomfortable questions.

## Power output: Some basic concepts

The graph below, which I’ve drawn based on data provided by the Cycling Power Lab website, shows the estimated time on the Tour’s first big mountain finish, Ax-3-Domaines, as a function of the power output produced for two riders, one at 64kg and one at 70kg.

For the uninitiated, the graph shows that mass matters in mountains. Climbers don’t want to carry weight, because at the same power output (say, 400W – follow the blue line up) the estimated time is around 1 min faster for the 64kg rider. That’s why relative power output matters – with some fineprint, it’s the power output produced per kilogram that tells in the mountains. That 400W performance corresponds to 6.25 W/kg for the 64kg rider, whereas it’s only 5.7 W/kg for the 70kg rider.

In order for the 70kg rider to match the 64kg, he’d need a similar relative power output (6.25 W/kg), which corresponds to around 435 W (red dashed lines).

## pVAM, residuals and the Ferrari equation

Now, in terms of what is considered a reasonable performance, in addition to historical comparisons, we can also begin to model what power outputs are considered ‘reasonable’.

Over at the Veloclinic, you can find what has been defined as the pVAM, or Predicted Vertical Ascension meters. This metric, which is expressed as meters per hour, originates with the notorious Dr Michele Ferrari, but it’s a contribution he made to the sport that is actually worth considering.

Basically, what Ferrari would have done over a few years, with many cyclists, is derive an equation that models the climbing rate in vertical meters per hour against the measured power output. VAM is easy to calculate – if a climb starts at 200m and tops out at 1200m, the vertical climb is 1000m. If that’s done in 40 minutes, VAM is 1500 m/hour.

Once an equation for the relationship between power output and VAM exists (this is a simple regression analysis), you would no longer need to measure the power output directly. Instead, you could estimate it based on the known climbing rate.

That climbing rate, or VAM, is a function of a few things, but primary among them is the gradient – a steep climb will, for the same power output, produce a higher VAM, and that’s why Ferrari’s equation looks as follows:

*Calculated power output (W/kg) = [VAM (m/hour)] / [((2 + (% grade/10)) x 100)]*

Example: if VAM = 1600m/h, and the gradient is 7.46%, then the equation is:

W = 1600/(((2+7.46/10)*100)) = 5.83 W/kg

Now, important to realize is that on longer climbs, we would expect that the power output will be lower, and thus our expectation for VAM should also be lower. Similarly, altitude affects performance, and so the greater the altitude of a climb, the lower the predicted power output and VAM.

The Ferrari equation does not account for this, but at Veloclinic, a new equation for pVAM has been estimated. It is as follows:

*pVAM = 2885.17 + 416.825 ln Gradient – 0.06197 VClimb – 0.08796 Altitude (as per Doc, Veloclinic)*

That equation, incidentally, was derived from an analysis of climbing times in the period 2008 to present, and so it assumes (probably not entirely correctly) that the performance is “unaided” by doping. Certainly, I think it’s a reasonable assumption when compared to the period before it, 2002 to 2007, which Doc has worked out as a dVAM, or predicted VAM with doping (all of this is explained, if you can bypass the strange language, at this link), but not altogether ‘pure’.

Nevertheless, the genius of this method is that it allows every climb to be predicted, accounting for the factors that most strongly influence the performance. Then, the actual performance can be compared to predicted performance to reveal historically strong or weak performances. Doc at Veloclinic does this, and he calls it the residual. If a performance is better than the prediction, you will get a positive difference/residual (for example, a guy climbs at 1640 m/h when the pVAM was 1600 m/h, the residual is + 2.5%).

One need not even convert VAM to power output for this to have meaning – the metric is basically the same as time, provided the climb is measured accurately and length and vertical height gain are known at specific time points.

Doing this, one can now draw the following graphs, again showing the estimated time for Ax-3-Domaines, this time as a function of pVAM (left panel) or calculated power output (right panel). You may have to click to enlarge.

So, the method predicts a time of 24:17 for the 8.9km climb, based on a pVAM of 1640 m/hour. That corresponds, per Ferrari’s equation, to a relative power output of 5.98 W/kg, which is near enough that 6 to 6.2 W/kg limit you may have heard of so often.

That limit, incidentally, is something I’ve written on a great deal in the past, but will refer you to this post and the links it contains for more detail on this subject.

Briefly, however, the premise here is that in order to produce 6.2 W/kg or higher for longer than about 30 minutes requires physiology that is, frankly, not seen in normal situations. That doesn’t mean it’s not possible, but to illustrate, in order to ride at this kind of power output, a cyclist must have a VO2max that is tremendously high, in combination with an exceptional efficiency, and the ability to sustain upward of 85% of VO2max for those 30 minutes or more, at the end of a 5 hour stage. The combination of physiological factors does not, in my opinion, exist in order to validate power outputs above 6.2 W/kg for those durations.

On shorter climbs, in the range of 10 to 20 minutes, it is absolutely expected that these power outputs will be recorded. But on the HC climbs that end the big mountain stages, I would be very, very skeptical of anything above those values.

The pVAM method accounts for the length, altitude and gradient, and the physiological method explains some of the implications of those power outputs, which I don’t believe are reasonable. Thus, if the Tour is clean (and how big an ‘if’ that is depends on your point of view, your disillusionment with the sport and your cynicism), then we expect to see power outputs in the range of 5.9 W/kg to 6.1 W/kg for the HC climbs.

## Looking ahead – what to expect from analysis here on The Science of Sport

During the Tour, what I’ll do ahead of each mountain stage is graphically show you the expected times, and the expected power outputs. Then, after the stage, once the data are in, we can compare each climb to the various predictions, and describe again the insights the performances offer.

Again, I can’t stress enough that this is not done with judgment of performance in mind. It’s partly because the study of the limits of performance is fascinating, and partly because we can develop informed, insightful opinions on the state of the sport by understanding the power output. I confess, upfront, and will continue to do so as the race develops, that these are imperfect methods, involving estimates and assumptions. Where possible, I will provide actual SRM data to validate the models (or reveal their inaccuracies), and hopefully by the time the Tour rolls into Paris in just under 3 weeks, we’ll all be better for it.

It is a busy time of year for me, so I will also apologize in advance if I can’t keep up with very stage of the race – these long posts obviously draw significant time away from other responsibilities, so no guarantees! I will however guarantee that I’ll share brief thoughts and the insights and analysis of others over at our Twitter and Facebook pages, so do follow us there if you would like more frequent, shorter thoughts during this 100th Tour de France.

Until the Pyrenees begins on Saturday, and our first prediction series, enjoy the racing!

Ross

###### Addendum – the accuracy of Ferrari’s VAM

The accuracy of this method is an important question. Based on some data sent to me recently, I plotted Ferrari’s estimated power output against actual SRM data and found a relatively strong correlation (r = 0.86) between the two. The range of error was between -5% and +5%, which means that on any given climb, the estimate by Ferrari could be around 20 W different to the SRM value. That’s likely because Ferrari’s method is a function of time, and so a tailwind would lower the time, raise the VAM and hence the calculated power output, leading to over-estimation. Similarly, a head wind (or solo effort which is denied some drafting benefit) might lead to under-estimate. These are, to repeat,all reasons why this kind of analysis must be done sensibly and always with context and specific situations in mind.

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