Problems with the critical speed model: Can power laws predict running performance better?

The critical speed model—also known as critical velocity, CV, or critical power—is a powerful concept for understanding what running speeds are sustainable at a metabolic steady-state and what speeds are not.

Critical speed is not without its detractors, though, and the critical speed model is certainly not without its flaws.

I just posted a huge article on understanding the science of critical speed, critical velocity, and critical power for runners. That article goes in-depth on what critical speed is, how the model works, and how you can use it in your training.

I was originally planning on including the major criticisms of critical speed as a part of that article, but it’s long enough as it is.

So here, separately, is a summary and analysis of the main problems with critical speed (and, by extension, critical power) as a model for endurance performance, plus some rejoinders as to why alternative models, like the power law model of performance, are not always better.

The critical speed model suggests “infinite” endurance below critical speed

Even though the modern view of the critical speed model does not see CS as an indefinitely-sustainable speed, a simple look at the hyperbolic fit clearly shows this shortcoming: critical speed will overpredict performance at event durations longer than about 20 minutes.

Let’s take a look again at a critical speed model. This time, I’ll fit it to data from Sifan Hassan’s 2019 season, where she ran distances from 1500m up to the half marathon. Remember, the critical speed model is only supposed to be fit to data from performances lasting between two and 20 minutes.[1]

Here’s a model fit to her 1500m, mile, 3k, and 5k times from that season:

Hyperbolic plot of critical speed model for an elite female runner.

As expected, the critical speed model fits these performances pretty well.

However, problems arise when we look at Sifan Hassan's 10k and half marathon performances that season:

The critical speed model predicts, incorrectly, that Hassan’s HM pace should be nearly the same as her 10k pace, when in fact she (as expected) slows down about 5% from 5k to 10k, and from 10k to the half marathon.[2]

Proponents of critical speed argue that critical speed is primarily a boundary between different domains of exercise, and speeds below critical speed are not indefinitely sustainable, but cause fatigue for different reasons than speeds above critical speed.

This is a pretty reasonable argument, and I cover the evidence for the “different types of fatigue” idea in this section of the main article on critical speed.

Critical speed overpredicts performances in events lasting less than two minutes

The critical speed model also does not perform well in very short events, suggesting better capabilities than you really possess in an event like the 400m or the 600m.

Here’s another critical speed model fit to a collegiate middle-distance runner active over distances of 400m to the mile. As recommended, I fit the critical speed model to her 800m, 1000m, 11500m, and mile times.

Critical speed model for middle distance runner overpredicting performance in the 600m and 400m

The critical speed model pretty seriously overestimates performance in the 600m and 400m.

Supporters of the critical speed model point out—fairly, in my view—that very short events start becoming affected not just by your glycolytic energy reserve (the D’ parameter), but also by the biomechanical limitations of muscle force production.

Once muscle fibers start contracting extremely quickly, their ability to produce force decreases dramatically (this is the “force-velocity relationship”). The critical speed model does not factor in these fundamental properties of muscle, which could be why it performs poorly at shorter events.

Power laws predict running performance better than critical speed (sometimes)

There are competing models of running performance as a function of distance based on power laws, which are equations of the form:

    \[s = S*t_{lim}^{1/F-1}\]

s is still running speed and t_{lim} is still time, but now the two parameters in the model are S and F, sometimes called a “speed factor” and “fatigue factor.”[3]

Old-school runners will recognize this equation as the “Riegel model,” which was popularized in the late ‘70s and early ‘80s for predicting running performance (often with F fixed at 1.06) and still crops up from time to time online.

Critical speed models vs. power law models

Here’s a comparison of the critical speed model and the power law model fit to Sifan Hassan’s data from earlier:

The fit is better for the 10k, and off by about the same amount for the half.

And here’s a comparison of a power law model to a critical speed model for our mid-distance runner from above.

For a fair comparison, I fit both models to the same performances (black dots). The white dots (10k and HM, 400m and 600m) are “new” data that the models did not use to learn their parameters.

Unlike critical speed models, power laws clearly do predict event performances below two minutes and above above 20 minutes pretty well, though they still break down for performances below one minute in duration (and in practice, often fail for >60 min races too) [1].

Problem: running power laws are not interpretable

However, power laws have a big problem: S and F are completely uninterpretable.

These parameters don't mean anything. Sifan Hassan’s “Speed factor (S)” is 10.1 and her “Fatigue factor (F)” is 1.089. What does that mean? Who knows! And don’t even ask what units those are measured in!

To be fair, if we had a lot of runners, we could perhaps rank them according to their S and F factors, or look at correlations among S and F factors across runners of different types.

Still, the power law model makes no biological predictions, other than that runners will slow down as the run duration gets longer. Moreover, the power law relationship implies nothing about where the maximum metabolic steady-state should occur!

The critical speed model is interpretable and makes testable predictions

In contrast, both CS and D’ from the critical speed model have clear and obvious interpretations: CS is the fastest speed that can be sustained at a metabolic steady-state, and D’ is a finite energy reserve.[4]

Moreover, CS and D’ make testable, and mostly correct, predictions. If Sifan Hassan runs a bit slower than than 5.56 m/s (4:49/mi), she’ll be at a metabolic steady-state. So, her coach could pretty confidently prescribe 2k repeats at 5:00/mi knowing these would produce a metabolic steady-state.

Ditto for our middle-distance runner—her critical speed is 4.11 m/s, which is 6:31/mi. Suppose she’s heading into summer break to start up cross-country training. She should be able to handle longer tempo-style workouts (repeats or continuous runs) at ~6:40/mi while maintaining a metabolic steady-state.

Of course you’d probably take a break after track, so you’d want to dial this pace back more in the real world, but for the sake of argument we’ll ignore that for now.

This, in my mind, is the real strength of critical speed: I can come up with any mathematical function to predict running performance. But only critical speed makes testable predictions about the body’s physiological response to different speeds.

The critical speed model works surprisingly well when used correctly

Speaking of cross-country, that same middle distance runner from above really does run cross-country. For fun, I plugged her 6k XC PR into both models. Remember, both models only use her 800m to mile performances to fit their predicted curve.

Critical speed predicts XC performance better than a power law

Amazing!

I was honestly shocked at how well the critical speed model predicted 6k XC performance for this runner. However, I’m not shocked at all that it outperforms the power law model.

Now, to be fair, this athlete probably couldn't run that 6k XC time during the track season: she's not specialized for the distance. You might be able to hand-wave some explanations about 6k XC being slower than a road or a track race, though.

In any case, let’s take a look at what the power law is actually predicting.

At a race duration of 30 minutes, the power law relationship predicts that our mid-distance runner should be capable of 3.36 meters per second.

You probably don’t know running speeds in meters per second, but if I told you that’s 8:00/mi pace, your intuitions should click here—that doesn’t make any sense!

I too find it extremely implausible that this 5:35 miler would be incapable of running faster than 8:00/mi for 30 minutes at an all-out effort. She probably does easy runs not much slower than that with her teammates in the fall.

The critical speed model fails in a predictable way

The nice part about the critical speed model is that its mathematical form “bakes in” some very specific knowledge about how humans typically fatigue, so it fails in a predictable way—because it only accounts for the biochemical, peripheral fatigue that occurs above critical speed, it will reliably over-predict performance in events over 25 minutes or so.[5]

Power laws, though, are in a sense too flexible: they bake in less biological prior knowledge, so can behave unpredictably both above and below the bounds of their ‘training data.’

For me at least, the marginally better performance at 400m and 600m are not worth the huge errors at 6k.

Critical speed does not explain optimal pacing, but power laws do

The critical power model does not “penalize” going faster early in a race, so going out too fast shouldn’t penalize your performance—when clearly it does in the real world.

Of note, proponents of power law models claim that they give a clear mathematical explanation of why even splits are the optimal pacing strategy [2]. However, it’s not clear to me that this explanation has explanatory power—it hasn’t been tested against real intermediate splits from races.[6]

Counterpoint: Power laws do not explain interval workout recovery

An important weakness of power laws is that they can’t account for or accurately model recovery in interval workouts or unevenly-paced.

The view of D’ as a finite energy source offers an easy way to predict recovery during an interval session, which is a more advanced application of critical speed that I’ll cover in a future article.[7]

The short version, though, is that a pretty simple half-life model of D’ recovery explains the recovery of performance abilities quite well, making critical speed useful for planning interval workouts and for more unusual sprint/recovery-type events, like you sometimes see in cycling.

Critical speed also provides a great explanation of the recovery that happens immediately after an all-out effort.

Here’s a fascinating finding described in a 2018 paper by Mark Burnely and Andy Jones—the studies were on cyclists, but I’ll translate them into running terms.

Let’s say you got on a treadmill and set the speed to your 3k pace—well faster than your critical speed—and ran until you were so exhausted you had to stop.

If you immediately bumped the speed down to a slower pace that was nevertheless still greater than your critical speed (say, 5k pace), you’d only be able to hang on for a few more seconds until you had to stop again.

However, if you bumped the speed down to a pace below critical speed (say, marathon pace), you’d be able to keep going for ten minutes or more until you finally had to stop again.

This finding is much more in line with the critical speed model than the power law model. Under the power law’s model of fatigue, once you’re done, you’re done—you shouldn’t be able to continue any longer, at any speed.

Recap

Critical speed is a powerful model, both for predicting race performance and for understanding the body’s response to different running speeds. However, it is not perfect.

Critical speed only predicts performance accurately for events from around two to 20–25 minutes in length. Moreover, if all you care about is accurate race performance prediction, a power law might do a better job across a wider range of speeds.

The critical speed model also does not explain why even splits are the best way to run a fast time, which is something that does come out of power laws.

The real strength of critical speed, though, is not in its race prediction capabilities, but in its ability to predict what speeds are sustainable at a metabolic steady-state, and what speeds are not. For this task, the critical speed model is the best approach in physiology to date, and it does surprisingly well when used correctly.

As long as you keep its limitations in mind, critical speed is more useful and more interpretable than power law models.

Footnotes


[1] Moreover, proponents of critical speed would argue that critical speed isn’t supposed to be used to make predictions for races over ~20-25 minutes in length anyways—though sometimes it is anyways.

[2] In fact, the “5% rule” predictions, which form part of the rationale for percentage-based training, are nearly perfect for Sifan Hassan. This does not surprise me at all, since the 5% rule works best for well-trained, well-rounded runners like Hassan.

[3] Power laws are also equivalent to “log-log” models of speed and time; you can estimate S and F from linear regression of log(t_{lim}) on the x axis and log(s) on the y axis. To get S you just do exp(\beta_0) and to get F you do 1/(\beta_1+1) if \beta_0 is your regression intercept and \beta_1 is your slope.

[4] In the main article, I pointed out how D’ being a distance really means it’s an amount of energy. Another view is that D’ represents the amount of extra distance you can cover in a given duration, on top of the distance you could’ve covered for that amount of time running at CS. It’s a mouthful to explain, but mathematically it’s pretty simple.

[5] It’s plausible to me that endurance performance is made up of several different “regimes” or “domains,” with performance in each domain being dominated, but not wholly determined by, a particular set of fatigue responses, and that the zoomed out view of performance from ~400m to the marathon and beyond, spanning many domains of fatigue, ends up approximating something like a power law. This wouldn’t be too different from the famous Galton board, where many different +1 / -1 discrete increments end up approximating the Gaussian (normal) distribution when you zoom out.

[6] This should be easy to do, by the way: just grab 400m split times from a big meet on FlashResults or similar, then test the model to see if it can predict finish times of people who get out too fast!

[7] The reason this doesn’t work for power laws is that they do not have the constant-area property that the critical speed hyperbola has. The authors of one of the main broadsides against critical speed claim they are working on a modified power law model that incorporates recovery, but have not published it yet.

About the Author

John J Davis, PhD

I have been coaching runners and writing about training and injuries for over ten years. I've helped total novices, NXN-qualifying high schoolers, elite-field competitors at major marathons, and runners everywhere in between. I have a Ph.D. in Human Performance, and I do scientific research focused on the biomechanics of overuse injuries in runners. I published my first book, Modern Training and Physiology for Middle and Long-Distance Runners, in 2013.

2 thoughts on “Problems with the critical speed model: Can power laws predict running performance better?”

  1. Hi John,

    I'm a practicing coach using both the critical speed paradigm and Riegel power law.

    I use CS to profile runners maximum metabolic steady state as a surrogate measure instead of lab test. In a few weeks, I'll be going to the lab myself to compare field predictions with a lab MLSS test to see how close or how far.

    For reasonably accurate race performance prediction, a power law helps. I've seen much more "sensible" marathon predictions coming out of power law because if you input data from atleast 3 race performances spread across 5K, 10K and 21.1K, the resulting fatigue factor is individual to that runner and yields a conservative marathon prediction. Its not perfect but fills a gap that the CS model cannot.

    I'll fill up some words on this on my blog in a week or two. Thank you for the post.

    -Ron
    http://www.georgeron.com
    http://www.instagram.com/coach.ronnieg

    Reply
    • Interesting that you find power laws useful for marathon training! Do you find that the "fatigue factor" is pretty constant as a given runner develops? I often use 10k pace or 5k pace as an initial (but approximate) estimate of marathon pace using the "5% rule." Often, I see athletes improving their marathon fitness while their 10k fitness and HM fitness do not change (at least based on their ability to do 10k pace or HM pace in workouts). My interpretation of this has always been that their marathon pace is getting closer to their 10k and HM paces, even though those aren't improving. Do you find this as well, or does the fatigue factor stay pretty constant?

      Reply

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