Bayesian 1RM Calculator

Here’s an idea that I had that I’d like to try to implement when I have some more free time. If I never get around to it, I still reckon it’s worth doing:

Building a Better 1RM Calculator: Bayesian Inference of Lifting Ability
Accommodating Variation between Individuals and Lifts and Uncertainty in Measurement and Inference

Motivation: AFAIK, most (if not all) internet one-rep-max calculators give you point estimates and fail to take into account a lot of factors (even what lift you’re doing!). Estimating 1RMs is a highly uncertain business, so why not try to preserve some of that uncertainty when making predictions?

Methods: Bayesian regression is one of the workhorses of statistics. Here, I’d average across a lot of plausible multilevel linear models (probably with a Gaussian likelihood) fitted with HMC (stuff like quadratic approximation won’t work with a hierarchical model structure). Inference of 1RM would make use of the entire joint posterior distribution using posterior predictive simulation.

To do this, I’d need to collect survey data on (people can skip questions if they want and we could impute missing data later):

Outcome Variables: Known, tested 1RMs, but also ask how certain people are in these (since Bayes can accommodate measurement uncertainty), if it’s on a “good day”, “bad day”, or an “average/any day”, their range of motion, and if any form breakdown occurred? Ask for a variety of lifts (maybe all the powerlifting and olympic weightlifting ones to start?)

Key Predictor Variables: Known, tested 3,5,8,10,12,15,20RMs (??), with the same extra bits as above. Use these to predict the 1RMs by fitting a variety of multilevel linear models with multiple predictor variables and interactions (alternatively, we can just have 2 main effects from here — weight lifted and # of reps — with an interaction between them). Multicollinearity and nonidentifiability are fine for prediction in a Bayesian framework (though not chain performance, which we’d check; we’re not interested in anything to do with causation and just care about getting good estimates of 1RM), and using approximations of out-of-sample deviance (e.g. WAIC, which is most applicable because it lets me use non-multivariate normal posteriors and adaptively regularizing, non-uniform priors. Other model comparison and averaging devices could also be used, though) we can accommodate model uncertainty and avoid overfitting to our training data.

Other Plausible Predictor Variables: These can help further “personalize” the estimates and potentially improve accuracy in prediction. Some variables that are likely to be relevant include: Gender (or morphological sex assigned at birth?), Height, Weight, PED use, Other Supplements (e.g. pre-workout, caffeine, creatine, etc.), Experience (years lifting, average number of hours per week spent lifting, powerlifting competition participation?). Instead of needing a complex biomechanical model that takes into account difficult-to-measure variables (e.g. muscle fiber composition, length of levers, w/e) we can get at some of the underlying variation in causal mechanism if it varies across some of these predictors. Again, we’d not be interested in process here, just prediction.

Even More Predictor Variables?: Country? Ethnicity? Other history with athletics in strength or cardio/conditioning/endurance (e.g. experience running)? Intermembral index? Other plausibly relevant anthropometrics? BF%? “Build”? Injury history?

Additional survey questions? Maybe ask people to guesstimate how exceptional their lifts are for their demographic using both qualitative and quantitative metrics (e.g. novice-intermediate-advanced-elite and percentile, respectively). See how responses vary across actual performance and who’s “self vs. other”-assessment is most accurate (e.g. look to see if high performers systematically underpredict their exceptionality and low performers systematically overpredict; sort of a lifting Dunning-Kruger Effect?).

Incentive to complete initial survey: In addition to people giving their data out of the goodness of their hearts and in the hopes of ultimately having access to a better 1RM calculator, reward them for their information by telling them where in the distribution for their demographic they actually are (given the current data).

Applications: Ultimately use all this to estimate some X% credible interval (e.g. 88% HPDI) of their 1RM; or, better yet, just give them the entire posterior predictive distribution. Can also predict other X-RMs, not just 1RMs, but while that’s generally less interesting, it can still be useful for planning out super customized periodization down the line.

Also let them ask, “Given the training data, my information, and the models under consideration, what is the probability that I will be able to lift at least ABC lbs for my 1RM?”

And “Given all my information, the training data, and models, how high should my X-RM that I might have >Y% chance of hitting Z for my 1RM”?

Outliers: One easy and obvious outlier criterion – if a person’s claimed lifts are breaking world records for their weight and gender, they’re probably lying, and if they aren’t, wouldn’t necessarily be representative. We’d only interested in excluding lying liars, not “statistical outliers”

Other Considerations: Issues with self-report vs actual testing – which is actually more reliable? Self-report might be prone to exaggeration, but we can take that into account. Experimentally collected data might not be the most representative of real world conditions and would take a lot of effort and a loooong time to collect. Are there any publicly available datasets upon which the many existing 1RM calculators are based? What’s included in those datasets?

(photocredit: Wikimedia Commons)

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