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The hot hand—in darts!

Roland Langrock writes:

Since on your blog you’ve regularly been discussing hot hand literature – which we closely followed – I’m writing to share with you a new working paper we wrote on a potential hot hand pattern in professional darts.

We use state-space models in which a continuous-valued latent “hotness” variable, modeled as an autoregressive process, is assumed to affect throwing success probabilities. We find strong but short-lived correlation in the latent (hotness) process, which we claim provides clear evidence for a hot hand phenomenon in darts. Right now, we’re implementing the model in Stan, since we want to incorporate random effects to account for potential player heterogeneity regarding the hot hand effect.

I replied:

Regarding hotness etc., we’ve considered two sorts of models:
1. continuously varying underlying ability (thus, the hot hand is a “correlational” phenomenon)
2. making a shot increases the probability of making the next shot (a “causal” model)
It would make sense to include both factors, of course, but either one of them is hard enough to estimate on its own, for reasons discussed on blog: (a) the binary outcome is so noisy that you’d need lots of data, but if you have a long series, then hotness won’t last so long, and (b) the state variable can only be measured from noisy 0/1 data, so it’s hard to tell whether you’re hot or not just from data on recent successes.

With darts it should be much easier to study this because you have continuous data on how close the shot is to its intended target. For basketball we’ve talked about doing something similar using 3-d ball-tracking data, but that would take a lot of work!

7 Comments

  1. Tom Passin says:

    “We find strong but short-lived correlation in the latent (hotness) process, which we claim provides clear evidence for a hot hand phenomenon in darts”.

    That’s exactly the kind of effect I would expect based on my personal experience back when I used to play ping-pong seriously. Of course it’s all subjective when you are playing, but I would get strings of successful forehand smashes, for example – it seemed like I couldn’t miss. That would last for five or eight shots, then back to normal. Same for fast killer serves, and for returns with a lot of English on the ball. While it was happening, I experienced some subtle difference in the way my muscles felt. And I could pretty well count on making the plays, to the point that I might actually change my tactics.

    I know that doesn’t prove anything, but it’s very consistent with short-lived serial correlations. My personal speculation about this effect (if real) is that your brain learns how to tune its processing to improve the signal to noise ratio of the sensor-nerve-muscle systems. But that’s so hard to do, and the underlying biological systems are so noisy and variable, that this improvement doesn’t last long. While it’s working though, wow! look out.

  2. MJT says:

    i like the make shot generating another shot idea.
    I gave a 15 minute talk on the “Hawkes Hand” model.
    It runs with that idea to describe temporal clustering in made shots.

    https://mikejacktzen.wordpress.com/2018/08/08/the-hawkes-hand/

    If anyone’s interested.

    PS, what made me want to revisit the hot hand, was daniel lakelands great summary in the past blog discussions

    https://andrewgelman.com/2017/04/02/gilovich-doubles-hot-hand-denial/#comment-457291

    • I’m glad you liked my summary.

      I do have an issue with Andrew calling: “continuously varying underlying ability” a “correlational” phenomenon, whereas “making a shot increases the probability of making the next shot” a “causal” model (and yes, I’m aware he put these words in scare quotes).

      It’s clear to me that there are causal models that explain continuous temporal variations in ability. For example the military decided to use amphetamine during WWII because it enhanced soldier’s ability to fight under adverse circumstances like lack of sleep and high stress etc. Taking the drug caused a temporary (ie. time-varying) change in a certain ability. The same is obviously true in the opposite direction for drinking alcohol and driving.

      So a continuously varying ability is only “correlational” if you don’t have a causal model for how it comes about. But anyone who has done any kind of human performance: music, sports, acting, public speaking, etc knows that there are times when you feel confident and capable, and other times when you look like Brazil vs Germany in world cup 2014…

      I actually think of “making a shot increases probability of making a shot” as a much less likely causal explanation for the phenomenon than “being ‘in the zone’ makes you take and hit a lot more shots” where ‘in the zone’ is some kind of way that you feel because your brain is doing a good job of controlling your body, and knows it.

      I actually am starting a project with a group that studies stroke rehab, and one of their models for walking is that you have a kind of clock that expects your foot to impact the ground at a certain time, and when your gait is altered due to stroke, your body is thrown off by a lead or lag between the expected impact and the actual impact. That’s the kind of thing that makes sense to me in terms of “hot” performance where your expected vs actual outcomes are consistently very well aligned.

      Playing soccer with my kids at their practices, I know even before I watch the ball when I’ve hit a ball so that it will arc gracefully right to the intended target, and when I’ve hit it so that it will go “about a radian off of where it was supposed to go” as Feynman put it in “Surely You’re Joking Mr Feynman”.

      • a reader says:

        I think you are misunderstanding Gelman’s use of correlation model vs causal model, although I see why it’s a little unclear.

        I think when AG says correlation model, he means something like an autoregressive model, in which there is an *unobserved* factor that is affecting a sequence of events, and the only evidence we have that the unobserved factor exists is that we see it as a correlation between recent events, as the unobserved factor is expected to change slowly. Note that in this kind of model, if we observed this factor, the next observation could be independent of the previous outcome, conditional on the unobserved factor. I believe when AG says causal model, he means something like a moving average model, where the next observation is conditioned on the previous outcome *only*.

        To me, it makes to refer to this a causal vs correlation model; in the autoregressive model, if you changed the very last outcome with all else remaining the same (including the unobserved factor), then this has no impact on the distribution of the next outcome. Note that this is (basically) the counter-factual definition of having no causal impact. With the moving average outcome, if you changed the very last outcome, this will change the distribution of the next outcome, so the last outcome does have a causal effect on the next outcome.

        • The unobserved skill variable clearly feeds back on itself through the outcomes. When I take a shot and feel that my effort went well and observe that the shot goes in, I take more shots and I maintain my feeling of hotness.

          But the same is true of near misses… The observed outcomes for the player are continuous. It’s only the data analyst that sees them as discrete.

          The problem with the model of making a shot affecting the probability of the next shot is it’s ridiculously simplistic and not a model of reality, it’s a model of a low bitrate binary measure of a complex multidimensional bio feedback system.

  3. Mark says:

    Of course there’s a hot hand in darts! I have known this for a long time, but just called it something a bit different… a “drinking window”. You start off the night completely sober (possibly) and can’t hit the broad side of a barn. A few beers in and, voila, you’re nailing triple 20s! But, then you keep drinking… another variable for your models! (I know, I know, “professional darts”)

  4. Nat says:

    From an outsider’s perspective, it seems fairly obvious that sometimes an individual may perform better in one game than another. Therefore, it also seems pretty obvious that an individual may perform better at one period in a single game than during another period. If the null hypothesis is that nothing is changing over time, then that seems implausible before we even collect data or run an analysis.

    I know my own performance is highly variable over time, particularly in activities that I do not regularly participate in. I thought one of the main characteristics of professional players was that they have much more consistent performance as a result of extensive practice. I bumble around (consciously and/or inadvertently changing my technique) and can sporadically achieve good performance or sporadically achieve bad performance. If you want to detect “hot hand” you should see me throw darts…

    I think I must be missing a key point of this “hot hand” discussion

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