I’m afraid this doesn’t help me understand why Warren said that the definition of “irrelevant side effects" is about subjective priorities.
The “evidential basis” for what I imagine to be the views of movement scientists (and hangers on like Eetu, Martin, Warren and you) is the extremely strong negative response to the disturbance of my analysis. You guys are definitely controlling for the power law not being an irrelevant side effect of control.
So if the ability to see both sides meets Fitzgerald’s test of a first-rate intelligence, movement scientists don’t pass the test any better than I do.
I doubt you have any evidence for that. All Eetu and I have ever been controlling for (I can’t speak for Warren) is the correct usage of standard mathematics. If your power-law theory had used correct mathematics, I doubt there would ever have been an issue.
My mathematical analysis of the power law is exactly the same as that of Maoz et al. You have only used the “bad math” argument against me, not Maoz et al, which means that their math was not a disturbance to what you are controlling for.
The only difference between my analysis and that of Maoz et al was the conclusion. This is pretty strong evidence that what you really don’t like is my conclusion – that the power law is an irrelevant side effect of control – not my math – that log V = -1/3 log C + 1/3 log D, which is exactly the same as the Maoz math
So it’s pretty clear to me that you are controlling for the power law NOT being an irrelevant side effect of control. What you (and your acolytes) think it is is beyond me.
Could you send a link to the Maoz et al paper? I can’t find it in google scholar. I remember looking at it a while back and not thinking it relevant to the discussion, so I’d like to have another look at it.
I didn’t see a link or an attachment in your reply, Rick, but a search on Maoz in our Discourse interface turned up this Dropbox link to “Noise and the two-thirds power law” in an August 1 post of yours.
You should stick with PCT on this - find controlled variables. The phenomenon is that people slow down in curves in most “natural” movement (“natural” meaning not too slow or too fast). The phenomenon is VERY RELEVANT because any model that purports to explain movement needs to also slow down in curves just like the people do, or not slow down, just like the people do.
The relationship between speed and curvature is either controlled or not, so the procedure should be directly disturbing this relationship and looking for the response from the participants: if they maintain it stable, it might be controlled. If it is not controlled, go for the next hypothesis of what is controlled and repeat the procedure.
Learn the basics of PCT, do some experiments, build a model, then you can understand what is relevant and what is irrelevant.
In principle, if you disturb a side effect there will be no resistance to the disturbance, yes.
How to disturb the speed of movement along a curved path without disturbing control of movement and/or control of curving? I don’t see an obvious way to tease these apart.
That’s assuming that movement and curving are CVs and speed is not.
Or is speed controlled, subject to disturbance due to control of curvature?
Yeah, it is tricky, it took me a few years to get to the bottom of it.
If the relationship between speed and curvature is the controlled variable, then this relationship should stay the same even when you change the one of the variables. Curvature is a spatial variable, pure geometry. Take any random path, you can change the speed as you wish - only if you move VERY SLOWLY. You can draw the letter “O” very, very slowly with a constant speed of 1mm/s at all times. You can draw ANY shape at a constant speed of 1 mm/s.
In that case, the relationship is NO CORRELATION. Constant speed regardless of curvature. I think we can control that even with a direct disturbance like a weight on your hand. Maybe up to a pound or two of mass in the hand.
In the low ranges of speed, we could even go faster in curves and slower in flat parts. Imagine driving a car with 0.1 meter/s in the straight parts and like 1 meter/s in the curves parts. We can change the speed as we wish - even if there is wind, and slopes in the road.
If we are going faster, on average, we have to slow down in curves, and we cannot maintain the same curvature (the same geometric path) with the same relationship to speed as before. We have to slow down in curves to stay on the path. This is what happens in the “natural” movement.
One way to test for controlled variables is to guess at what they are and then put them in a model and see how well the model fits the data. I was able to demonstrate how this is done using object interception data collected by Dennis Shaffer and I described the results of that testing in this paper. The main result was that a model that controlled vertical optical velocity fit the data somewhat better than one that controlled vertical optical acceleration. I used this approach to testing for controlled variables to model movement behavior that follows the power law.
A control model that successfully fits curved movement data – including yours --will show the apparent “slowing down through curves” power relationship between velocity and curvature. I describe a control model that fits power-law-conforming movement data in the behavioral illusions paper. That model perfectly accounts for cursor movements and the mouse movements that produced them. In both cases, the movement appears to slow down through curves, although there was nothing in the model that was designed to produce that result. The apparent slowing down though curves is a side-effect of control that results from the mathematical relationship between velocity, curvature and affine velocity.
Actually, I did all those things. But thanks for the tip.
That was good! Also not a formal TCV, but ok. It did not explain why do people slow down in curves in other types of movement, like handwriting or drawing ellipses, where the power law was initially found.
Apparent slowing down? I had real slowing down in my experiments. Raw speed and curvature plots over time show an obvious correlation, not merely an apparent correlation.
If you had only “apparent” slowing down - that means you had a statistical artifact, and there was no need to make a model to explain the findings - the statistics explain it all.
If the slowing down was real, then the model might explain it as a side effect of controlling the cursor-target distance. But you had some VERY slow movements, didn’t you?
The cursor-target distance control model produces much bigger ellipses than people in the same task, so it does NOT account for behavior or the power law in this task.
The correlation is real; a result of the mathematical relationship between velocity, curvature and affine velocity. What is apparent is that the slowing down occurs in response to the curvature.
I made the cursor movements in the Figure below at a speed of about 8 cm/s. The mouse movement were made over a much smaller distance, so the speed was ~2cm/s.
I don’t known if those movements would be considered very slow but the position control model accounted for both movements with great accuracy. And in both cases there was a negative power relationship between curvature and speed for both participant and model
Yes, but your model with the higher order phase lock system accounted for both the slow and high speed movement, right? If so, you should see about the same power relationship for the model as found for the participant in each condition.
Nope. There is only the correlation between speed and curvature. No affine velocity in the analysis.
You said: “apparent “slowing down through curves” power relationship betweeen velocity and curvature”
So, you made a mistake: the correlation is real, but the interpretation might be a mistake?
Yep, too slow. If you draw whole ellipse in one second, that might be showing a real power law. This is the situation where people CANNOT draw elliptic shaped without slowing down in curves. And this is real slowing down, not just apparent.
Yes, of course. But, as Maoz et al showed, it’s the correlation between the variable omitted from the regression, affine velocity, and the variable included , curvature, that determines how much beta (the power coefficient relating curvature to velocity) will deviate from -1/3. The fast movements are closer to an ellipse, where the correlation between curvature and affine velocity will be close to zero. So beta will be close to -1/3.
It’s not clear from this what my mistake was but you correctly state what I meant to say: the correlation between curvature and velocity is real, but the interpretation of that correlation might be a mistake.
Ah, now I think I see why you think that. You get a beta <>-1/3 and a low R^2 with the slow movement and a beta close to -1/3 and a high R^2 with the fast movement. Is that it?
If you do multiple regression with logC and logVa as predictors, and logV as the criterion variables, you will always get the coefficients from the formula, regardless of the correlation between C and V.
The mistake was saying that the correlation was apparent. This would imply that people do NOT slow down in curves. In fact, in fast movements - they do slow down in curves.
The main thing is the R^2, the fit to the power law. If the trajectory has an R^2 < 0.75 in my analysis - this means it is not a power-law movement.
Empirically, slow elliptic movements did not have a good fit to the power law, and fast movement did. Betas had different values, depending on the speed, but this is irrelevant because of the low R^2.
Yes, it was implied. They showed how the observed beta would be affected by the correlation between curvature and affine velocity, the latter being the variable that is never included in the regression used to evaluate the fit to the power law.
It might have helped if we had called affine velocity something other than the “omitted” variable because calling it that implies that power law researchers have been purposefully omitting that variable from their analysis. And, of course, they haven’t. We should have just called it a “third variable” that influences the observed beta in the regression used to evaluate fit to the power law.
Correct.
I don’t think I said that the correlation is apparent; if I did, then that would definitely have been a mistake. But I think I said that the correlation is real; what is apparent is the appearance that people slow down in curves.
Maybe our problem is just linguistic. The term “slow down” implies (to me, anyway) that the slowing is done actively, presumably in response to the increase in curvature. But what we see in the correlation between speed and curvature is just an association between these variables. This association is more accurately described as “movement is slower through high curvature than through low curvature segments of a movement”. If this is what you meant when you said that people “slow down through curves” then you were, indeed, describing something that is real (the correlation between speed and curvature) and not apparent (active slowing in response to curvature).
OK, thanks. I’ll try to find out what, if anything, our analysis implies about the relationship between R^2 and observed beta.
