Hot Hands, a Paradox, and one reason why it’s bad to combine within-subject data

The ‘Hot Hand’ phenomenon is a popular belief (applicable to many domains from sports to gambling) that players who were successful in their most recent attempt(s) have increased odds of being successful in their next attempt — they are on a so-called ‘hot streak’ or have a ‘hot hand’. The statistical validity of this belief can be investigated using actual data. Indeed, it has been. For example, Tversky and Gilovich (1989) investigated the hot hand belief in basketball.

Here however, we are not scrutinizing the hot hand belief, but rather using this framework and dataset to reveal the presence of a Simpsonian Paradox. The definition of this paradox will precipitate from the following example…

In the 1996/97 NBA jam seasons Michael Jordan shot a pair of free throws on 338 occasions. MJ made both 251 times, missed both 5 times, made only the first 34 times, and made only the second 48 times. These data are presented in the table above, as are the same data for Dennis Rodman, and also their combined numbers.

Let Phit and Pmiss denote the proportion of first shot hits followed by a hit, and the proportion of first shot misses followed by a hit, respectively. These proportions for Jordan and Rodman, along with their combined numbers are:

Jordan
Phit = 251 / 285 = .881
Pmiss = 48 / 53 = .906

Rodman
Phit = 54 / 91 = .593
Pmiss = 49 / 80 = .612

Combined
Phit = 305 / 376 = .811
Pmiss = 97 / 133 = .729

Notice that, contrary to the hot hand phenomenon, MJ actually shot worse after he made the first of two freethrow shots. The same goes for Rodman.

So both players are actually worse on their second freethrow, on attempts when they’ve made their first shot.

Combining MJ and Rodman’s freethrow data together, the opposite is true. This is the Simpsonian paradox.

Thoughts on Special Relativity

If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest.  However, there seems to be no such thing, whether on the basis of experience or according to Maxwell’s equations.  From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest.  For how, otherwise, should the first observer know, i.e., be able to determine, that he is in a state of fast uniform motion?  One sees that in this paradox the germ of the special relativity theory is already contained.  Today everyone knows, of course, that all attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of time, viz., of a simultaneous, unrecognizedly was anchored in the unconscious.  Clearly to recognize this axiom and its arbitrary character really implies already the solution to the problem.

…thought 16 year old Albert Einstein as he scribbled it down in his notebook.

Coding Behavioral Experiments in Matlab using a Touch Device like KORG padKONTROL

Korg padKontrol nanoPad2

Korg padKontrol nanoPad2

In my previous post titled “Using Matlab with a KORG padKONTROL USB MIDI Input Device“, I went over the initial preparations necessary for utilizing a KORG padKONTROL midi studio controller in Matlab as a multipurpose input device using a tool called padKONTROL Editor Librarian. Once all the controller triggers are customized to your liking (and can talk to Matlab), it’s time to make use of this newly gained functionality. For this demo, I’m going to run through an example of how this device was used in two different behavioral psychology experiments. The first experiment was examining reaction times to various tones, and the second was a stop-signal experiment that made use of the pressure sensitivity in the KORG padKONTROL tap pads.

These are just examples of the many ways this KORG padKONTROL midi studio controller can be used as a versatile multipurpose input device. Here’s even a Matlab Simulink design that can make use of the padKONTROL to control robotic devices:

KORG Simulink

Let’s start by looking at some code…

To be continued…