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.

The Donald

A vain man childishly fishing for the adoration of strangers. Being rich isn’t enough; he wants to be universally loved for his imperfections, too. Slurping at the popularity spigot, The Donald’s need for ego gratification is double the size of any ordinary adult. Never in history has such a supercilious dope been more adored. It is beyond reason. Perhaps aware of this, and sensing defection, Trump whips out his phone… 140 characters is all it takes. Tweet.

He glances out the window of his private jet at the destruction beset a once beautiful US island; it disgusts him. He retreats to his phone to check emails, but years of operant conditioning and muscle memory take over. He taps on the bird icon. “30k likes! Only.“. Trump works best knowing his base is in a frothy outrage. “Better do one more…

That’s the one! Mmmmm. That one had everything.

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.