Chance News 106
Quotations
Forsooth
You really are old as you feel
"Some of the subjects 'aged physiologically not at all [while] at the other extreme there were folks aging two to three times as much.'"
Submitted by Chris Andrews
See amazing comments here, as bloggers try to figure out the arithmetic.
Submitted by Margaret Cibes at the suggestion of Sarah Bedichek
More on the hot hand
In Chance News 105, the last item was titled Does selection bias explain the hot hand?. It described how in their July 6 article, Miller and Sanjurjo assert that to determine the probability of a heads following a heads in a fixed sequence, you may calculate the proportion of times a head is followed by a head for each possible sequence and then compute the average proportion, giving each sequence an equal weighting on the grounds that each possible sequence is equally likely to occur. I agree that each possible sequence is equally likely to occur. But I assert that it is illegitimate to weight each sequence equally because some sequences have more chances for a head to follow a second head than others.
Let us assume, as Miller and Sanjurjo do, that we are considering the 14 possible sequences of four flips containing at least one head in the first three flips. A head is followed by another head in only one of the six sequences (see below) that contain only one head that could be followed by another, making the probability of a head being followed by another 1/6 for this set of six sequences.
TTHT Heads follows heads 0 time THTT Heads follows heads 0 times HTTT Heads follows heads 0 times TTHH Heads follows heads 1 time THTH Heads follows heads 0 times HTTH Heads follows heads 0 times
A head is followed by another head six out of 12 times in the six sequences (see below) that contain two heads that could be followed by another head, making the probability of a head being followed by another 6/12 = 1/2 for this set of six sequences.
THHT Heads follows heads 1 time HTHT Heads follows heads 0 times HHTT Heads follows heads 1 time THHH Heads follows heads 2 times HTHH Heads follows heads 1 time HHTH Heads follows heads 1 time
A head is followed by another head five out of six times in the two sequences (see below) that contain three heads that could be followed by another head, making the probability of a head being followed by another 5/6 this set of two sequences.
HHHT Heads follows heads 2 times HHHH Heads follows heads 3 times
An unweighted average of the 14 sequences gives
- [(6 × 1/6) + (6 × 1/2) + (2 × 5/6)] / 14 = [17/3] / 14 = 0.405,
which is what Miller and Sanjurjo report. A weighted average of the 14 sequences gives
- [(1)(6 × 1/6) + (2)(6 × 1/2) + (3)(2 × 5/6)] / [(1×6) + (2 × 6) + (3 × 2)]
- = [1 + 6 + 5] / [6 + 12 + 6] = 12/24 = 0.50.
Using an unweighted average instead of a weighted average is the pattern of reasoning underlying the statistical artifact known as Simpson’s paradox. And as is the case with Simpson’s paradox, it leads to faulty conclusions about how the world works.
Submitted by Jeff Eiseman, University of Massachusetts
Predicting GOP debate participants
Ethan Brown posted this following link on the Isolated Statisticians list:
- The first G.O.P. debate: Who’s in, who’s out and the role of chance
- by Kevin Quealy and Amanda Cox , "Upshot" blog New York Times, 21 July 2015
Because of the large number of declared candidates (16 and growing at the time of the article), Fox News has limited participation in its August 6 debate to those who meet the following criterion
Must place in the top 10 of an average of the five most recent national polls, as recognized by FOX News leading up to August 4th at 5 PM/ET. Such polling must be conducted by major, nationally recognized organizations that use standard methodological techniques.
But of course, polls are subject to sampling error. The NYT article uses simulation to illustrate how this could affect participation. Supposing the latest polling averages represent the "correct" values, they simulate 5 additional polls (as Ethan noted, this is a bootstrapping approach). The results demonstrate that this can affect who's in and who's out, as well as the location on the stage for those who do make the cut.
The Washington Post maintains a State of the debate widget that updates the current top 10 based on the most recent polling results. For July 30 we see
Note that the top figure is not a bar chart; it represents the positions of the candidates on the stage.
Update
Update
Who gets into the Republican debate: Rounding could decide
by Kevin Quealy, "Upshot" blog, New York Times, 3 August 2015
The article points out that Fox has not specified if or how it will round the polling figures. For example, the 9th, 10th and 11th places in the earlier post above are Chris Christie (3.3%), John Kasich (2.7%) and Rick Perry (2.3%). These are all within a one percentage point range, but Christie and Kasich currently in, and Perry out. More recent data have Kasich at 3.3% and Perry at 2.7%, a tie if both are rounded to 3%.