# Sandbox

## Help for the again brain?

Omega-3 supplements and exercise have no protective effect on the aging brain, studies find
by Susan Perry, Minneapolis Post, 26 August 2015

Reporting on the results of two randomized trials published in JAMA, Perry writes, "Neither exercise nor omega-3 supplements has a protective effect on the brains of older adults, according to the results of two large randomized controlled studies published Tuesday [August 25, 2015] in JAMA." The details of the two studies may be found (behind a pay wall) at

For the omega-3 study, Perry points out that

Half of the [approximately 3500] participants were given supplements with omega-3, a fatty acid found most abundantly in fish, but also in flaxseed, walnuts, soy products and a few other plant sources. The other half took a placebo. All had their cognitive skills tested before the study started and then twice more, at two-year intervals. At the end of five years, no difference in cognitive abilities was found between the groups.

With regard to the exercise study, she reports that

After two years, the [1635] participants’ cognitive function was assessed through a series of tests. No significant differences in scores were found between the two groups [exercise or health education].

Do these results mean that exercising and eating nutrient-rich healthful foods (not supplements) is a waste of time and effort for older adults?”

She concludes,

Absolutely not, as the authors of a JAMA editorial that accompanies the studies stress. While these two studies "failed to demonstrate significant cognitive benefits, these results should not lead to nihilism involving lifestyle factors in older adults," they write.

Discussion

1. For what it is worth, the nutrient study (omega-3 and other supplements) reported p-values of .66 and .63 regarding the difference between treatment and control. What sort of reaction follows from those numbers?
2. For what it is worth, the exercise study (of several different cognitive tests) reported p-values of .97 and .84 regarding the difference between treatment and control. Again, what sort of reaction follows from those numbers?
3. The studies mentioned in JAMA involved elderly people who presumably by virtue of age have already declined. The editorial asserts, “It is likely the biggest gains in reducing the overall burden of dementia will be achieved through policy and public health initiatives promoting primary prevention of cognitive decline rather than efforts directed toward individuals who have already developed significant cognitive deficits.” The editorial lauds life-long adherence to the so-called Mediterranean Diet as a way of staying mentally and physically healthy. For a more caustic look at the Mediterranean Diet see this discussion from Chance News 92.

Submitted by Paul Alper

## 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 a way 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.

A head is followed by another head six 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.

A head is followed by another head five times in the six 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.

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

### Comment

Sequence
of tosses
Number of H
in first 3 tosses
Number of H
followed by H
Number of HH
in first 3 tosses
Number of HH
followed by H
TTTT 0 0 0 0
TTTH 0 0 0 0
TTHT 1 0 0 0
THTT 1 0 0 0
HTTT 1 0 0 0
TTHH 1 1 0 0
THTH 1 0 0 0
THHT 2 1 1 0
HTTH 1 0 0 0
HTHT 2 0 0 0
HHTT 2 1 1 0
THHH 2 2 1 1
HTHH 2 1 0 0
HHTH 2 1 1 0
HHHT 3 2 2 1
HHHH 3 3 2 2
Total 24 12 8 4

## Percent change vs. percentage point change

Mike Olinick sent a link to the following exchange:

The poor in college
by John S. Bowman, William Brigham, and Frank Robertson, The New York Review of Books, 4 June 2015

These are responses to two earlier articles by Christopher Jenks:

The war on poverty: Was it lost?, 2 April 2015
Did we lose the war on poverty?—II, 23 April 2015

## Some math doodles

$P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)$

$\hat{p}(H|H)$

$\hat{p}(H|HH)$

## Accidental insights

My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.

While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.

As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.

Piece Sq. Inches  % of Total
1 43.25 31.9%
2 35.25 26.0%
3 23.25 17.2%
4 14.10 10.4%
5 7.10 5.2%
6 4.70 3.5%
7 3.60 2.7%
8 3.03 2.2%
9 0.66 0.5%
10 0.61 0.5%

The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”

My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.

Submitted by William Montante