Picking the right metric is everything – the case of railroads

Metric #1 – ton miles: miles x weight  – really important. Railroads carry more freight than any other mode of transporation – 41% versus 40% for trucks.
Metric #2 – tons (tonnage), not so important: 15% v 70% for trucks.
Metric #3 – value of freight: even less important 4% versus 70% for trucks.
Another example – valuing companies. Perhaps the most common metric is the price-t0 earnings ratio. But this says absolutely nothing about either the balance sheet or the quality or sustainability of earnings. A good metric adjusts for all these factors.
YOUR TURN: Can you think of other examples?

Should blood pressure targets be lowered? what is the math behind the claim?

EXCERPTS FROM INTERESTING ARTICLE BY A PHYSICIAN IN THE NEW YORK TIMES:
1.) “Under New Guidelines, Millions More Americans Will Need to Lower Blood Pressure.” This is the type of headline that raises my blood pressure to dangerously high levels.
For years, doctors were told to aim for a systolic blood pressure of less than 140. (The first of the two blood pressure numbers.) Then, in 2013, recommendations were relaxed to less than 150 for patients age 60 and older. Now they have been tightened, to less than 130 for anyone with at least a 10 percent risk of heart attack or stroke in the next decade. That means that nearly half of all adults in the United States are now considered to have high blood pressure.
2.) The new recommendation is principally in response to the results of a large, federally funded study called Sprint that was published in 2015 in The New England Journal of Medicine. Sprint was a high-quality, well-done study. It randomly assigned high blood pressure patients age 50 and older to one of two treatment targets: systolic blood pressure of less than 140 or one of less than 120. The primary finding was that the lower target led to a 25 percent reduction in cardiovascular events — the combined rate of heart attacks, strokes, heart failures and cardiovascular deaths. Relative changes — like a 25 percent reduction — always sound impressive. Relative changes, however, need to be put in perspective; the underlying numbers are important. Consider the patients in Sprint’s high target group (less than 140): About 8 percent had one of these cardiovascular events over four years. The corresponding number in the low target group (less than 120) was around 6 percent. Eight percent versus 6 percent. That’s your 25 percent reduction.
3.) The effect was small enough that The New England Journal used a special pair of graphical displays used for health events that occur rarely. One display focused on those participants suffering the cardiovascular events (8 percent versus 6 percent); the other shows the big picture — highlighting the fact that most did not (92 percent versus 94 percent).
4.) Oh, and did I mention that to be eligible for Sprint, participants were required to be at higher-than-average risk for cardiovascular events? That means the benefit for average patients would be even smaller.
But the problem with using Sprint to guide practice goes well beyond its small effect. Blood pressure is an exceptionally volatile biologic variable — blood pressure changes in response to activity, stress and your surroundings, like being in a doctor’s office. In short, how it is measured matters. For the study, blood pressure was taken as an average of three measurements during an office visit while the patient was seated and after five minutes of quiet rest with no staff members in the room.
When was the last time your doctor measured your blood pressure that way? While this may be an ideal way to measure it, that’s not what happens in most doctors’ offices. A blood pressure of 130 in the Sprint study may be equivalent to a blood pressure of 140, even 150, in a busy clinic. A national goal of 130 as measured in actual practice may lead many to be overmedicated — making their blood pressures too low. One of the most impressive findings in Sprint was that few patients had problems with low blood pressure like becoming lightheaded from overmedication and then falling. But one of the most important principles in medicine is that the effects seen in a meticulously managed randomized trial may not be replicated in the messy world of actual clinical practice.
Serious falls are common among older adults. In the real world, will a nationwide target of 130, and the side effects of medication lowering blood pressure, lead to more hip fractures? Ask your doctors. See what they think. Let me be clear: Using medications to lower very high blood pressure is the most important preventive intervention we doctors do. But more medications and lower blood pressures are not always better for everyone.
I suspect many primary-care practitioners will want to ignore this new target. They understand the downsides of the relentless expansion of medical care into the lives of more people. At the same time, I fear many will be coerced into compliance as the health care industry’s middle management translates the 130 target into a measure of physician performance. That will push doctors to meet the target using whatever means necessary — and that usually means more medications.
So focusing on the number 130 not only will involve millions of people but also will involve millions of new prescriptions and millions of dollars. And it will further distract doctors and their patients from activities that aren’t easily measured by numbers, yet are more important to health — real food, regular movement and finding meaning in life. These matter whatever your blood pressure is.
YOUR TURN: What do you think?

Mammograms and Physician Statistical Illiteracy

“The probability that a woman of age 40 has breast cancer is
about 1%. If she has breast cancer, the probability that she
tests positive on a (first) screening mammogram is 90%.
If she does not have breast cancer, the probability that she
tests positive is 9%. What are the chances that a woman who tests
positive actually has breast cancer? Many doctors who were presented
with this common medical situation got the answer wrong –
wildly wrong…The answer most commonly given by physicians
was 90%.” The real answer: 10%.
–Gerd Gigenrenzer, Calculated Risks: How to Know When Numbers
Deceive You
More examples here:

Base rate fallacy

Base rate neglect is a specific form of the more general extension neglect.
YOUR TURN:
Do you have a favorite example of the Base Rate Fallacy?
Any other favorite statistical fallacies?
Any cool math at all to share?
Favorite proof perhaps?

Simpson’s Paradox

What hospital to go to: A or B? Well, hospital A’s patients survive 90% of the time and hospital B’s survive 10% of time.
Clearly hospital A is the right choice. Well, not necessarily. Hospital B’s patients could be a much sicker group to begin with. Should you buy stock in company A or company B? Well, A has higher margin and faster growth, So clearly company A. Well, not necessarily. Company A also has too much debt and its growth is all from acquisitions. Company B’s growth is organic and its balance sheet is debt free. Similarly, is there gender discrimination in at college x. Well, say 60% of women who apply are admitted while 90% of men are. Clearly there is discrimination, right? Not necessarily. Perhaps the women are applying to more competitive departments. It is extraordinarily easy to come to the wrong conclusion based on incomplete data. Demagogues love twisting your emotions with data that sound compelling when critical, granular data is omitted.
THE MOST FAMOUS EXAMPLE (from Wikipedia)

UC Berkeley gender bias

One of the best-known examples of Simpson’s paradox is a study of gender bias among graduate school admissions to University of California, Berkeley. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.[14][15]
Applicants Admitted
Men 8442 44%
Women 4321 35%
But when examining the individual departments, it appeared that six out of 85 departments were significantly biased against men, whereas only four were significantly biased against women. In fact, the pooled and corrected data showed a “small but statistically significant bias in favor of women.”[15] The data from the six largest departments is listed below.
Department Men Women
Applicants Admitted Applicants Admitted
A 825 62% 108 82%
B 560 63%  25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 373  6% 341 7%
The research paper by Bickel et al.[15] concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry).

Simpson’s Paradox: How statistics can mislead

What hospital to go to: A or B. Well, hospital A’s patients survive 90% of the time and hospital B’s survive 10% of time. Clearly, hospital A is the right choice. Well, not necessarily. Hospital B’s patients could be a much sicker group to begin with. Should you buy stock in company A or company B. Well, A has higher margin and faster growth, So clearly company A. Well, not necessarily. Company A also has too much debt and its growth is all from acquisitions. Company B’s growth is organic and its balance sheet is debt free. Similarly, is there gender discrimination in at college x. Well, say 60% of women who apply are admitted while 90% of men are. Clearly there is discrimination, right? Not necessarily. Perhaps the women are applying to more competitive departments. It is extraordinarily easy to come to the wrong conclusion based on incomplete data. Demagogues love twisting your emotions with data that sound compelling when critical, granular data is omitted.