After some frustration in trying to explain the issues in PACE to yet another person, I suddenly realized they did not have experience with any distributions except Gaussian and binomial, or any applications which used more than vague and woolly reasoning of the "maybe it is, and maybe it ain't" variety. The example below was dredged up from ancient memory. I hope it shows how you can make valid statistical arguments with good predictions. There is far more available on-line, or in textbooks. I hope this will have the concrete feel that gets lost before page 50 of most textbooks.
Assume you have done experiments which tell you air is composed of molecules, and you suspect air pressure is simply due to random impacts of many molecules. You can do a simple experiment to gain some insight into the distribution of velocities.
You inflate a small balloon, and hang it in the middle of a sealed room. After you exclude drafts and convection currents of air, the motion of the balloon becomes imperceptible. You also note that it takes a definite minimum pressure to inflate the balloon. Once it is inflated it approaches a spherical shape. At the very least, wrinkled balloons become smoother as inflated.
Together these things tell you a good bit. First, the mean of all molecular velocities striking the balloon must be close to zero, otherwise it would move. Second, there must be a definite, non-zero absolute (root-mean-squared) value for the deviation of velocities from zero. Third, the pressure of molecules inside and outside the balloon is the same in all directions (isotropic). One thing you didn't even think to question is that both room and balloon are three dimensional.
With this starting point, you can derive a distribution of velocities. The simplest possible assumption is that the component velocities along three axes at right angles will be independent, each Gaussian with the same mean (zero) and standard deviation. The assumption the 3-dimensional distribution is isotropic leads directly to derivation of a
Maxwell-Boltzmann distribution for velocity magnitudes. This can be confirmed experimentally.
(Many people see "turning the crank" to get from those assumptions to the distribution as impossibly forbidding. It is less difficult than turning the crank to predict the motion of the Moon from first principles to an accuracy appropriate for observation by the mark 1 human eyeball. Many millions of people took the recent prediction of an exceptional full moon for granted.)
What can we learn from this? First, you started with very good reason to believe the distribution could be completely described with two parameters. You even had a value for one of them. Second, you did not assume the normal distribution applied directly to any possible measurement you might make. You might have trouble making separate independent measurements of component velocities of individual molecules. The magnitude of the vector velocity leads to directly observable effects like ionization or chemical reactions.
While the Maxwell-Boltzmann distribution may look kinda, sorta
Gaussian for large values of standard deviation, it is not. Not every smooth, one-hump distribution is Gaussian. The influence of three dimensions is important even when component velocities are Gaussian and statistically independent. The number of dimensions matters even when they are not immediately visible. The space in which this takes place has both a particular (Euclidean) geometric structure and dynamics.
My personal prejudice about reasoning in medical and psychological research is that somebody should have learned something about the space, geometry and dynamics describing health and illness after about 200 years of measuring various things in isolation. My own visits to doctors are less than reassuring.
My doctor looks at a printed list of laboratory measurements describing my physical condition. This might refer to a space with some 20 dimensions, or a number of different spaces with smaller dimensions. Instead, each measurement seems to exist in isolation. Even when a measurement falls outside of published norms, it is often ignored. It could be laboratory error, or a meaningless variation. In some cases, doctors will send me back to the lab repeatedly to get numbers they can ignore. I suspect the mark 1 human eyeball is saying "he don't look sick".
Just as a mathematical exercise,
compare the volume of a bounding box with a bounding sphere as the number of dimensions goes up. In three dimensions, the volume outside the sphere is almost equal to the volume inside. In four dimensions and above, volume outside dominates. I don't know what shape describes health, (apparently nobody does.) I do know nature seems averse to boxes with square corners. It is entirely possible the bounding box approach misses the vast majority of health trends before irreversible damage takes place. Increasing the number of lines on a page for doctors to ignore is a great way to increase costs without corresponding benefits.
If you believe I am talking up purely hypothetical concerns take a look at trends in health-care costs and admissions through emergency rooms. Is the profession getting better or worse at prevention of expensive interventions? Are realized cost savings the result of prevention, shifting burdens or denial of claims?
