William M. Briggs

Statistician to the Stars!

Since the last worked so well, and nothing exceeds like excess, there will be an extended e-holiday at the blog for at least the next month. New and guest posts are in the queue. Since the blog will not be monitored, the spam filter has been tightened to enormous degree: many comments will thus linger in limbo. Email will not be seen during this time. St Thomas, ora pro nobis.

A Simple (But Not Short) Stats Question

From reader JH:

Thank you again for taking the time to write your blog. It is exceptionally thought-provoking.

My job involves much of the six sigma “capability” studies and such. I have lots of tools available to “quantify” our measurement data. But I’m wondering now if much of this approach isn’t baloney.

Given my questioning of our corporate orthodoxy, I decided to try a different approach to testing one aspect of a part. I’m curious what embedded fallacies I may have indulged in doing so?

Let’s say I want to demonstrate that the torque required to damage something is > 80. I have a capable measuring system and a representative sample and my uncertainties are (to our knowledge) randomly distributed (or presumed to be, given they are unknown).

The first measurement comes in at 280. Statistical value? Worthless. It’s a single data point.

The second data point comes in at 270. Ok, perhaps slightly less worthless. I could theoretically calculate my mean (275) and standard deviation (5) from a whopping two data points. Indulging corporate orthodoxy, I could say that my goal of > 80 is a whopping 39 standard deviations below my mean, and generate some impressively high probability that all future parts are going to be > 80. From two little points.

Has anything actually been demonstrated? If I shift the “mean” down to the lower-95% T-value (which is ~230.077 for these two points), can I claim ‘there is at least 95% probability that the population mean is > ~230.077’? If so, that still doesn’t let me calculate P{x>80} unless I assume some kind (Gaussian kind?) of distribution.

It feels wrong, but I can’t clearly articulate the source of error beyond assuming a “normal” distribution just because. Who knows what the actual distribution is?

Terrific set of questions. First, there is no actual distribution. Statistical, or rather probability, distributions are purely epistemological, i.e. measures of information, and are not properties of actual things.

Now this torque you are measuring. It actually was, once, 280, and another time 270. You have thus conclusively demonstrated the torque is, or can be, greater than 80. That is, given the measurements and assuming the measurements are without error, the probability the torque can be greater than 80 is 1. You are certain.

Questions like this next one are entirely different: M = “The next measured torque will be greater than 80.” The probability of that given just your two measurements—and nothing else—is, as is obvious, greater than 0. How much greater than 0 cannot be quantified unless you are to make (what are probably) ad hoc assumptions. Or the probability doesn’t have to be quantified, but can be made sharper if you were to add implicit information about these kinds of torques. Something like, “Given my experience of these kinds of machines, 280 and 270 are way above 80”, and then the probability of M with that implicit premise and given the two measurements is “high”.

Again, to say how high is “high” is requires ad hoc assumptions. Saying a normal distribution represents uncertainty in torque measurements is one such assumption. Then you can say the probability of M given this ad hoc assumptions, and given the two measurements, but leaving out the implicit expert knowledge about your experience.

This is all fine because, as is proved in this one-day best seller, all probability is conditional. Probability is not a property of any system, which is why there are no correct distributions to use—unless the probability can be derived from information you know or assume to be true about the process at hand. That kind of information appears to be missing here.

So, yes, Pr(M > 80 | x = 280, 270, and assuming a normal distribution with certain known central and spread parameters 275 and 7.07) = 1 (or, rather, .999 with about 160 or so 9’s). That probability will be less if you assume you don’t know the parameters and they are instead estimated from the data (something like .999999).

These are the correct numbers given these assumptions—and no other assumptions.

Instead of a normal, you could have used your ad hoc freedom to use any of hundreds of other standard distributions, and none of these would be any more correct. That is, they all would have been correct. Conditionally correct. Since the distribution is not derived, or deduced, from known causal principles about the process, that’s the best you can do.

Unless you bring in outside, expert knowledge. We saw above how that works: and it works well. Problem is, the hunger for quantification. Management wants a hard number, not an opinion. It rarely matters were the hard number comes from; that it is hard is what counts. This is why Six Sigma is so beloved. It gives precision where precision is desired. Not that it is giving useful precision, or numbers from which excellent decisions will be made.

The final answer is—drum roll, please—there is no answer. Unless you’re willing to live with expert knowledge and non-quantified probabilities, there is no way to come to numerical probabilities without making ad hoc assumptions.

You can use history, too, as expert knowledge. For instance, you’ve found normal distributions to work well in the past, hence you use them again. This is weakly ad hoc.

Summary Against Modern Thought: The immortality of the soul, Part I

This may be proved in three ways. The first...

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles. All posts are under the category SAMT.

Previous post.

We’re finally at the immortality of the soul! Well, the start of the proof, anyway. This week is long, but it’s surprisingly easy given what we’d done thus far.

Chapter 79 That the human soul does not perish when the body is corrupted (alternate translation) We’re still using the alternate translation.

1 From what has been said, therefore, it can be clearly shown that the human soul is not corrupted when the body is corrupted.

2 For it was proved above that every intellectual substance is incorruptible. But man’s soul is an intellectual substance, as was shown. It therefore follows that the human soul is incorruptible.

Notes Where this incorruptible soul reposits is not here proved: like many metaphysical arguments, this one is a proof of existence; it is not a constructive proof. You cannot make your own soul! But you procreate new ones. (Forgive me).

3 Again, no thing is corrupted with respect to that wherein its perfection consists, for mutations in regard to perfection and corruption are contrary to one another. The perfection of the human soul, however, consists in a certain abstraction from the body. For the soul is perfected by knowledge and virtue, and it is perfected in knowledge the more it considers immaterial things, the perfection of virtue consisting in man’s not submitting to the passions of the body, but moderating and controlling them in accordance with reason. Consequently, the soul is not corrupted by being separated from the body.

Notes The soul “is perfected in knowledge the more it considers immaterial things”. Get your mind out of the gutter! This kind of perfection used to be well known in (what used to be called) higher education.

4 Now, it may be said that the soul’s perfection lies in its operational separation from the body, and its corruption in its existential separation therefrom. Such an argument misses the mark, for a thing’s operation manifests its substance and its being, since a thing operates according as it is a being, and its proper operation follows upon its proper nature. The operation of a thing, therefore, can be perfected only so far as its substance is perfected. Thus, if the soul, in leaving the body, is perfected operationally, its incorporeal substance will not fail in its being through separation from the body.

Notes But you lose the chance to further perfect your soul when you hand in your dinner pail. Plan accordingly.

5 Likewise, that which properly perfects the soul of man is something incorruptible; for the proper operation of man, as man, is understanding, since it is in this that he differs from brutes, plants, and inanimate things. Now, it properly pertains to this act to apprehend objects universal and incorruptible as such. But perfections must be proportionate to things perfectible. Therefore, the human soul is incorruptible.

6 Moreover, it is impossible that natural appetite should be in vain. But man naturally desires to exist forever.

This is evidenced by the fact that being is that which all desire; and man by his intellect apprehends being not merely in the present, as brute animals do, but unqualifiedly.

Therefore, man attains perpetual existence as regards his soul, whereby he apprehends being unqualifiedly and in respect of every time.

7 Also, the reception of one thing in another accords with the recipient’s manner of being. But the forms of things are received in the possible intellect according as they are actually intelligible; and they are actually intelligible according as they are immaterial, universal, and consequently incorruptible. Therefore, the possible intellect is incorruptible. The possible intellect, however, is part of the human soul, as we proved above. Hence, the human soul is incorruptible.

8 Then, too, intelligible being is more permanent than sensible being. But in sensible things that which has the role of first recipient, namely, prime matter, is incorruptible in its substance; much more so, therefore, is the possible intellect, which is receptive of intelligible forms. Therefore, the human soul, of which the possible intellect is a part, is also incorruptible.

9 Moreover, the maker is superior to the thing made, as Aristotle says. But the agent intellect actualizes intelligibles, as was shown above. Therefore, since intelligibles in act, as such, are incorruptible, much more will the agent intellect be incorruptible. So, too, then, is the human soul, whose light is the agent intellect, as we have previously made clear.

10 Again, a form is corrupted by three things only: the action of its contrary, the corruption of its subject, the failure of its cause; by the action of a contrary, as when heat is destroyed by the action of cold; by the corruption of its subject, as when the power of sight is destroyed through the destruction of the eye; by the failure of its cause, as when the air’s illumination fails through the failure of its cause, the sun, to be present.

But the human soul cannot be corrupted by the action of a contrary, for nothing is contrary to it; since, through the possible intellect, it is cognizant and receptive of all contraries. Nor can the human soul be destroyed through the corruption of its subject, for we have already shown that it is a form independent of the body in its being. Nor, again, can the soul be destroyed through the failure of its cause, since it can have no cause except an eternal one, as we shall prove later on. Therefore, in no way can the human soul be corrupted.

Notes The “it can have no cause except an eternal one” is the constructive part of the proof.

11 Furthermore, if the soul perishes as the result of the body’s corruption, then its being must be weakened through the debility of the body.

But if a power of the soul is weakened for that reason, this occurs only by accident, namely, in so far as that power has need of a bodily organ.

Thus, the power of sight is debilitated through the weakening of its organ—accidentally, however.

The following considerations will make this point clear. If some weakness were attached to the power through itself, it would never be restored as the result of the organ’s being restored; yet it is a fact of observation that, however much the power of sight may seem to be weakened, if the organ is restored, then the power is restored. That is why Aristotle says, in De anima I [4], “that if an old man were to recover the eye of a youth, he would see just as well as the youth does.”

Since, then, the intellect is a power of the soul that needs no organ—as we proved above—it is not weakened, either through itself or accidentally, by old age or any other bodily weakness. Now, if in the operation of the intellect fatigue occurs, or some impediment because of a bodily infirmity, this is due not to any weakness on the part of the intellect itself, but to the weakness of the powers which the intellect needs, namely, of the imagination, the memory, and the cogitative power. Clearly, therefore, the intellect is incorruptible. And since it is an intellective substance, the human soul likewise is incorruptible.

Notes And the “imagination, the memory, and the cogitative power” are, as all known, often corrupted.

12 This conclusion also comes to light through the authority of Aristotle. For he says in De anima I [4] that the intellect is evidently a substance and is incapable of being destroyed. And it can be inferred from what has been said already that remark of Aristotle’s cannot apply to a separate substance that is either the possible or the agent intellect.

13 The same conclusion also follows from what Aristotle says in Metaphysics XI [3], speaking against Plato, namely, “that moving causes exist prior to their effects, whereas formal causes are simultaneous with their effects; thus when a man is healed, then health exists,” and not before—Plato’s position, that the forms of things exist prior to the things themselves, to the contrary notwithstanding.

Having said this, Aristotle adds: But we must examine whether anything also survives afterwards. “For in some cases there is nothing to prevent this—the soul, for example, may be of this sort, not every soul, but the intellect.” Since Aristotle is speaking of forms, he clearly means that the intellect, which is the form of man, remains after the matter, which is the body.

14 It is also clear from these texts of Aristotle that, while he maintains that the soul is a form, he does not say it is non-subsistent and therefore corruptible—an interpretation which Gregory of Nyssa attributes to him. For Aristotle excludes the intellective soul from the generality of other forms, in saying that it remains after the body, and is a certain substance.

Notes The next paragraphs all rely on the Bible and may be skipped by heathens and heretics.

15 The doctrine of the Catholic faith is in agreement on these matters. For in the work On the Teachings of the Church there is this statement: “We believe that man alone is possessed of a subsistent soul, which continues to live even after divesting itself of the body, and is the animating principle of the senses and powers; nor does the soul die with the body, as the Arabian asserts, nor after a short period of time, as Zeno would have it, because it is a living substance.”

16 This eliminates the error of the ungodly, in whose person Solomon says: “We are born of nothing, and after this we shall be as if we had not been” (Wis. 2:2); and in whose person again Solomon says: “The death of man and of beasts is one, and the condition of them both is equal: as man dies, so they also die: all things breathe alike, and man has nothing more than beast” (Eccle. 3:19). For

Solomon clearly is not speaking in his own person but in that of the godless, since at the end of the book he adds in a decisive manner: “Before the dusts return into its earth, from whence it was, and the spirit returns to Him Who gave it” (Eccle. 17:6-7).

17 Furthermore, there are myriad passages of sacred Scripture which proclaim the immortality of the soul.

To Undergo Chemotherapy Or Not

Headline: “How I used math to conquer my cancer” by Michael Kaplan.

Gist: fellow named Reitzen, 45, discovered he had kidney cancer.

“The doctor came up to my house and had fantastic bedside manner,” says Reitzen, now 57. “He told me that the tumor was larger than the kidney itself, which would necessitate removing my kidney and the lymph nodes around it.”

But Reitzen didn’t want a body part removed — unless data showed it to be absolutely necessary — so he sought a second opinion.

So Reitzen used “data mining”, i.e. he conducted a search of the Internet, to find the kind of surgeon he wanted. He found two. One “thought that it would be better for me not to lose [the kidney] because I had high blood pressure…He had no bedside manner, but I liked his opinion of there being a 70 percent chance that I would not be left with just one kidney.”

The cancer was duly hacked out. “But the oncologist/hematologist who gave him the original diagnosis suggested chemotherapy to avoid a recurrence.”

Using decision-theory math, Reitzen took into account the likelihood of surviving with and without chemo. With the chemo, there was an average life expectancy of 8.1 years; without it, he would be looking at 7 years.

But taking chemo’s side effects into account, his quality of life would be only 70 percent, which he based on information from health-related quality-of-life studies on chemotherapy patients. Without the chemo, he would have no side effects and 100 percent quality of life. After doing the math, quality-adjusted life years came to 5.7 with chemo and 7 without.

“Would you want to exchange a 15 percent life-expectancy increase for a 30 percent drop in quality of life?” he asks.

Quality-adjusted life years? Yet another attempt to quantify the unquantifiable. Wikiwik’s dry statement puts it best: “To be dead is associated with 0 QALYs, and in some circumstances it is possible to accrue negative QALYs to reflect health states deemed ‘worse than dead’.”

Reitzen’s calculation is simple: 8.1 years of “expected” life left under chemo times 70% (0.70) of “worthy living” equals 5.7 QALY. And 7 years “expected” life without chemo times 100% (1) of full life equals 7 QALY. Since 7 is greater than 5.7, the calculation says to pick no chemo.

Which was his choice: no chemo. “Ten years later, Reitzen is cancer-free, with two functioning kidneys, and did not have to endure the misery of chemotherapy. The treatment that he was initially offered has been deemed ineffective for kidney cancer.”

To which we say, God bless him. But do note, and do pause, at the statement The treatment that he was initially offered has been deemed ineffective for kidney cancer. And then recall that the “expected” 8 years he was to live with that now-deemed-ineffective treatment was calculated on the belief (or assumption) that the treatment was effective. All probability is conditional.

While Reitzen routinely makes critical investment decisions with the guidance of math, and found comfort in his cancer-by-the-numbers strategy, doctors don’t think his approach to treatment should be relegated only to statistics-loving professionals.

To which we say capital-A-men. A doctor’s “loss” and “gain” are different, and sometimes far different, than a patient’s. The internal “calculations” the doctor uses to recommend a specific treatment might be best in his mind and for him, but not for the patient. Only the patient can decide what a “worthy” life is.

Now the real problem is only partially the attempt to quantify the unquantifiable. A life’s worth cannot be put to a number, and doing so can only be the crudest of approximations. Collapsing a life’s worth to a single number necessarily strips away vast amounts of information. And that means bad decisions can be made.

The second difficulty is, even is all can be quantified, all should be quantified in its fullness. When it was believed (by at least that one doc) that chemo would work, this doc’s calculation said patients would live an “expected” 8 years. That number is found by multiplying the probability (conditional, as all these probabilities are, on the doc’s belief in the treatment) a patient would live one year by 1, the probability the patient would live 2 years times 2, and so on, the result being a weighted average.

Collapsing these probabilities to one “expected” number again strips away information. The whole swath should be presented to the patient—not necessarily at the finest levels. Do we really need information per year 10, 20 years out?

Well, you get the idea. What is most crucial to grasp is not all this numeric mumbo jumbo, but that not accepting harsh treatment is an solid option. Though it cannot be quantified, a shorter life without the brutal suffering caused by a treatment can be much better than one with it. Especially when it is, as it always should be, recalled that All men are mortal, etc.

In New Zealand, “Transgender” Wins Weightlifting Contest, River Becomes Person

In New Zealand, “Transgender” Wins Weightlifting Contest, River Becomes Person:

Now I ask you, is it strange in a country where a woman pretending to be a man wins a weight-lifting contest, that that country would declare a river to be a person?

Wait. It might have been a man pretending to be a woman. You can never tell in these “transgender” stories which part of Reality has been affronted.

What we do know is that the New Zealand Herald reported a “transgender” person named Hubbard won a weight-lifting competition, besting the second-place finisher by hosting about 40 additional pounds.

Now this was either a man pretending to be a woman, and therefore this is a story that a man lifted more weight than a woman, which falls under the Dog-Bites-Man category. Or it was a woman pretending to be a man, in which case it must have been a woman juiced on various drugs, like anabolic steroids and testosterone, to make her competitive with real men. And that makes it a story of performance-enhancing legal-illegal drug use.

Both stories are depressing.

As is the story that New Zealand’s Parliament has recognized the Whanganui River as a legal person. Yes, the river, also called Te Awa Tupua, is to be treated the same as hot dog hawkers and college professors. According to BioEdge:

Riverine personhood is an untested concept in a Western legal system. According to the government, Te Awa Tupua will now have its own legal personality with all the corresponding rights, duties and liabilities of a legal person. Lawyers say that the river cannot vote and cannot be charged with homicide if people drown in it. But it will have to pay taxes, if liable. The gender of the river is unspecified at the moment.

How this riverine person will pay taxes is something to be watched. Maybe in the spirit of Finders-Keepers, the river will offer up rings and other jewelry lost by actual people while swimming? But will swimming be allowed? You can’t swim in an actual person, it unfortunately cannot go without saying, but can you swim in a riverine person?

That brings up the natural question: How will we know Mr—or Miss—Whanganui’s opinion about swimming? We don’t even know his or her’s preferred “gender”. Obviously, like in Hubbard’s case, people are free to call themselves whatever “gender” they wish. Thinking anything else is rank bigotry. But we at least have the advantage of asking Hubbard’s opinion whether she is a he or he is a she, or whatever. We can assume Whanganui gurgles, as all rivers do, but who speaks river and thus who can tell us about Whanganui preferred gender?

[…hydromancy?…]

If you are a man or woman or in-between, you can click here and read the rest.

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