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Category: Statistics

The general theory, methods, and philosophy of the Science of Guessing What Is.

March 1, 2018 | 13 Comments

Infinity — Part 1 of N

We’ve discussed infinity before, but since the subject in inexhaustible, we’re discussing it again.

Infinity comes in sizes, as many readers know. What’s less appreciated is how these sizes relate to questions in epistemology, physics, even theology. We’ll explore some of these facets in future articles. For now, some basics.

The first and smallest infinity is the set of natural numbers. This infinity is not small. All together now: Just how big is it? We are be tempted to say it is incomprehensibly big, and there is some truth to that, but the sticking point is that difficult word comprehensible.

Let’s see if we can get a feel for this tiniest of infinities. Count 1, 2, 3, and keep going. Never stop. Eventually we get to 1010, which is 10 billion. How far is that from the end? Infinitely far away. Take that billion and make it the exponent, i.e. 10billion. How far is that one from the end? Same answer: an infinite distance.

Now that might seem like a lot of zeros after the 10, but it’s a pittance, not even a drop. Exponents are not very handy for counting really large numbers, so let’s work with tetrations. They look just like exponents, except the superscript is on the other side. So 210 = 10^10^10, and 310 = 10^10^10^10, and so on.

We’re getting really big here. Consider B = billion10, which is 10^10^… a billion times. Big number! Bigger than we will ever need for any counting of physical objects. But it’s still infinitely far from the end. Next try BB = B10, which is 10^10^… not just a billion times, but B times. This is so huge it can’t be well thought of. But it’s still tiny compared to the tiniest infinity.

Well, we can keep going, BBB = BB10, BBBB = BBB10, et cetera. Do that B times, then do it BB more times, then BBB more times, and then…you get the idea. You’ll eventually stop, and come to a finite number that is beyond anybody’s ability to grasp (if B isn’t already). But whatever this number is, it is still infinitely far away from the smallest infinity.

What I’m trying to imbue in you is an appreciation how mind-bogglingly big the first infinity is. The number is so large that numbers less than it are all we would ever need if we’re interested in counting (and, yes, many mathematicians call this infinity a number).

After this “simple” counting infinity probably comes what are called, in a playful fit of whimsy, real numbers. I say “probably” because nobody knows for sure if there is another kind of infinity between the counting kind and the so-called continuum, where the reals live. That there is no differently sized infinity between the counting numbers and the reals is called the continuum hypothesis, which most believe is true, but nobody knows how to prove.

Anyway, one way to think about the reals, is to take any two counting numbers, like 1 and 2, and imagine stuffing an infinite number of numbers in between. Count these “stuffing numbers” however you like. Then take any two of these next to one another, still inside 1 and 2, and stuff another infinity of numbers in between them. Like 1.1000000001 and 1.1000000002, and stuff an infinity between them. And keep doing this for any two successive numbers.

You can go on packing numbers into the gaps like this until you come to a point where you have formed a dense flood of numbers, the succession infinitesimally increasing.

The problem, which may be obvious to you, is that if you’re not careful, this infinity doesn’t seem as big as the counting infinity. That’s because it’s impossible, at least for me, to envision what an infinitely dense succession of numbers look like, when I can’t even tell you what BBBBBBBBB10 looks like. Best I can do is to tell you that the number of natural numbers between 1 and BBBBBBBBB10 is infinitely smaller than the number of reals between 1 and 2.

Strangely, counting big natural numbers is hard, yet working with reals is easy. That ease produces in mathematicians a sort of hubris, or rather, forgetfulness. We’re so used to calculating with reals that we forget just how impossible large the continuum is. The forgetfulness arises when we try and apply real-number equations to things in existence. Are there any actual objects that correspond to the continuum? Depends on what you mean by “actual.”

Skip that question—for now—and think of this. As big as the natural counting infinity is, and as infinitely larger are the reals, there are more infinities larger still. They are comprehensible only in the sense we know they exist, and because we know minimal things about them, such as their ordering. But I don’t think anybody grasps what these numbers are really like. Not when we can’t even say what billion Bs tetrated to a billion BBBB10 is like.

So how many sized infinities are there? And what might this have to do with a proof of God’s existence?
Great question. We’ll do that another time. For those who are adept at math and want to read more, I recommend the paper “Infinite Sets and Infinite Sizes” by the very aptly named Gary Hardegree.

February 27, 2018 | 6 Comments

Some Doctors Want More People Taking Antidepressants

There are calls for “at least a million more Britons” to be put on antidepressants. This is odd because Britain’s National Health Service already “prescribed a record number of antidepressants” in 2016.

That represented “a massive 108.5% increase on the 31 [million] antidepressants which pharmacies dispensed in 2006.” In the States, one estimate is that 12% are already on these drugs.

Still, the clarion for ever more drugs was signaled after the results from a new statistical analysis were announced.

The study was “Comparative efficacy and acceptability of 21 antidepressant drugs for the acute treatment of adults with major depressive disorder: a systematic review and network meta-analysis.” It was led by Andrea Cipriani and published in The Lancet.

Do antidepressants alleviate or ameliorate the suffering caused by acute major depressive disorder? In some cases, the analysis says the answer appears to be yes. Which means that in some cases, the answer appears to be no. This is another way of saying that antidepressants don’t always work, or do not work for all people all of the time.

And that means that, at least for some, placebos are as “effective” as the active chemicals in antidepressants. The authors admit “Depressive symptoms tend to spontaneously improve over time and this phenomenon contributes to the high percentage of placebo responders in antidepressant trials.”

Placebos, it should go without saying, do not carry any risk of side effects. Actual drugs do; about which, more in a moment.

Caution Over the Results

Now this was not an original study, but a re-look at old studies called a “meta-analysis.” As a statistician, I only often half-jokingly say that meta-analyses are conducted to “prove” what individual studies could not. If the results from individual researches were clear and robust, meta-analyses would hardly be needed. On the other hand, a meta-analysis can provide a vantage individual studies cannot. The limitations of the method must be kept in mind.

Only studies that treated acute depression were examined here. What about side-effects? Cipriani cautioned “that some of the adverse effects of antidepressants occur over a prolonged period, meaning that positive results need to be taken with great caution, because the trials in this network meta-analysis were of short duration.”

The result of the meta-analysis indicate antidepressant effectiveness is not strong, classed as medium to small effect sizes. The authors warn “Given the modest effect sizes, non-response to antidepressants will occur.” Meaning not all who are given drugs will react to them.

Now the study’s reported statistical measures are highly specialized and take definite meaning only inside a mathematical system. The details are too technical to go into, but naive use of reported measures can exaggerate effectiveness.

If you’re not taking so many pills you can’t see straight, click here to read the rest.

February 21, 2018 | 15 Comments

Science Is Not The Most Important Subject

Stream: Science Is Not The Most Important Subject

What’s with all the kowtowing to science among religious folks?

As soon as a scientist, or science cheerleader, starts talking about the “unbridgeable” divide between religion and science, a Christian apologist trots out and pleads “There is no contradiction between science and Christianity.”

Well, there isn’t. But the Christian has the wrong attitude. There is no need of meek acceptance of science’s superior ground. Science does not hold the hill. It is down in the valley boasting big. Christians need to recognize this. When a scientist starts waving his slide rule around in a menacing manner, the Christian should say What is wrong with you people?

The Limitations of Science

Science is terrific. But isn’t everything. It isn’t even most things. Knowing the weight of a neutrino won’t tell you why stealing is a sin. Neither can positing some mathematical formula for altruism and selfish genes tell you why men cooperate. All arguments along this line are circular or invalid, anyway. They either assume what they want to know, like that rape is wrong. Or they assume that alone among men, the scientist has escaped the pull of his biology and can tell you how things really are.

Look. Figuring how to create a magnetic monopole won’t get you into Heaven. It won’t keep you out, either. So why are scientists so combative about religion?

The suspicion—more like the raw, rabid hope—of some scientists is that a culture which embraces science will eschew religion. Science will allow humanity to leave its infancy behind and lead it to a bright, happy future where everybody goes around chatting about the reproductive habits of newts.

But not discussing why it’s right wrong to kill yourself. Scientists figure they can handle those tough questions themselves, and then tell the rest of us their “discovery.” This is a vain hope.

The Unmeasurable Cannot Be Measured

Science can speak only about the measurable properties of things. That’s it. Nothing more. About elementary fermions science is teeming with a lot o’ news. It has many cheerful facts about your brainwaves when you take a snooze.


You can click here and observe the words at the link, but science will never tell you why they are important.

February 19, 2018 | 6 Comments

Did A Man Really Breastfeed A Baby?

Stream: Did A Man Really Breastfeed A Baby?

Something’s not right.

A paper in the journal Transgender Health reports that a man injected with a myriad of chemicals was able to temporarily breastfeed a baby.

The press is not surprisingly reporting the event uncritically. For instance, The Guardian calls the event a “breakthrough”.

There are reasons to doubt the study, however, as we shall see.

Reported Details

Here are the salient details. A thirty-year-old man desirous of breastfeeding presented at the Mount Sinai Center for Transgender Medicine and Surgery.

The man was at that time was in a “feminizing hormone regimen”, taking spironolactone, estradiol, and micronized progesterone. He was also taking occasional clonazepam for a “panic disorder”
and zolpidem for insomnia.

Presumably because of the long use of hormones, and without augmentation surgery, the man’s breasts appeared well developed.

To induce lactation, the researchers:

(1) increased estradiol and progesterone dosing to mimic high levels seen during pregnancy, (2) use of a galactogogue [a lactation-inducing drug] to increase prolactin levels, (3) use of a breast pump with the speculation that it would increase prolactin and oxytocin levels, and (4) subsequent reduction in estradiol and progesterone levels, with the intention of mimicking delivery.

Potentially Dangerous Drugs

The galactogogue was domperidone, which is now banned in the United States. The FDA said:

The serious risks associated with domperidone include cardiac arrhythmias, cardiac arrest, and sudden death. These risks are related to the blood level of domperidone, and higher levels in the blood are associated with higher risks of these events. Concurrent use of certain commonly used drugs, such as erythromycin, could raise blood levels of domperidone and further increase the risk of serious adverse cardiac outcomes.

Domperidone is used on-label as a digestive aid. A listed side effect is “swelling of the breasts or discharge from the nipple in men or women.”

The man was able to secure domperidone from Canada.

How Much Milk?

After one month of treatment, the man “was able to express droplets of milk”. His drug dosages were increased, and after three months of treatment “the patient was making 8 oz [one cup] of breast milk per day.”

The baby in question finally arrived weighing 6 pounds, 13 ounces.

Here are the reported results:

The patient breastfed exclusively for 6 weeks. During that time the child’s pediatrician reported that the child’s growth, feeding, and bowel habits were developmentally appropriate. At 6 weeks, the patient began supplementing breastfeedings with 4-8 oz of Similac brand formula daily due to concerns about insufficient milk volume. At the time of this article submission, the baby is approaching 6 months old.


This is were the suspicion we haven’t learned the whole story begins.

Newborn babies weighing 6-7 pounds require about 14-17 ounces of breast milk per day. This is double what the paper reports the man capable of producing.

Normally developing babies at six weeks need somewhere north of 24-30 ounces of milk daily. The paper reports the baby’s diet was only then supplemented by 4-8 ounces of formula. This means the man must have consistently been producing at least 20 ounces of milk per day!


Pull up a bottle, and click here to read the rest.