Statistics, Obamas, and Internet Memes

Statistics made visible!!

The Principle of Statistical Inference

One of the many reasons for doing Statistics, besides the fact that it is more fun than one may have sitting down, is that it enables one to draw tentative conclusions about unexamined populations based on carefully examined samples.  In fact, the difference between probability, statistics, and process control can be visualized by means of a sample taken from a box full of beads, some percentage of which are red and the others white.

1. Probability. Knowing the proportion of red beads in the box, how many red beads will show up in a sample of n beads?
2. Statistics.  Knowing the number of red beads in a sample of n beads, what is the proportion of red beads in the box? 
3. Process Control.  Is there a box?



Item 2 is the question of inference, illustrated nicely in an article by the eminent statistician, Donald Wheeler, using a box containing 4800 beads, of which 10% are yellow.  He takes ten successive samples of n=50 beads, counting the number of yellows, for a grand total sample of n*=500 beads per experiment.  He repeats this 20 times and plots the cumulative proportion yellow.  As expected, as the sample size gets progressively larger, the cumulative proportion converges (and the 90% confidence band shrinks as a function of the square root of n*).

Cumulative % of yellow beads in successive samples

Oddly enough, it converges on 11% not on the true value of 10%.  In fact, the true (census) value is outside the confidence interval!  This is even more odd when we consider that the cumulative sample was n*=10,000 while there were only 4800 beads in the box to begin with.  In effect, Wheeler has examined every single bead, twice.  Plus a few more.  And still overestimated by a full percentage point, a bias of one part in ten.

TOF (I hear you say), how can this be? 

The reason is simple.  The real world never measures up to the mathematical one.  All of the calculations are predicted on assumption, two of which are:

• The sample is a random sample
• There is a population being sampled from.

A random sample of locations.  If this were 'Iraq,
these would have been radiation monitoring; it it were
global warming, these would be weather stations.

A Good Random Sample is Hard to Find
A random sample is generally considered one in which every unit in the population has a known (typically, an equal) probability of entering the sample.  But the problem is this: We never have a random sample; only the results of applying a sampling and measurement method. 

In the case of Wheeler's bead box, the method was a mechanical, spring-loaded paddle with fifty holes following a vigorous shaking of the box.  I have an identical box (somewhere, I can't lay my hands on it at the moment, so it may be in the storage locker) and I have conducted similar exercises in training classes and at SF cons.  In the more famous example of the Literary Digest Straw Poll of 1936, people who possessed neither telephones nor automobiles had almost zero chance of entering the sample; those who had both had double the chance.  Plus, there was an enormous non-response.  Even though the respondents totaled 2.4 million people, the fact that they chose to respond may have biased the results.  There is no justification for an a priori assumption that non-respondents have the same opinions as respondents. 

In the case of the bead box, it is entirely possible for the agitation of the box to leave the bulk of the beads inside relatively undisturbed, and so the sampling paddle consistently scoops up many of the same beads over and over.  I have accomplished this deliberately on some occasions.

A stratified sample.  This is more like it, whether in 'Iraq
or for weather stations.  Alas, weather stations are situated
for reasons other than estimating global temperature.

Trouble Comes in Bunches, So Take Your Sample in Bunches

In the illustration, above left, a total of 48 locations have been selected out of 400 locations.  Suppose this were a country and the dots represent weather stations.  How do we estimate the temperature of the whole ball of wax from the 48 locations?  We could simply average them, keeping the previous yellow-bead warning in mind.  But then we notice that there are 16 "zones" or regions of 25 locations each.  There are two zones with only one location while two other zones have six locations each.  If each zone ("stratum") were the same, there would be no problemo.  A simple random sample is sufficient.  But what if there were the possibility of significant variation among zones?  Remember the problem of instrumentation and sampling bias.  In those zones with multiple stations, the stations can be cross-checked against one another, and the results adjusted accordingly.  But what if the single temperature station in those two zones were malfunctioning?  Its bogus reading would be assigned to the whole zone. 

A stratified sample allocates the sample proportionately among all strata, then samples at random within the stratum.  In this case, 3 per zone.  So if a county has one-third each Republicans, Democrats, and Independents, then one-third of the sample ought to be from each group.  And so on. 

Is There a Box?

Item #3 up above may have seemed a trifle Zen, but the question Process Control asks is essentially whether the contents of the bead box remain the same from time to time (or equivalently, from group to group).  Process control samples are therefore stratified by time intervals.  Wheeler presents an example of batch weights in kilos exiting a blending operation.  The cumulative mean is:

Cumulative average batch weight after blending.

which you can see does not converge on any particular value, biased or not.  The reason is simple.  Unlike the bead box, this process does not have a single mean value being repeatedly estimated.  Here is a time-plot of the data, plotted on an XmR chart (Individual X and Moving Range).  The moving range provides an estimate of the short-term variation used to calculate the limits of random variation.

Batch weights on upper chart.  Range between consecutive batches on lower chart.
Lower chart is used to calculate limits of random variation on upper chart. 
There is no "process" and it's getting worse. 

As you can see, the data are not compatible with a single process mean.  Therefore, the aggregate data does not comprise a single population for purposes of inference.  As Wheeler puts it, the data have multiple personality disorder.  (BTW, We have seen much of this before, regarding global temperature anomalies.)

Now, I told you that to tell you this.

The halo is a nice touch.

Obama-rama-bo-bama, Banana-fana fo-fama.
Fee, fie, fo, fum.  Something smells.  But what cannot be done by the People's Representatives openly in Congress Assembled must be imposed stealthily by autocratic orders from the Besserwissers.  Yes, I speak of the recent central government command to employers to supply contraceptives for their female employees.  (Though not condoms for their boyfriends.  Go figure.) 

This power grab was evidently made not so much to control the birth of untermenschen as to assert the Executive's authority to order private citizens to buy Stuff the Executive thinks is Really Kool.  (cf. Obamacare wrt buying insurance).  It has nothing to do with whether contraceptives are a good idea; nor with whether they are legal, nor with whether lots of people want them.  It does have to do with the Omnicompetent State instructing a religious body as to which of its activities are "truly" religious and which are not.  That is explicitly forbidden by the First Amendment to do so.  As Jefferson said, “To compel a man to subsidize with his taxes the propagation of ideas which he disbelieves and abhors is sinful and tyrannical.”

But this president said, long before his election, that the Constitution was an "obstacle" to doing the right thing and has on more than one occasion expressed the wistful desire to rule by decree - though quickly backing off after doing so.  And after worrisome applause by his audience.  The appeal of fascism did not die with the 1920s and 30s. 

Why is this man laughing?

Now, the old encyclical Humanae Vitae warned of four trends that would result from freely available contraception.  These can easily be seen as raving delusions of a "slippery slope."  He predicted: 

• a general lowering of moral standards throughout society; 
• a rise in infidelity; 
• a lessening of respect for women by men; and 
• the coercive use of reproductive technologies by governments.

As you can see, none of these...  Er, um...

Good for health;
best if taken with sauerkraut

OK, well, but good idea or not, nonetheless it's legal.  Well, it’s legal to buy pork chops; but should the government compel Jewish organizations to serve them in the cafeteria because of the protein value of America's Other White Meat™?  Gun ownership is legal, but should the government compel Quaker meetings to provide a free ammo employee plan or pay for their membership in shooting clubs because a well-regulated militia is necessary for a free republic?  Would it matter if lots of Jews have eaten pork or even that some Quakers have gun licenses?

(Don't laugh.  What if the One Ring falls into the Wrong Hands; that is, if the Other Party wins an election and inherits the power now being arrogated?)

But how dare the Church bar contraceptives for non-communicants?  

Contragestion

Well, of course, she doesn't.  She merely says that she will not pay for them.  That a citizen has the right to purchase contraceptives does not entail the right to send someone else the bill.  Membership in a health club would also be "good" for health, but that does not impose on you the obligation to pay for my membership.  “Reproductive rights” are no more impaired by such a refusal than “digestive rights” would be if others failed to pay your Weight Watchers dues for you.

BTW, the parallel with digestive rights is instructive.  Both are the result of a collision between the desire to indulge an appetite (sexual or gustatory) without getting fat.  The notion that one ought to be able to eat whatever one likes without gaining weight is generally perceived as a failure of the intellect and no one supposes that the government ought to mandate health insurance coverage to include emetics, so as to enable bulemia.  

But TOF (I hear you say), what has all this to do with that wondrous subject, "Statistics," on which we started?  TOF is glad you asked.

98% of Catholic Women

Sample conclusions apply only to that
population whose members had a
random chance to enter the sample.

A basic rule of statistical inference is that the conclusions of a sample apply only to the population from which the sample came.  If the Literary Digest sampled only telephone owners, a legitimate conclusion cannot be drawn about all voters, only those who have telephones.  If a survey covered only left-handed Iowans, no conclusion can be drawn about right-handed Michiganders.  Get the picture?

Now, one meme that has been repeated in the current foo-foo is that “98% of Catholic women use contraceptives.”  How this obligates the Catholic church to pay for them is unknown, since Church dogma is not determined by popular vote, but by either revelation or (as in this case) natural reason.  Since the Church has been described as "a hospital for sinners, not a country club for saints," we would expect that there are many at least nominal members who do not adhere to the moral law.  I know I haven't.  I bet 98% have pilfered office supplies from their place of work, too; but that does not suggest the government should mandate the underwriting of kleptomania.

However, the 98% figure is bogus.  It comes from Figure 3 in a Guttmacher Institute study of the kinds of contraceptives women choose.  Now, the mission of the Guttmacher Institute is to propagandize the use of contraceptives, and their studies should be viewed in that light.  However, this particular study, though statistically primitive, does not itself make the claim attributed to it by the statistically illiterate. 

The 98% seems suspicious.  What of the elderly?  What about nuns?  What about the proverbially fertile Catholic mother?  Do they comprise only 2% of the Church?

Remember what we said that the results of sample S can only be projected onto the population P from which it was randomly selected?  Ignore for a moment the issues related to methodology, randomness, etc.  What was their population? 

We discover that the study was restricted to "women at risk for unintended pregnancy." [emph. added].  They defined this group as those:

• aged 15-44 
• who were “sexually active” in the three months prior to the survey 
• but were not pregnant, postpartum or trying to get pregnant 

Fits Guttmacher profile

IOW, it excluded any woman participating in the Darwinian effort to colonize the future.  Excluded are Catholic women who are married, trying to have a baby (or at least open to the possibility), nuns and other virgins, and any woman older than 44 years or younger than 15.  This may actually exclude a fair number of "Catholic women" from the population.   

So the study tells us only that 98% of women of child-bearing age who want to have sex without having babies use some form of birth control.  That qualifies as a sort of “d’uh” moment.

(Remember, Guttmacher focused on this group because their interest was centered on which form of birth control different groups used.  It was not they who made the unwarranted inference to "all Catholic women.") 

BTW, you will notice the criteria also excludes anyone sexually active in the preceding three months, using contraception, but who became pregnant anyway.  That would be an interesting number.  

Catholic women.  On the Pill?

Now what about those "Catholic women."  Figure 1 in the Guttmacher report provides a breakdown of religious participation.  We find that only 30% of the “Catholic women” in their study reported attending church weekly, versus 11% who said “never” and 29% who said less than monthly.  IOW, 40% of those claiming to be Catholic are either Easter Bunnies or never attend Mass.  It’s unclear how “Catholic” such women really are.  But it certainly seems as if they are less likely to have been touched by catechesis. 

Lagniappe

But what about the Public Interest in controlling the birth of untermenschen?  The poor are having Way Too Many of Them versus Not Enough of Us.  I mean, dude, Darwin himself worried about this. 

With savages, the weak in body or mind are soon eliminated; and those that survive commonly exhibit a vigorous state of health. We civilised men, on the other hand, do our utmost to check the process of elimination; we build asylums for the imbecile, the maimed, and the sick; we institute poor-laws; and our medical men exert their utmost skill to save the life of every one to the last moment. There is reason to believe that vaccination has preserved thousands, who from a weak constitution would formerly have succumbed to small-pox. Thus the weak members of civilised societies propagate their kind. No one who has attended to the breeding of domestic animals will doubt that this must be highly injurious to the race of man. It is surprising how soon a want of care, or care wrongly directed, leads to the degeneration of a domestic race; but excepting in the case of man itself, hardly any one is so ignorant as to allow his worst animals to breed.

-- Charles Darwin, The Descent of Man, and Selection in Relation to Sex (2nd ed., 1882) pp. 133-134.

So we don't want those worst animals breeding, now do we?

But a funny thing happened on the way to the birth rate.  In the US, the birth rate has been declining steadily, a sine wave on a decaying exponential trend, since at least 1820.  You remember, do you not, that that was the year when President Monroe instituted a government program supplying cheap contraceptives to the poor frontiersmen and slum dwellers.  

US Birth Rates, showing effectiveness of Govt Contraceptive program
instituted in 1820 by President Monroe.  "Replacement level" is at 21
per 1000 people.  Thank goodness for immigration. 

Oh, wait.  There was no such program.  Yet the birth rates did come down.  Very strange.  How could people do things without being told by the government, financed by taxes, and led by the vanguard of the besserwissers?  TOF must plumb this mystery; but he has already overstayed your patience, and will leave that for another day.  But today's takeaway is this: Why do some people suppose that people today need to be managed by the government, when their grgrgrandparents did not?