CHAPTER IV-3
MATHEMATICS AND STATISTICIANS
"It's a gift to be simple..."
Old Shaker hymn, quoted by Chatterjee and Price, 1991,
frontispiece).
The headnote above is quoted in a statistics text. But in
mathematical statistics, simplicity is given only lip service.
Its breach is honored and rewarded with publication and
professional status in the statistics (as in other academeic)
profession. The real professional gifts are rigor, elegance,
sophistication. Indeed, a propensity for simplicity often is not
a gift professionally, but a curse.
The resampling approach to statistical practice is an
important contemporary example. By almost every conceivable
test, resampling is simpler than conventional methods. Yet for
decades statisticians would have nothing to do with resampling,
and even at present they do not embrace it as the tool of first
resort for everyday problems.
A ILLUSTRATION TO FOCUS THE DISCUSSION
Consider as an example a study that offers "A New Confidence
Interval Method..." (Peskun, 1993). It works with the following
example: 36 of 72 (.5) taxis surveyed in Pittsburgh had
visible seatbelts, whereas 77 of 129 taxis in Chicago (.597)
had visible seatbelts. The assigned task (whether or not it
would be more sensible to think in terms of a test of a
hypothesis is left aside) is to derive a confidence interval for
the difference of .097 between the proportions.
Peskun's new method provides a 95 percent confidence
interval of -.237 to .047, to be compared with three other
methods he cites whose results are, respectively, -.240 to .046,
-.251 to .057, and -.248 to .054.
The density of the four pages of mathematics that enter into
Peskun's derivation must be seen to be believed.
In contrast, consider how resampling handles the problem:
1. Construct urns with proportions like those observed int
the two cities.
2. Draw samples with replacement from each urn of same size
as actual samples.
3. Compute the observed difference between the experimental
trial results.
4. Repeat steps (2-3), graph the results, and count off the
confidence interval.
For those with a taste for efficiency, the following
RESAMPLING STATS computer program mimics the by-hand steps given
above:
URN 36#1 36#0 pitt Pittsburgh 36 seatbelts, 36 no seatbelts
URN 77#1 52#0 chic Chicago 77 seatbelts, 52 no seatbelts
REPEAT 15000
SAMPLE 72 pitt pitt$ Draw 72 taxis with Pittsburgh probability
SAMPLE 129 chic chic$ Draw 129 taxis with Chicago probability
MEAN pitt$ p Compute experimental proportion in Pittsburgh
MEAN chic$ c Compute experimental proportion in Chicago
SUBTRACT p c d Find difference in experimental proportions
SCORE d z Record difference
END
HISTOGRAM z Show histogram of 15,000 trials
PERCENTILE z (2.5 97.5) k Confidence interval from histogram
PRINT k
k = -0.23934 0.043605
1000+
+ *
+ * *
F + * * *
r + * * *
e 750+ *****
q + *********
u + *********
e + ***********
n + ***************
c 500+ ****************
y + ****************
+ *******************
* + ******************* *
+ *********************
Z 250+ ************************
+ **************************
+ ****************************
+ *******************************
+ ***************************************
0+--------------------------------------------------------------------
|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
Vector z
The result of 15,000 simulation trials requiring 5 minutes
on a desk computer- a confidence interval from -.239 to .044 -
agrees with Peskun's interval (-.237 to .047) but is even
narrower (which may or may not constitute greater precision).
But much more important than the apparent accuracy (assumed from
the close agreement), the resampling treatment is transparent to
the extent that any user understands exactly what s/he is doing,
in contrast to the equational mystery of Peskun's Normal-
approximation method which inevitably involves such quantities as
pi and e, as a result of using Stirling's formula.
The four methods Peskun cites, as well as the resampling
method, agree to the second decimal point. It is hard to imagine
any application in which the user would care about differences at
that level of accuracy, because such differences are likely to be
dwarfed by the variation in other aspects of the empirical
research study and its data collection and analysis; even
indicating that degree of accuracy may well mislead a reader .
These are some of the ways in which resampling is simpler
than the conventional method:
1. Simpler to understand. Even a freshman or sophomore in
an introductory statistics class - indeed, seventh graders - can
understand the resampling manipulation shown above. How many who
are potential users of this device and not professional
mathematical statisticianscan understand the mathematics in the
ASA article? (Leave aside that the underlying logic of
confidence intervals itself is unintelligible or meaningless to
some statisticians of the stature of Leonard Savage and Ronald
Fisher.)
2. Simpler to perform. The resampling treatment requires
far fewer manipulations than does any conventional treatment,
unless the conventional user simply pushes the button on a
computer and calls for a complex black-box program.
3. Logically simpler. The resampling treatment requires
fewer assumptions.
4. Conceptually simpler. The resampling approach is
theoretically simpler because it does not require a count of the
size of a sample space and of one or more partitions of it, which
is the nub of the problem in any probabilistic calculation, and
therefore underlies every probabilistic statistic somewhere in
its derivation.
***
Why does everyone not handle basic problems the resampling
way? Mathematical statisticians have been antagonistic to
resampling - and perhaps more generally, to simplicity - for two
main reasons, one intellectual-esthetic and the other nitty-
gritty issues of self-interest. The latter issue is discussed in
the context of teaching statistics in Chapter 00.
Simulation fulfills your need when you aim to get a specific
answer to a specific question; this is the situation in
scientific research and in decision-making in business and
elsewhere. Analytic methods provide general answers to classes
of questions; this is the aim of mathematicians in their
mathematical work. The pedagogical problem is that teaching is
in the hands of mathematicians who impute their needs to the
students who have entirely different needs, and hence answer
questions that are not being asked, with explanations that are
not understood; hence the teaching does not even have general
cultural value to the students let alone meet their intellectual
and practical needs. The end result is that the students are
turned away from the subject and lose the chance to gain this
valuable knowledge, while the mathematical teachers are
frustrated and disgusted by the apparent stupidity of the
students.
ESTHETICS, MATHEMATICS, AND STATISTICS
A mathematician, like a painter or a poet, is a maker
of patterns...The mathematician's patterns, like the
painter's or the poet's, must be beautiful (C. H.
Hardy, A Mathematician's Apology, pp. 84-85, italics in
original)
Tobias Dantzig pinpointed the key element of thought better,
the attitude of the user of statistics]:
Between the philosopher's [and the scientist's]
attitude toward the issue of reality and that of the
mathematician there is this essential difference: for
the philosopher [or scientific researcher] the issue is
paramount; the mathematician's love for reality is
purely platonic.
The mathematician is only too willing to admit that he is
dealing exclusively with acts of the mind. To be sure, he
is aware that the ingenious artifices which form his stock
in trade had their genesis in the sense impressions which he
identifies with crude reality, and he is not surprised to
find that at times these artifices fit quite neatly the
reality in which they were born. But this neatness the
mathematician refuses to recognize as a criterion of his
achievement: the value of the beings which spring from his
creative imagination shall not be measured by the scope of
their application to physical reality. No! Mathematical
achievement shall be measured by standards which are
peculiar to mathematics. These standards are independent of
the crude reality of our senses. They are: freedom from
logical contradictions, the generality of the laws governing
the created form, the kinship which exists between this new
form and those that have preceded it.
The mathematician may be compared to a designer of garments,
who is utterly oblivious of the creatures whom his garments
may fit. (1954, p. 235)
Greek philosophy, we must remember, was essentially
aristocratic. The methods of the artisan, ingenious and
elegant though they may appear, were regarded as vulgar and
banal, and general contempt attached to all those who used
their knowledge for gainful ends. (There is the story of the
young nobleman who enrolled in the academy of Euclid. After
a few days, he was so struck with the abstract nature of the
subject that he inquired of the master of what practical use
his speculations were. Whereupon the master called a slave
and commanded: "Give this youth a chalcus, so that he may
derive gain from his knowledge.") (Dantzig, p. 117)
The activity and knowledge of mathematics have always had a
special appeal to many - a fascination that can be most
absorbing, and even pass over into religious belief and practice
(as seen in such religious disciplines involving mathematics as
the numerology of kabbala in mystic Judaism). We also see this
fascination in mathematical puzzles, as Martin Gardner notes;
There is a fascination about recreational mathematics
that can, for some persons, become a kind of drug.
Vladimir Nabokov's great chess novel, The Defense, is
about such a man. He permitted chess (one form of
mathematical play) to dominate his mind so completely
that he finally lost contact with the real world and
ended his miserable life-game with what chess proble-
mists call a suimate or self-mate. He jumped out of a
window. It is consistent with the steady disintegra-
tion of Nabokov's chess master that as a boy he had
been a poor student, even in mathematics, at the same
time that he had been "extraordinarily engrossed in a
collection of problems entitled Merry Mathematics, in
the fantastical misbehavior of numbers and the wayward
frolic of geometric lines, in everything that the
schoolbook lacked."
The moral is: Enjoy mathematical play, if you have the
mind and taste for it, but don't enjoy it too much.
Let it provide occasional holidays. Let it stimulate
your interest in serious science and mathematics. But
keep it under firm control (1966, p. 9).
An interesting sidelight here concerns the number pi (and
also e) which frequently appear in connection with probability
and statistics, and especially in the Normal Appoximation. Many
people have been impressed by that. For example, the Nobel-
prize-winning physicist Eugene Wigner wrote thusly about pi and e
(see introduction to Chapter 00 [Normal distribution], as well as
other complex numbers:
Certainly, nothing in our experience suggests the
introduction of these quantities. Indeed, if a
mathematician is asked to justify his interest in
complex numbers, he will point, some some indignation,
to the many beautiful theorems in the theory of
equations, of power series, and of analytic functions
in general, which owe their origin to the introduction
of complex numbers. The mathematician is not willing
to give up his interest in these most beautiful
accomplishments of his genius, (Wigner, p. 224).
Resampling - aside from the "real mathematics"<1> that
constitutes the state-of-the-art investigations of resampling
techniques - offends mathematical statisticans because it does
not meet the main criteria of what has always been considered
truth in mathematics. Monte Carlo simulations of all kinds butt
up against the fundamental attitude of the mathematics profession
toward non-proof-based methods. As S. Stigler put the matter in
a related connection: "Within the context of post-Newtonian
scientific thought, the only acceptable grounds for the choice of
an error distribution were to show that the curve could be
mathematically derived from an acceptable set of first princi-
ples" (1986, p. 110, italics added). This may be related to
Mosteller's comment somewhere that the bootstrap (and presumably
all resampling) is "anti-intuitive" [? see history file], whereas
to the layperson resampling certainly is much more intuitive than
is the formulaic method.
Statisticians also often reject simple methods because it
seems prudent to do so.
...profitable, practical operations-research work is
often arrived at by so simple, even crude, an analysis
that very often its author himself would hesitate to
write a paper about it for publication in a learned
journal. And if he did it might well be rejected for
its simplicity and lack of mathematical sophistica-
tion. ...learned periodicals and journals value mathematical
refinement and academicism for their own sake..."
(Singh, 1972, p. 17)
Mathematical physicist John Barrow invented a revealing
scenario about proof-based mathematics. He imagined what might
happen if we were to receive a response from Martians to an
Earth-transmitted extra-terrestrial messages. Those messages
depend heavily upon mathematics, on the assumption that that will
be the easiest for the Martians to decode. Barrow writes first
about the excitement:
There is great excitement at NASA today. Years of
patient listening have finally borne fruit. Contact
has been found. Soon the initial euphoria turns to
ecstasy as computer scientists discover that they are
eavesdropping not upon random chit-chat but a systemat-
ic broadcast of some advanced civilisation's mathemati-
cal information bank. The first files to be decoded
list all the contents of the detailed archives to come.
Terrestrial mathematicians are staggered: at first they
see listings of results that they know, then hundreds
of new ones including all the great unsolved problems
of human mathematics....
Soon, the computer files of the extraterrestrials'
mathematical textbooks begin to arrive on earth for
decoding and are translated and compiled into English
to await study by the most distinguished representa-
tives of the International Mathematical Congress.
Mathematicians and journalists all over the world wait
expectantly for the first reactions to this treasure
chest of ideas.
Then he writes about the next peculiar reaction:
But odd things happened: the mathematicians' first
press conference was postponed, then it was cancelled
without explanation. Disappointed participants were
seen leaving, expressionless, making no comment; the
whole atmosphere of euphoria seemed to have
evaporated.
After some days still no official statement had been
made but rumours had begun to circulate around the
mathematical world. The extraterrestrials' mathematics
was not like ours at all. In fact, it was horrible.
They saw mathematics as another brance of science in
which all the facts were established by observation or
experiment.(Barrow, 1992, pp. 178-179)
The key is the disappointment. Terrestrial mathematicians
are not excited by a method that simply offers answers or
solutions. The method must also meet aesthetic tests to be
acceptable. It is here that resampling fails.
The connection between the esthetic of mathematics, and the
rejection of simplicity, can be seen in the comment of a referee
on one of my papers: "Most of us [presumably, statisticians]
would be concerned about the implication that users should always
[I do not say "always"] use self-invented ultra-simple
techniques" (italics added). Here we see in starkest terms the
issue that separates the mainstream thinking in the statistics
profession and the approach suggested here -- whether simplicity
is better. I believe that simplicity -- and re-creation of the
technique from first principles (rather than taking a technique
off the shelf) is part-and-parcel of simplicity -- reduces the
chance of using an unsound technique. But to that referee and
the others s/he refers to, simplicity is unaesthetic, and perhaps
threatening. The attitude is that with which G. Stigler
characterizes science, "The ...work should be pursued with non-
vulgar instruments" (1973, quoted by Fisher, 1986, p. 78).
Dantzig contrasts the outlook of the mathematician with the
person who has practical interests:
Greek philosophy, we must remember, was essentially
aristocratic. The methods of the artisan, ingenious
and elegant though they may appear, were regarded as
vulgar and banal, and general contempt attached to all
those who used their knowledge for gainful ends. (There
is the story of the young nobleman who enrolled in the
academy of Euclid. After a few days, he was so struck
with the abstract nature of the subject that he
inquired of the master of what practical use his
speculations were. Whereupon the master called a slave
and commanded: "Give this youth a chalcus, so that he
may derive gain from his knowledge.") (Dantzig, p.
117)
The irrational quantities can be expressed through
rational approximations to any desired degree of
accuracy...
Such methods enable one to "trap" the irrational
number between two sequences of rational numbers, of
which the first is consistently "less" than the
irrational, and the second consistently "greater."
And, what is more, the interval between these rational
approximations may be rendered as small as one
desires.
Well, what further can be desired? The physicist,
the engineer, the practical man generally are fully
satisfied. What the physicist requires of his
calculating methods is a degree of refinement which
will permit him to take full advantage of the growing
precision of his measuring devices. The fact that
certain magnitudes, like **2, **, or e, are not
expressible mathematically by means of rational numbers
will not cause him to lose any sleep, as long as
mathematics is furnishing him with rational
approximations for such magnitudes to any accuracy he
desires.
The position of the mathematician with respect to
this problem is different. (Dantzig, p. 107)
There also was the problem at first of resampling have been
suggested from outside of statistics. Even the greatest of
scientists sometimes look down their noses at upstarts without
the proper credentials. Consider this lovely quote by James
Clerk Maxwell (by way of Conant, 1965, pp. 39, 40) about
Alexander Graham Bell:
When about two years ago news came from the other side
of the Atlantic that a method had been invented of
transmitting by means of electricity the articulate
sounds of the human voice so as to be heard hundreds of
miles away from the speaker, those of us who had reason
to believe that the report had some foundation in fact
began to exercise our imagination, picturing some
triumph of constructive skill--something as far
surpassing Sir William Thomson's siphon recorder in
delicacy and intricacy as that is beyond a common bell
pull. When at last this little instrument appeared,
consisting, as it does, of parts every one of which is
familiar to us and capable of being put together by an
amateur, the disappointment arising from its humble
appearance was only partially relieved on finding that
it was really able to talk...
Professor Graham Bell, the inventor of the telephone,
is not an electrician who has found out how to make a
tin plate speak, but a speaker who, to gain his private
ends, has become an electrician.
Two more quotations from commentators on science are
relevant to the subject of the initial rejection and now the slow
and grudging acceptance of resampling by the mathematical-
statistical profession: 1) "It is the essence of a profession
that the skills required therein are not possessed by those
without" (Goodwin, 1973, cited by Fisher, p. 80). 2) "Is not
every new discovery a slur upon the sagacity of those who
overlooked it?" (Jewkes, 1991, p. 11). <2>
The main points of divergence between the statistics
profession and me, then, are three-fold: 1) I am a Martian, in
Barrow's phrase, providing simple new methods but not rigorous
proofs; proofs are of the highest value to the profession, while
the simplicity is anathema. 2) As mathematicians, statisticians
aim at generality (embodied in formulae), whereas I aim at
specific solutions (embodied in numerical answers and the
procedures to obtain them). 3) I seek methods that are so simple
and transparent that those who are not mathematical statisticians
can use them correctly and with full understanding, but the
statistics profession honors methods that require the
intercession of a professional statistician.
THE VALUE OF DEXTERITY WITH MATHEMATICAL REASONING
This section picks a fight with the most self-assured and
prestigious sub-collection of these people - those who pride
themselves on being clever with mathematical thinking and puzzle-
solving.<3> The only conceivable counter-force that might support
the argument being mde here includes those who think themselves
clever yet know that that are not good at mathematical closed-
system thinking. But this group tends to be intimidated in these
matters by the mathematically-clever types, and therefore are not
likely to come out in support.
Let's be clear about the supposed equation in
mathematicians' minds of cleverness and mathematical thinking.
For example, a famous book of mathematical puzzlesstarts its
first page as follows: "To see how good your brain is..."
(Kordemsky, 1972, p. 1). And in Francis Galton's early
discussion of intelligence, he wrote: "There can hardly be a
surer evidence of the enormous difference between the
intellectual capacity of men, than the prodigious differences in
the numbers of marks [the grades] obtained by those who gain
mathematical honours at Cambridge" (1869, quoted in Herrnstein
and Boring, 1965, p. 416).
A professorial colleague of the author's teaches elementary
statistics (his department limits him to that course) and is not
fully witted by any practical test I know of. His behavior
borders on the bizarre in business and personal matters. He
apparently has never produced a useful piece of research work in
his several decades of university employment. He is not even
amiable or amusing. By no sensible measure that I know of could
his "intelligence" (whatever one means by that) be considered
even borderline average, except for his capacity to manipulate
mathematical symbols. Yet he contemptuously refers to his
university students as stupid because they don't follow the
formulas he writes on the blackboard.
Not only do I not know of any empirical evidence to believe
that those who think in mathematical terms (whatever that means)
are generally better thinkers than those who don't, but I can't
find much reason to believe that it ought to be so. "When was
the latest year that is the same upside down?" is Kordemsky's
(1972, puzzle 39, p. 15). So you figure out that 1961 is the
answer, either by some shortcut logical process or by trying
individual years going backwards. So what? What does finding
the answer prove? Perhaps there is some imagination shown in
working out a system for finding the number, but is this "better"
in any way than figuring out a system to water the garden more
efficiently? Or to set up a new club for neighborhood children?
In another puzzle (Kordemsky, p. 4), a troop of soldiers
stands by a river with no boat or bridge. Two boys are nearby in
a rowboat that is big enough for the two of them, but only for
one soldier. How can you move the soldiers across the river?
The book's solution has to do with moving combinations of boys
and soldiers. But what about making a rope and pulling the boat
back and forth? The mathematical approach shuts out such ideas
that go outside the facts that have been given. Such closed-
system thinking certainly can be useful, but it certainly is not
the whole of good thinking, and it may well direct people away
from creative open-system thinking.
Another flaw with treating mathematical puzzles as a test of
the goodness of one's brain is that solving many math puzzles is
simply a matter of knowing mathematical rules. For example, the
solution of one well-known puzzle hinges on remembering that one
is not allowed to divide by an algebraic expression that equals
zero. This has nothing to do with having a brain with good
general capabilities; it has to do only with knowing mathematics
(and it might just as well be the rules of chess or of bingo).
Amos Tversky once remarked that nothing is surer than that
people err in their thinking. In a valuable body of research,
cognitive psychologists point out the large defects in people's
thinking about issues that require statistical understanding, and
they document these effects in controlled experiments. For
example, people do not recognize the effect of sample size upon
the extent of variability (as was the case even with the great
John Graunt; see Chapter 00), and they do not estimate posterior
odds well when Bayesian thinking is called for, as well as
misapplying a variety of heuristics.
In response, the psychologists diagnose the problem as a
lack of "intuition" deriving from a poor grasp of statistical
theory, and with the aim of "improving the quality of thinking"
(Kahneman and Tversky, 1982, p. 494) they suggest more and better
statistical training. For example, Tversky and Kahneman note
that when asked how many different different committees of k
members can be formed from a group of n people, subjects will
guess that there are more possible 2-person than 5-person in a
group of 10 persons. They then say that "One way to answer this
question without computation is to mentally construct..." (p. 12,
italics added), and they go on to suggest a mode of imagining.
Lewis Carroll produced an entire book full of "Pillow
Problems" that he solved in bed without paper and pencil. He
wrote that he did not take special pride in having solved them
that way, but is that believable?
Should we consider a person a better carpenter who chooses
to build a house only with old-style hammer and saw rather than
using the entire array of modern tools that are available? Or
playing a piano concerto with one hand? A tour de force may be
amusing and impressive, but it displays art and not productive
power.
Kahneman and Tversky show subjects the series 8 * 7 *...1,
and 1 * 2 *...8 for five seconds and ask them to compute the
answer, finding different answers for the two modes of
presentation. But why not focus on the problems people make
when they have plenty of time to solve them? (Perhaps there is a
hint here of the idea that cleverness is associated with being
"quick" or "fast" in one's thinking.) Certainly these authors
would agree that the better scientist is not the one who thinks
up conclusions or research ideas more rapidly, but rather the
scientist who gets better conclusions and more important research
ideas. Always our aim should be to make people more sound in
their thinking, as tested by their effectiveness with regard to
the world of objects, people, and events, rather than making them
more clever (although admittedly, being seen as clever can
enhance one's effectiveness with people.)
Nisbett et. al., say that "reasoning is based on models",
and recommend that people learn to "call to mind a statistical
heuristic" (1982, p. 448). And as a way of teaching people that
there will be more variability in the percentage of babies born
in a small hospital than in a large hospital they suggest that "a
correct answer can be elicited in a series of easy steps" of a
Socratic nature (1982, p. 500).
I submit that the essence of the problem is not the lack of
well-trained intuition but lack of a more important mental tool:
the habit of actually (not in the imagination) simulating the
situation at hand so as to estimate the relevant probabilities.
That is, we should advise people not to rely on mental
operations, and not to try to improve their thinking with
statistical theory directly. Instead, we should teach them the
heuristic "Try it". Instead of asking people whether the
percentage will vary more in large and small hospitals, ask them
the probability that the proportion will be greater than (say) 60
percent with daily samples of 15 and 45 babies, and then teach
them to simulate the situation either with coins or with a simple
computer program. The ironic part is that even though the
subjects do not need to understand the theory to arrive at a
sound answer, they are likely to learn from the results the very
theory that would help them deal with the problem correctly by
mental computation alone. But learning the theory and improving
one's intuition as a result of the doing simulations is just a
bonus; the main goal - which should always be kept in mind - is
arriving at sound answers to the questions that arise in a
person's work and personal life.
Let's be specific. One of the most-used problems in the
study of cognitive psychology is K and T's taxi question. It
goes like this:
A cab was involved in a hit-and-run accident at night.
Two cab companies, the Green and the Blue, operate in
the city. You are given the following data:
(a) 85% of the cabs in the city are Green and 15% are Blue.
(b) a witness identified the cab as Blue. The
court tested the reliability of the witness under the
same circumstances that existed on the night of the
accidentally and concluded that the witness correctly
identified each one of the two colors 80% of the time
and failed 20% of the time.
What is the probability that the cab involved in the
accident was Blue rather than Green?(Tversky and
Kahnemann, p. 157)
Subjects mostly guess quite wrongly, the median answer
being .8 whereas the correct answer is .41.
A person who chooses to experiment rather than ratiocinate
may proceed as follows:
1. Assign numbers 1-85 as Green taxis, 86-100 as Blue
taxis. Because 80 percent of each group will be identified
correctly and 20 percent incorrectly , assign numbers 1-68 as
correctly identified Green taxis, 69-85 as incorrectly-identified
Green taxis (as Blue), and numbers 86-97 and 98-100 as correctly-
and incorrectly-identified Blue taxis respectively.
2. Choose x integers randomly between 1 and 100 and record
them.
3. Count the number of integers between 69-85 and between
86 and 97.
4. Compute the ratio of the number of integers between 86
and 97 to the sum of those 69-85 and 86-97.
One may ask why one should bother to experiment when one can
go directly to step 4. But the fact is that people do not manage
to go directly to step 4 because doing so requires an
understanding that most people demonstrably do not possess. And
that understanding is not necessary when one models the process
directly and proceeds from step 1 to step 4. I do not argue that
the latter process is better than former; I "merely" say that
that it produces more correct answers.
Such simulation is feasible in just about every situation.
And it may not even take longer than ratiocination. Consider for
another of the psychologists' pet problems (from Kahneman and
Tversky), the variability in proportions of boys and girls born
in hospitals of various sizes:
A certain town is served by two hospitals. In the
larger hospital about 45 babies are born each day. As
you know, about 50 percent of all babies are boys.
However, the exact percentage varies from day to day.
Sometimes it may be higher than 50 percent, sometimes
lower.
For a period of 1 year, each hospital recorded the
days on which more than 60 percent of the babies born
were boys. Which hospital do you think recorded more
such days?
Here is how it may be handled with the following computer
program:
REPEAT 1000 Begin the first of 1000 trials
GENERATE 200 1, 2 a Create a sample of 200 boys or girls
COUNT a 1 b Count the number of boys
DIVIDE b 200 c Normalize result as percent boys
SCORE b z Record the result of the trial
END End the 1000 trials
HISTOGRAM z Plot the results of the 1000 trials
By changing the sample size of babies from 200 to 20 and
2000 - by simply substituting one of those numbers in the second
and fourth numbers in the program above one can immediately the
effect of changing the sample size on the variability.
The cognitive psychologists ask: Why do people get these
questions wrong? I answer on a different dimension than they do,
saying "Because people do not experiment, but rather try to think
unaided by concrete trials".[1]
Nor is this change in dimension of the answer simply ducking
the "real" question. Once I watched an Indian in a market of the
city of Jubalpur carve bed legs on a lathe. The four legs for a
single bed came out far from identical because the lathe worker
worked by eye rather than using a patterned guide. It makes
sense to say that the reason that the legs were uneven is that he
did not use a guide, rather than saying that his eyes and hands
are fallible. (When I suggested using a guide, the lathe worker
said that it was a good idea. Maybe in the future. The shop was
still new. I asked how new. Nine years old, I was told.)
Why do people raticinate and not experiment? Because we
teach them the former and not the latter, and because we put a
high value - and the measure of "intelligence" - on the former
rather than the latter.
***
A comment of the philosopher Collingwood pertains to the
mathematically clever: The tail-less dog praises tail-
lessness.
**FOOTNOTES**
[1]: Nothing said here is intended to suggest that the
cognitive psychologists' study of the causes of biases is not
valuable. It certainly is important for us to understand biases
if only to anticipate and counter them in non-quantitative
situations where one cannot resort to experimentation. But I
would suggest to the cognitive psychologists that it is
worthwhile to teach people to seek concrete representations for
abstract questions even in non-quantitative situations; the
fathers-and-sons puzzle in Chapter 00 is a powerful example.
Another example is estimating the area of a figure with the
Archimedean method or by counting squares under the curve.
Indeed, the cognitive psychologists have long shown the way
in just such a fashion when they instruct people to estimate odds
with various devices such as hypothetical bets, or by simply
checking for consistency by estimating the negative of an event
and calculating whether the two probabilities add to unity. (I
have not, however, come across any research on improvement in
skills from using such methods.)
copied into refstats. take out this page for book, but keep for
ref
REFERENCES
Barrow, John D., Pi in the Sky: Counting, Thinking and Being
(New York: Oxford UP, 1992)
Chatterjee, Samprit, and Bertram Price, Regression Analysis
by Example (New York: John Wiley, 1991).
Gardner, Martin, New Mathematical Diversions From Scientific
American, ( New York: Simon & Schuster, 1966)
Hardy, C. H., A Mathematician's Apology. Peskun, Peter
H., "A New Confidence Interval Method Based on the Normal
Approximation for the Difference of Two Binomial Probabilities",
JASA, Vol 88, #422, pp. 656-660
Stigler, George J., "The Adoption of the Marginal Utility
Theory", in R. D. Black, A. W. Coats and Craufurd D. W. Goodwin
(eds), The Marginal Revolution in Economics: Interpretation and
Evaluation (Durham: Duke U. Press, 1973)
Stigler, Stephen M., The History of Statistics (Cambridge:
Harvard U. Press, 1986).
Bin Cum Center Freq Pct Pct -------------------------------------------- -0.38 2 0.0 0.0 -0.35 1 0.0 0.0 -0.34 7 0.0 0.1 -0.33 3 0.0 0.1 -0.32 7 0.0 0.1 -0.31 13 0.1 0.2 -0.3 8 0.1 0.3 -0.29 26 0.2 0.4 -0.28 29 0.2 0.6 -0.27 47 0.3 1.0 -0.26 55 0.4 1.3 -0.25 107 0.7 2.0 -0.24 134 0.9 2.9 -0.23 167 1.1 4.0 -0.22 195 1.3 5.3 -0.21 260 1.7 7.1 -0.2 269 1.8 8.9 -0.19 402 2.7 11.5 -0.18 386 2.6 14.1 -0.17 527 3.5 17.6 -0.16 528 3.5 21.2 -0.15 568 3.8 24.9 -0.14 721 4.8 29.7 -0.13 686 4.6 34.3 -0.12 840 5.6 39.9 -0.11 770 5.1 45.1 -0.1 910 6.1 51.1 -0.09 758 5.1 56.2 -0.08 929 6.2 62.4 -0.07 715 4.8 67.1 -0.06 711 4.7 71.9 -0.05 620 4.1 76.0 -0.04 593 4.0 80.0 -0.03 545 3.6 83.6 -0.02 477 3.2 86.8 -0.01 418 2.8 89.6 0 307 2.0 91.6 0.01 346 2.3 93.9 0.02 227 1.5 95.4 0.03 222 1.5 96.9 0.04 101 0.7 97.6 0.05 91 0.6 98.2 0.06 74 0.5 98.7 0.07 63 0.4 99.1 0.08 37 0.2 99.3 0.09 32 0.2 99.6 0.1 20 0.1 99.7 0.11 16 0.1 99.8 0.12 8 0.1 99.9 0.13 8 0.1 99.9 0.14 4 0.0 99.9 0.15 3 0.0 100.0 0.16 3 0.0 100.0 0.17 1 0.0 100.0 0.2 1 0.0 100.0 0.23 2 0.0 100.0
Note: Each bin covers all values within 0.005 of its center.
ENDNOTES
**ENDNOTES**
<1>: This is the term that William Kruskal used in 1984
correspondence with me in distinguishing between a) the simple
uses of resmampling and the bootstrap of the sort described
here, and b) the sophisticated uses and explorations of the
characteristics of these techniques that began with the work of
Efron and colleagues in 1979.
<2>: This discussion may reflect the fact that much of my
professional life has been a tussle with the mathematical
attitude - people I consider deductionists. In discussion of
such issues as causality, duopoly theory, and of course
resampling, they have always sought to arrive at conclusions by
the logical analysis of closed systems, excluding rich variables
which would make the analysis intractable mathematically, and
also excluding any judgments which would make it impossible to
close the system. My tendency always is the opposite. And I
always begin with concrete examples, and work my way toward the
abstract generality, whereas many deductionists praise the
opposite route (even if they themselves often do not follow it.)
In the same spirit, I tend to verify matters by using two or
more methods to arrive at an answer, and if they agree, I need
not worry about which is the better method. I find the
deductionists often preferring to argue about which is the
correct method.
Perhaps my not being attracted to abstraction and deduction
is due to my lack of capacity to manipulate equations, and this
in turn seems due to my inability to remember the meanings of
notation. This may be connected with my preference for working
with concrete entitities. I do not share the ability of phyicists
and others to forget the content of an equation and manipulate
it symbolically to see where it arrives. (I am startled at how
so many physicists have no sense of the "philosophical" content
of Einstein's special relativity, but place great confidence in
the results of their manipulations of its equations.) It makes
sense to me, and seems confirmed by observation, that working
with contentless symbols often leads to error.
<3>: More generally, this section challenges the most
powerful collection of persons in the world - those who think
that they are clever. This set is also one of the largest
collections of persons in the body of humanity, including almost
everyone who went to college and a good many who never went to
school at all.
Most specifically, the argument in this section challenges