Nobel Prizes and
the Stock Market
In her book, "Beyond Terman  contemporary longitudinal
studies of giftedness and talent", Rena Subotnik discusses the prior
environments that predispose toward Nobel Prizes. I
won't say anything about "old boy" networks.
There's an article
in this month's issue of Smart Money that confirms what I had deduced from the
calculations I ran a few weeks ago: that the stock market has reached the
highest valuations in his history by a factor of, at its peak, twotoone. The
author says,
"As I wrote here
on May 10, the stock market, in December 1996, launched into a prolonged
period of consistently high valuation. Ever since then, the prices that
investors have been willing to pay for stocks have been far greater in relation
to average expected earnings than ever before in postwar history. Ironically,
this period of high valuation began almost immediately after Federal Reserve
Chairman Alan Greenspan warned for the first time of "irrational
exuberance." Only during the panics of October 1998 and September 2001 did
the markets return — and then only briefly — to anything like what Greenspan
would consider "rational." Those brief moments proved to be bottoms.
We might be approaching such a moment right now."
The author concludes that the stock market is going to turn
up again in due course, and we're going to have business as usual. I certainly
can't gainsay him, but I'm still concerned.
Stock valuations in terms of the Federal Reserve's yield formula may not be
outofline, but the formula itself is of recent vintage, and may possibly be
something that won't guide investments twenty years from today. One of its
vulnerabilities seems to me to be that of putting all one's eggs in one basket.
If pricetoearnings ratios are the defining measure of value in pricing
stocks, then pricetoearnings ratios become the sole target for spin and window
dressing, and in fact, earnings scandals seem to be the concern du jour
that's holding back the stock market right now.
Agreement
of the LnNormal Model with WISC Manual
John Scoville has found a table in the
administration manual for the Wechsler IntelligenceScaleforChildren (WISC)
that converts StanfordBinet IQs to Wechsler IQs. He writes that it, "basically
agrees with the Sare data."
I've ordered a copy of the WISC user's guide,
and should receive it within a day or two. But that's a very important data set in establishing the validity of the lnnormal model.
One
important fact about this lnnormal relationship is that it would
seem to me to be allornothing. There aren't any parameters to
jiggle to make it fit except the standard deviation of the logarithms
(which John has determined is very close to, if, possibly, slightly
below 0.15).
The first step will be to buttress the link
between children's StanfordBinet IQs and the Wechsler
IQ scale. (The Wechsler IQ scale is a deviation"IQ"
scale with a standard deviation of 15.) Then it will be time to
look at how far this can be pushed.
In Defense of
Deviation "IQs"
One of the motivations for cleaving to a deviation"IQ"
is that in the mentalage/chronologicalage formula, chronological
age can't be as easily defined as it is for children. The rate
of rise of intelligence slows down markedly after puberty, as
it does with height, with "g" peaking in the twenties,
and crystallized intelligence continuing in some cases to rise
into midlife. Then it slowly declines, as shown in this age curve. Since it's based upon
percentiles, the adoption of deviation IQs allowed the testtakers
to assign the same IQ to someone who was 75years old that they would
to that same individual at 25, even though, on average, the 75yearold
would score only 90% as high as she did at 25. (Of course, the
same approach could be taken with StanfordBinet IQs by upping
the chronological age divisor somewhat between 16 and 25, and
then lowering it gradually with rising age until, by age 75, the
chronologicalage divisor were 90% of what it had been at 25.)
DeviationIQs
are Relative; RatioIQs are Absolute
Mental age gives us an absolute meterstick
against which to measure IQs (although not as meaningful a scale
as I would like), whereas deviationIQs are relative, defined by percentile
rankings. I could assign percentiles to a population whose IQs
lay between 100 and 101, and then go on to assign deviationIQs
from 0 to 200 to the people whose IQs lie between 100 and 101. Now, there are
claims that deviation "IQs" are the only true IQs. After all, after
correcting for the difference in standard deviations (15 vs. 16), Wechsler IQs
are only a point or two lower than SB IQs over the range from 70 to 130, and
that's the range for which David Wechsler certified his test. The growing
difference above an IQ of 130 between IQs delivered by the two tests may simply
be caused by uneven rates of maturation. Maybe the brighter the child, the
sooner the child stops growing mentally. Maybe adult intelligence covers a
considerably smaller range of mental capacities than childhood intelligence.
With deviation IQs, there's no way to tell. And who cares what happens above IQ
130, anyway? We've shown that above an IQ of about 120, there are no significant
differences in life outcomes.
Or so it might be argued.
Patrick Wahl's
LnNormal Distribution Simulators
"First, from Eric Weisstein's Math World pages:
http://mathworld.wolfram.com/LogNormalDistribution.html
Second, from a Swiss institute I don't know:
http://www.inf.ethz.ch/~gut/lognormal/
and
http://www.inf.ethz.ch/~gut/lognormal/start_applet_800_600.html
The last link is a Java applet ... like the familiar simulation, where
falling balls bounce randomly through a regular array of
pins and pile up into a heaped normal distribution, except
that these parameters are multiplicative, and the
resulting heap is lognormal. Fun to watch, if you pick
the higher speeds. (You can't reset once it starts,
you have to refresh the screen and then enter new
parameters.)"
When Patrick sent his email, for some reason, I
wasn't able to get these to work, but now they're working fine. Thanks again,
Patrick.
Are
Geniuses Both Born and Made?
Between 1912 and 1926, the Lutheran High School
in Budapest produced four of the leading physicists and one of
the leading mathematicians of the twentieth century: Leo Szilard,
Eugene Wigner, John von Neumann, Edward Teller and Paul Erdos.
The proximate cause? An inspiring science teacher by the name
of László Rátz. In the meantime, the physics
department at the University of Rome was producing the "Italian
school" of geniuses, Enrico Fermi, Bruno Rossi, Bruno Pontecorvo,
and Emilio Segre. The secret? An inspiring department head (who
was also a senator).
These individuals were undoubtedly extraordinarily
gifted by birth, but they also were inspired by a mentor at an
age when they were making lifetime career decisions. Students
of this process observe that to transition from being a prodigy
to a trailblazing adult begins with family support, then mentoring
by a nurturing mentor, and finally, mentoring under an exacting
taskmaster (or taskmistress) who can teach the burgeoning genius
her/his craft.
I have published this
before, but it needs to go into the intelligence site map, as
does this.
The
Army General Classification Test (AGCT) Scores
I had
planned to present the Army General Classification Test data,
and to compare it with a lnnormal distribution, but upon closer
examination, I discovered that it gives Wechsler "IQs",
only it uses a standard deviation of 20 instead of 15(!?) In other
words, it converts the frequencies of occurrence to zscores,
multiplied by 20. Consequently, it can't be used to compare ratioIQs
with a lnnormal distribution.
Caveats
In
touting the implications of the thesis that the natural logarithms
of IQs are normally distributed, I want to be careful that I don't
get carried away, and present this as gospel. I'm enthusiastic
so far about the fit that seems to be occurring between the logarithms
of IQs and a normal distribution. At the same time, to the best
of my knowledge, this hasn't had the benefit of any review by
experts in the field of psychometrics, or if so, it hasn't been
at my behest. For all I know, lnnormal models of I distributions
may have been explored and abandoned 50 years ago. Also, there
are some hard questions that may be posed if we assume that IQ
distributions are lnnormal. For example, Terman, et al, observed
that the Concept Mastery Test rose by the equivalent of 5 or 6
points of IQ between the administration of the CMTA form of the
test in 1940 and the CMTT version in 1950. Both these tests are
tests of crystallized intelligence rather than fluid intelligence,
so these findings aren't at variance with the current idea that
vocabulary can rise into one's 50's and 60's. But how do we dovetail
this into a lnnormal model of IQs? The disparity between fluid
intelligence, which rises at 6 points per decade, and crystallized
intelligence, which, for a givenaged test subject, has risen
little or no enhancement over the decades, seems to me to force
a split between these two aspects of intelligence even though
factor analysis reveals only a single general factor, "g".
Does
Fluid Intelligence Decline with Age, or Is It Overtaken By Rapidly
Rising Norms?
Certain components of intelligence seem to
change with age. On the other hand, this might be simply a manifestation
of the rapid, 6pointperdecade rise of fluid intelligence, coupled
with little or no change in vocabulary, general information. or
arithmetic. This could make it appear as though people experience
decreasing fluid intelligence over the decades, when, in fact,
they're as bright as they ever were, but slower and slower compared
with younger folk, who are boosting the norms with sevenleague
strides at 6 points per decade.
Does
Cognitive Decline with Aging Vary Markedly from One Individual
to the Next?
I suspect that when it comes to the effects
of aging on cognitive powers, a lot will depend upon general health
and mental activity, and may vary markedly from one person to
the next. Arteriosclerosis and prodromal manifestations of Alzheimer's
Disease probably also enter in. You wonder if brain size and reserve
capacity play a role, also.
Last night's link to the table converting Wechsler
"IQs" to StanfordBinet IQs went in the wrong
direction. It's now been corrected.
Plotting
Plots
One interesting
corollary to this investigation of IQ distributions is that the Wechsler "IQ"
isn't an IQ at all but is the natural logarithm of the IQ! For example, a Wechsler IQ of 144 becomes
0.44 after subtracting 100 and dividing by 100 to convert it to
a fractional deviation. Taking the exponential of 0.44 yields
1.5527, which, when multiplied by 100 to convert it to an IQ becomes
an IQ of 155.27, or about 155.
A precision table
converting Wechsler "IQs" to StanfordBinet (true) IQs
may be found here.
The table below reproduces the comparison among
Dr. Terman's IQ distribution, a Gaussian IQ distribution, and
a lnnormal (Sare) distribution. These tabular results are then
plotted in the two figures below the table.

Wechsler "IQ" Range 




140144 
133136  637 
1,000 
160 
16% 
145149 
137140  264 
525 
150 
29% 
150154 
141143  100 
281 
134 
48% 
155159 
144146  34 
151 
64 
42% 
160164 
147149  11 
82 
43 
54% 
165169 
150152  3 
34 
27 
79% 
170174 
153155  0.77 
21 
20 
95% 
175179 
156158  0.18 
8 
8 
100% 
180184 
159161  0.04 
4 
10 
250% 
185189 
162164  0.0085 
2 
2 
100% 
190194 
164166  0.00125 
1 
2 
200% 
195199 
167169  0.00022 
0.664 
0 
0% 
200+ 
169+  0.00003 
0.336 
1 
300% 
*  This column shows the distribution of IQs that Dr. Terman must have expected to find, based upon a presumed Gaussian distribution of IQs in his subject population. I have used a standard deviation of 16 in calculating the Gaussian predictions because that's what I think Dr. Terman would have employed.
You can see
in the Figure below how the Terman distribution of IQ scores (dark
blue) differs from both the Gaussian and the lnnormal distributions.
The bar chart below affords a slightly different perspective on
this data.
The
Meaning of Regression to the Mean?
Regression
to the mean takes place at or near the cutoff boundary
In thinking about
the testretest regression to the mean that occurs, it penetrated
my numb skull today that regression to the mean must occur as
a form of downward diffusion of scores that occurs when we establish
a threshold score on a first test and then give the test again.
There will be a some test subjects who barely passed the threshold
the first time we gave them the test who will fall just below
the threshold the second time we give them the test. However,
for the most part, test subjects who scored far above the threshold
the first time they took the test will probably still be well
above the threshold the second time the take the test.
If we
retested the upper half of the population (IQ = 101+), the regression
would be away from the mean.
One interesting case would be that in which
we selected the entire upper half of the IQ distribution on our
first testeveryone with IQ's of 101 or above. If we tested them
again a few months later on the same test or on an equivalent
form of the same test, some of their scores on the second test
would have drifted below 101. In this case, the regression
would be not to the mean but away from the mean.
If we
retest an entire population on a different version of the same
test, there will be no regression to the mean.
If we retest an entire population, there can
be no regression to the mean. If we retest with an equivalent
form of the same test, the population will have to exhibit the
same distribution curve that it did with the first test (except
for statistical fluctuations, which will be minimized if the size
of the population is sufficiently large). Individuals within the
population may be expected to have slightly different scores than
they got the first time they took the test, but the overall envelope
must remain unchanged.
Groups
within a group that are selected by a lowerend cutoff will regress
toward the mean on a retest
Individual groups within a selected population
who were selected by a lowerend cutoff, like all those who scored
170+ on a childhood StanfordBinet test, will diffuse downward
below 170 on a retest, while other children who scored in the
160's (and below) will diffuse upward. So we would expect the
average score of the children who scored 170+ on their childhood
SB test to be less on a retest than it was on their first
test (although the overall number of children with IQ's of 170+
should stay the same. There will just be some different children
in the 170+ group following the second testing.
The
higher scorers among the 260,000+ schoolchildren screened in 192122
should still be present when the group was retested in later life.
This has important consequences for the Terman
Study. Since I think that the Terman screening probably identified
all or nearly all of the students at IQs above, perhaps, 160,
it follows that all of them would still be present during the
followup IQtesting that occurred in 1940 and in 1950.
Why
I'm Finding It Difficult to Get Complete Data
The Terman Group Test scores are cited in Dr.
Terman's first volume in his series, "Genetic Studies
of Genius", but with no translation table converting
them to IQ scores, Consequently, I've been unable to arrive at
numbers for his total set of Termites. I could only use the numbers
for the "main group", which constitutes a little over
40% of them. I've come across a few other numbers, such as the
fact that there were 77 children identified with IQs of 170 or
above, that 26 of the 77 were children with IQs above 180,
and that the highest IQ was either 201, or in Ellen Winner's,
"Gifted Children, Myths and Realities",
pg. 24, that the IQs in the group ranged from 135 to 196.
Update,
6182002: Today,
I found mention of the fact that the average IQ of the high school
students who took the Terman Group Test in 192122 "was 142.6,
but because of the low ceiling on this test we estimate that the
mean "was spuriously reduced by at least 8 to 10 points.
If so, the mean IQ of the total gifted group when selected was
not far from 151 in terms of 1916 StanfordBinet scores."
(Lewis Terman, Genetic Studies of Genius, Vol. IV., pg.
135).
Dr.
Terman Explains Later That He Really Wasn't Trying to Find All
the Eligible Children
In his 1940 followup volume, Dr. Terman doesn't
claim to have made a clean sweep of the entire 260,000+ children
that he screened. Instead, he says in 1940, he wanted the higher
IQs among the top 1%, though that's not the way it's' presented
in his first volume, published in 1925. In his first volume, Dr.
Terman spends several pages discussing steps taken to insure that
most of the children who were eligible for the study were included
in it. He mentions that spot checks were made at a school in Los
Angeles with 350 children and a school in San Francisco with 800
children, both of which which were resifted for eligible children.
No new entries were found in the Los Angeles school and ten new
children were picked up in the San Francisco retest. However,
he concludes that overall, about 90% of the children with IQs
above 140 were caught with their filter. (You wonder how he rationalized
the fact that they found only 629 children with IQs of 140+ in
the "main group" when a Gaussian distribution would
have predicted 1,050.)
Dr.
Ellen Winner on the Terman Study
Dr. Winner says (p. 24) that on the StanfordBinet,
"About two or three children out of a hundred have IQs of
130 or higher; only about one in a hundred has an IQ of 140 or
higher. Only one in ten thousand to thirty thousand will score
160 or higher, and only one in a million will score above 180."
As mentioned above, Dr. Terman found 26 children with IQs at or
above 180 out of 260,000 schoolchildren. Using Dr. Winner's frequency
of 1 child in 1,000,000 with an IQ of 180+, 26 children
would exhaust the quota for 26,000,000 children, or all
of the schoolchildren in the United States in 1921! I can only
guess at the number of children with IQs above 160, but I would
guess, considering the way the numbers are running, that it would
be about 200. Using Dr. Winner's frequency measure of 1 child
in 10,000 to 1 child in 30,000, that would exhaust a population
of about 2,000,000 to 6,000,000 schoolchildren, or about 8 to
24 times the size of Dr. Terman's sample.
Dr. Winner is one of our leading experts on
giftedness.
I don't mean to knock Dr. Terman or Dr. Winner
for the anomalies I seem to be finding. I'm highly impressed with,
and grateful for the research they've carried out. They've done
such good and important work that their results are widely quoted,
making it important that (in my opinion) what they say be reviewed.
Returning
to the Scene of the Crime
I keep returning to the Terman Study like a
detective to the scene of a crime. The Terman Study is still pivotal
in discussions of the intelligence and the highly gifted. There
are strange and major irregularities in the published Terman results.
Why
Didn't Dr. Terman Pursue or Proclaim His Major Discovery?
As far back as 192122, Dr. Terman and his
colleagues discovered that the IQs of his children didn't fit
a Gaussian distribution even approximately. Now when a scientist
finds something that deviates wildly from what theory would predict,
it's usually a great discovery. Normally, he will follow up his
anomalous findings with an investigation of the anomalies, or
if he doesn't, someone else will. Dr. Terman wrote in 1925 that
his data weren't fitting the expected normal distribution. And
yet, in 1940, when Drs. Terman and McNemar are reviewing the average
152 childhood IQ of the Termites, they describe this 152 IQ as
falling 3.15 standard deviations from the population norm. But
their own data showed that it didn't. A 152 IQ lies about 2.8
S. D. from the mean IQ of 100. And why didn't someone over that
18year period investigate the IQ distributions that must have
been pouring in from all the statebystate IQ testing that was
taking place?
Why
Didn't Anyone Question Dr. Terman's Strange Distribution of IQs?
Why hasn't anyone said anything about the gross
underrepresentation of Termites in the 140to155 IQ range? There
are only 20% more children in the 140144 IQ range than
there are in the 150154 IQ range. Didn't anyone question that?
Yet people run around blithely quoting the Terman Study and Dr.
Terman's results without any question being raised about his strange
data.
What
Was the National Intelligence Test?
Dr. Terman preselected his children with the
National Intelligence Test. What was the National Intelligence
Test? Was it some rehash of the Army Alpha? The year 1921 was
only five years after the introduction of the first intelligence
test, the 1916 StanfordBinet, and only three years after the
development and introduction of the Army Alpha intelligence test.
I never heard about a "National Intelligence Test" in
1945.
What
Was the Terman Group Test?
Dr. Terman rated his high school "Termites"
using the Terman Group Test (TGT). However, he doesn't provide
any IQ scores for the high school students . . only their
raw scores on the TGT. What is the Terman Group Test? What are
the IQ scores for the Terman Group Test? How was the Terman Group
Test normed? What ever became of it? Like the "National Intelligence
Test", I never heard of the Terman Group Test in 1945, although
many other tests were mentioned in the literature of that day.
Who
Else Has Tried to Correct for Ceiling Effects?
Dr. Terman "corrected" the IQs of
his oldest and/or brightest students for ceiling effects. No one
corrects IQ scores for ceiling effects today. How was that justified?
No,
Virginia, There Really Is No Grinch Who Stole Christmas
No, I don't really think there was any conspiracy
involved, except, perhaps, a "conspiracy" of 'If you
don't mention it, maybe they won't notice it", or "We
don't know what to do with it, so let's not bring it up"
And yet, it's one of those situations when, even though no conspiracy
was involved, it seems as though one might almost as well have
been.