CHANCE 59
Visual Revelations
Howard Wainer,
Column Editor
Until Proven Guilty: False Positives
and the War on Terror
Sam Savage and Howard Wainer
P
rint and electronic media have
beenfilledwithdebateconcerning
the tactics employed in the War
on Terror. These must invariably walk
thelinebetweenmaintainingcivilliberties
and screening for possible terrorists.
Discussions have typically focused on
issues of ethics and morality. Is it ethical
to eavesdrop on phone and email
conversations; to use ethnic profiling
in picking possible terrorists for
further investigation; to imprison
suspects without legal recourse for
indefinite amounts of time; to make
suspects miserable in an effort to
get them to reveal cabals and plots?
While we believe these are impor-
tantquestionstoask,wearesurprised
by how little of the debate has dealt
with the likely success of these tac-
tics.Giventheobvioussocialcosts,the
efficacy of such surveillance programs
must be clearly understood if a rational
policy is to be developed. Perhaps the
biggest barrier to public understanding
of this problem surrounds the issue of
falsepositives.Notonlyisthissubjectnot
intuitive, but getting it wrong can result
in counter-productive policies.
Bayesian analysis is the customary
tool for determining the rate
of false positives, and we
will illustrate its use on a
particularly vexing
analys
i
s
60 VOL. 21, NO. 1, 2008
problem: the use of wiretaps to uncover
possible terrorists living in our midst.
The use of such wiretaps has been hotly
debated. To evaluate the use of wiretaps,
we must consider both the chance we
will correctly identify a terrorist if we
haveoneontheotherendofthelineand
the overall prevalence of terrorists in the
population we are listening in on.
We shall start with the latter. How
many terrorists are currently in the
United States? (By terrorists, we mean
hard-core extremists intent on mass
murder and mayhem.) Let us assume
for argument's sake that among the
300,000,000 people living in the United
States, there are 3,000 terrorists. Or,
in other words, the prevalence is one
terrorist per 100,000. Once this case is
understood, it will be easy to generalize
for other numbers.
Now, consider a magic bullet for
this threat: unlimited wiretapping tied
to advanced voice analysis software on
everyone's line that could detect wouldFigure
1. A target whose total area is 2,999,970+2,970 and whose bull's eye has an area of 2,970, representing those terrorists who would
be identified by wire tapping. Courtesy of Sam Savage
Size of Bull's Eye=2,970
Size of Target=2,999,970+2,970=3,002,940
Chance of Bull's Eye=
2,970+3,002,940=0.1%
or 1 in 1,000
1% of 299,997,000=2,999,970
Falsely Reported Nonterrorists
99% of 3,000=2,970
True Terrorists
Target Represents All
Who Would Get Reported
beterroristswithintheutteranceoftheir
first three words on the phone. The
software would automatically call in the
FBI, as required, to arrest and question
those who triggered the terror detector.
Let's assume the system was 99% accurate.
That is, if a true terrorist was on the
line, it would notify the FBI 99% of the
time, while for nonterrorists, it would
call the FBI (in error) only 1% of the
time. Although such detection software
probably could never be this accurate, it
is instructive to think through the effectiveness
of such a system if it did exist.
The False Positive Problem
When the FBI gets a report from the
system, what is the chance it has identified
a true terrorist? There are two
possibilities. Either there has been the
correct report of a true terrorist or the
false report of a nonterrorist. Of the
3,000 true terrorists, 99%--or 2,970--
would be correctly identified. Of the
299,997,000 nonterrorists (300 million
minusthe3,000terrorists),only1%--or
2,999,970--would be falsely reported.
Thismaybethoughtofasatargetwhose
total area is 2,999,970+2,970 and whose
bull's eye has an area of 2,970, as shown
in Figure 1.
Thus, the chance of correctly identifying
a true terrorist is analogous to hitting
the bull's eye with a dart--roughly
one in a thousand, even with a 99%
accurate detector. If there were fewer
than 3,000 terrorists, this probability
would decrease still further. And even
if the number of terrorists went up tenfold
to 30,000, the chances of a correct
identification would still be only 1 in
100. What looked at first like a magic
bullet doesn't look as attractive once
we realize that about 999 out of 1,000
suspects are innocent. This is the "false
positives" problem.
But, of course, we started with an
extreme case for dramatic effect. In reality,
the bad news is that your terrorist
CHANCE 61
Accuracy of Screen
Prevalence
Number
Screened
per Actual
Terrorist
50% 60% 70% 80% 90%
100,000 100.0% 100.0% 100.0% 100.0% 100.0%
10,000 100.0% 100.0% 100.0% 100.0% 99.9%
1,000 99.9% 99.9% 99.8% 99.6% 99.1%
100 99.0% 98.5% 97.7% 96.1% 91.7%
10 90.0% 85.7% 79.4% 69.2% 50.0%
detector would be nowhere near 99%
effective. But the good news is that you
would be much more selective in who
you wire tapped in the first place. So,
in reality, the accuracy would be lower,
but the prevalence could be expected
to be higher.
Suppose you can sufficiently narrow
your target population to the point that
the prevalence is up to one actual terrorist
per 100 people wiretapped. Also,
assume a 90% effective test. That is, a
true terrorist will be correctly identified
90% of the time and an innocent person
10% of the time. The chance of a false
positiveisstill91.7%.Thatis,evenwhen
someone triggers an arrest, the odds are
11 to 1 they are not a terrorist.
Table 1 shows the probability that
someone implicated by a terror screening
system is actually innocent, based
on the prevalence of terrorists in the
screened population and the accuracy
of the test. In this table, the accuracy
is defined as both the probability a true
terroristisidentifiedassuchandaninnocent
person is identified as innocent.
Thereisnoneedforthesetwoprobabilities
to be the same, and later we provide
a calculator that will solve the general
case of asymmetric accuracy.
Upon calculating this table, even the
authors were surprised to find that if 1
person in 10 were an actual terrorist,
and if the screen were 90% accurate,
you would still have as many innocent
suspects as guilty ones.
For example, consider a battle with
insurgents who make up 10% of a population.
Table 1 implies that if you call
in air strikes on suspected enemy posi-
tionsbasedontargetingintelligencethat
is 90% accurate, you will be killing as
many civilians as combatants.
This result suggests arresting and
prosecuting terrorists in our midst is a
real challenge. A fascinating web site
at Syracuse University provides real
statistics in this area. According to the
Transactional Records Access Clearinghouse
at http://trac.syr.edu/tracreports/
terrorism/169, in recent federal criminal
prosecutions under the Justice Department
Program of International Terror-
ism,roughly90%ofthecasesbroughtin
have been declined for further action.
The Cost of False Positives
It is tempting for politicians to play
off our fears of horrific, but extremely
Table 1-- Percentage of False Positives by Prevalence and
Indicator Accuracy
unlikely, events. When they do, it is
easy for the nonstatistically trained to
fall for faulty logic. For example, a few
years ago, a supporter of an anti-missile
system for protecting the United
States from rogue states argued that the
specter of a nuclear weapon destroying
New York was so horrible that the U.S.
government should stop at nothing to
deter it. Oddly, he didn't bring up the
fact that it is much more likely such a
weapon would be delivered to New York
by ship, and that a missile attack--aside
from being much more expensive and
difficult--is the only delivery method
providing a definitive return address
for our own nuclear response. Interestingly,
recent studies have shown that an
effective program for detecting weapons
of mass destruction smuggled on ships
would cost about as much as an anti-mis-
silesystem.So,here,thequestioniseasy.
Should we defend against a likely source
of attack, or a rare one? But, in general,
the decision will be how much incursion
of our personal freedom we are willing to
endure per life saved.
In making this calculation, we must
be mindful of the extent to which the
harsh treatment of innocents can create
terrorists. For example, in a 2003
memo, then U.S. Defense Secretary
Donald Rumsfeld asked, "Are we capturing,
killing, or deterring and dissuading
more terrorists every day than
the madrassas and the radical clerics
are recruiting, training, and deploying
against us?" We must add to our cost
function the chance that the prosecution
of the innocent will bolster the
recruiting efforts of our enemies.
Is this probability algebra limited to
the War on Terror? Consider what would
be the likely outcome if we had universal
AIDS testing. Because the test is far from
perfect, we would surely find that the
numberoffalsepositiveswoulddominate
the number of AIDS cases uncovered.
And how much of the agony associated
withreceivingsuchanincorrectdiagnosis
would compensate for finding an otherwise
undetected case?
Our point is not that all testing,
whether for disease or terrorism, is fruitless;
only that we should be aware of the
calculus of false positives and use what-
everancillaryinformationisavailableand
suitable to shrink the candidate population
and probability of errors enough so
that the false positive rates fall into line
with a reasonable cost function.
The False Positive Probability
Calculator
Calculating the rate of false positives
involves comparing the ratio of areas in
thetargetofFigure1.Thisisusuallydone
with Bayes' formula, which is known to
all statisticians, but apparently few policymakers.
Thus, as a service to mankind,
we have created a false positive calculator
in Microsoft Excel, available for free
download at www.FlawOfAverages.com.
This calculator was created with
XLTree (see www.AnalyCorp.com). First, a
probability tree was generated based on
the prevalence and probabilities shown
in Figure 2. This tree was then inverted
(flipped) to perform what is known as
Bayesian Inversion. Figure 3 displays
the formula section of the calculator,
along with the other outputs in the dark
boxes and intermediate calculations in
grey boxes.
Last, we are certainly not the first
statisticians to use the tools of our trade
to look into the topic of terrorism.
Nor even the first to use the pages of
CHANCE to do so. The challenge is not
62 VOL. 21, NO. 1, 2008
False Positive Probability Calculator
Inputs
Probability of False Positive
Prevalence of Trait X in Screened Population
Probability That Trait X Is Correctly Detected
Probability That Lack of Trait X Is Correctly Detected
1 in 100
90%
95%
84.62%
Figure 2. False positive calculator, www.flawofaverages.com
9 4 . 0 5 %
Inputs
output
Other calculations
Probability Tree
Person screened
has Trait X
Person screened
is OK
Test is
Positive for X
negative for X
Probability of negative test
Probability of lack of X
Probability of X
Probability of positive test
Probability that trait X is
correctly detected
Joint probabilities
must sum to 100%
Probabililty that lack of trait X
is correctly detected
Probabililty of true positive
Probabililty of false positive
Probabililty of false negative
Probabililty of true negative
1.00%
99.00%
90%
10%
0.90%
0.10%
5%
95% 94.95%
4.95%
Test is
positive for X
negative for X
positive for X
negative for X
Person screened
has trait X
OK
has trait X
OK
15.38%
84.62%
0.11%
99.89%
94.15%
5.85%
Inverted Probability Tree
0.90%
4.95%
0.10%
0.40%
 Copyright 2007, Sam Savage Model Developed using XL Tree, available at www.AnalyCorp.com
Figure 3. Trees underlying the false positive calculator. Courtesy of Sam Savage
to convince other statisticians, but to
get decisionmakers to take both Type
I and Type II errors into consideration
when making policy. Toward this end,
we believe embodying these ideas in
a widely available calculator increases
the chances of holding decisionmakers
accountable. So, the next time you
become aware of a politician or bureaucrat
about to make a decision that may
bring more harm through false positives
than benefit though true ones, we
suggest you send them the link to the
calculator.
Further Reading
Banks, D. (2002). "Statistics for Homeland
Defense." CHANCE, 15(1):8­10.
Rumsfeld, D. (2005). "Rumsfeld's Waron-Terror
Memo." USA Today,
www.usatoday.com/news/washington/
executive/rumsfeld-memo.htm.
Savage, S. (In Preparation). The Flaw of
Averages. www.FlawOfAverages.com.
Stoto, M.A.; Schonlau, M.; and Mariano,
L.T. (2004). "Syndromic Surveillance:
Is It Worth the Effort?"
CHANCE, 17:19­24.
Wein, L.M.; Wilkins, A.H.; Baveja, M.;
and Flynn, S.E. (2006). "Preventing
the Importation of Illicit Nuclear
Materials in Shipping Containers."
Risk Analysis, 26(5):1377­1393.