Diagnostic testing in medicine refers to the direct measurement of a sign,
substance, response or change that is either a direct or a reasonably indirect
predictor of a disease, a disease predisposition, or a disease agent. Most diagnostic
test results are either continuous (such as an optical density reading using an
ELISA), ordered (such as serum neutralisation titers), or dichotomous (e.g. a
precipitate that is either present or not on an agar gel immunodiffusion assay). By
convention, diagnostic tests based on continuous or ordered results are frequently
dichotomised for decision-making purposes.
Depending upon its characteristics, costs and side effects, the use of a diagnostic
test may be confined to a comparatively small high-risk group, or it may represent a
screening tool for the population as a whole. In any case, although medical
diagnoses seldom achieve perfect certainty, for a diagnostic test to be useful, it
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should represent more than merely "an art of making conjectures".
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Sometimes, the posterior certainty about the occurrence of an event B naturally
influences the probability of occurrence of another event, say A (note that the
posterior certainty about the occurrence of B, of course, implies that the a priori
probability of B must have been larger than zero, i.e. P(B)>0). In the present
example, the information that the die shows at least 4 points reduces the "universe"
of the die from six possible outcomes (i.e. 1,2,3,4,5,6) to three (i.e. 4,5,6). The
stochastic nature of the die has remained unchanged, however, so that each of the
three remaining possible outcomes is still equally likely. This implies that the
'conditional probability' of an even number of points, given the die shows 4 points or
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more, equals 2/3.
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The second result on this slide helps solving a famous game theoretical puzzle that
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has confused many people over the last 100 years, including some famous
mathematicians. In the 'Monty Hall Problem', a player is shown three closed
doors. Behind one is a car and behind each of the other two is a goat. The player
is allowed to open one door, and will win whatever is behind the door. However,
after the player has selected a door, say no.1, but before opening it, the game
host (who knows what's behind the doors) opens another door, say no.2,
revealing a goat. The host then offers the player the option to switch to the other
closed door (i.e. no.3). Does switching improve the player's chance of winning the
car? The answer is yes! But why?
Whether the car is behind door no.1 (event A) must be independent of the host's
choice of door no.2 rather than no.3 (event B), so that P(A|B)=P(A). In other
words, the probability that the car is behind door no.1 stays at 1/3. The only
other possibility is for the car to be behind door no.3, which therefore has
probability 2/3 since probabilities sum up to unity.
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With the information on this slide, it is possible to calculate the relative hypertension
risk associated with a high lipid level. Since
P( A ∩ B C ) P( A) − P( A ∩ B) 0.25 − 0.17
P( A | B ) =
=
=
= 0.10
1 − P(B)
1 − 0.20
P(B C )
C
the relative risk, i.e. P(A|B)/P(A|BC), equals 0.85/0.10 = 8.5.
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Thomas Bayes was born in London around 1702. He was privately educated by his
parents and later went on to be ordained as a Nonconformist minister, like his father.
Although Bayes worked as a clergyman until he retired in 1752, he nevertheless
managed to make major contributions to mathematics, most specifically in the fields
of probability and statistics. Bayes died in Tunbridge Wells, Kent, in 1761, although
his burial tomb is located in Bunhill Fields Cemetery in London.
Thomas Bayes wrote a number of different papers but only two are known to be
published during his life, namely 'Divine Providence and Government Is the
Happiness of His Creatures' (1731) and 'An Introduction to the Doctrine of Fluxions,
and a Defence of the Analyst' (1736). Bayes is best known for his 'Essay Towards
Solving a Problem in the Doctrine of Chances', published after his death in 1763.
Contrary to common belief, however, this paper does not contain what is known as
'Bayes' Theorem' today, but only a special case of it.
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Bayes' Theorem enables a reverse, backward-looking assessment of the stochastic
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relationship between two events, say "cause" and "consequence". In other words,
if the probability is known with which A causes B, i.e. the conditional probability
P(B|A), then the probability can be calculated that B, once it has occurred, was
indeed preceded by cause A, i.e. the posterior probability P(A|B). These
calculations also require knowledge, of course, of the unconditional prior
probabilities P(A) and P(B).
Strictly speaking, Bayes' Theorem is not a mathematical theorem but rather
represents the simple arithmetic transformation of two conditional probabilities. It
therefore required no discovery in 1763 (and Bayes himself made no such claim),
or in 1774 when Pierre Simon de Laplace independently wrote down a similar
"principle". Many scholars even maintain that Bayes' Theorem reflects a pretty
modern way of thinking, so that the reason why it has never been found on a
Babylonian clay board is simply because there was no practical need for it. The
truth of this notwithstanding, in medical diagnostics the simple arithmetic of
Bayes's Theorem (or Bayes' Formula, for that matter) is of the highest practical
relevance.
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The denominator of Bayes' Theorem, i.e. the prior probability P(B) of consequence
B, is usually unknown. It is therefore replaced by the right-hand side of the so-called
'Law of Total Probability'. This mathematical law states that the prior probability of
an event B equals the weighted average of the two conditional probabilities of B
given either cause A or its complement AC, respectively.
Since the two events A and AC are exhaustive and mutually exclusive, they define a
decomposition, not only of the "universe" Ω, but also of event B. Formally
P(B) = P(B ∩ A ) + P(B ∩ A C ).
Now, if we apply the definition of the conditional probability to each individual term
in this sum, we get
P(B) = P(B | A ) ⋅ P( A ) + P(B | A C ) ⋅ P( A C ),
which completes the proof.
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In the context of diagnostic testing, the Law of Total Probability usually relates to
two mutually exclusive events, A and AC, representing the presence or absence of a
disease or disease predisposition, respectively. Event B signifies a diagnostic test
result so that Bayes' Theorem serves the purpose of computing the conditional
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probability of disease or disease predisposition, given the outcome of the test.
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The current practice of medical diagnosis (from Greek "dia": by, "gnosis":
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knowledge) is still dominated by views and theories set down in the early 1900's.
According to William Osler (1849-1919), the famous Canadian physician who
introduced bedside teaching into the medical curriculum, the role of a doctor is to
identify disease and to understand how it may be prevented or cured. Osler's own
emphasis in this respect was on the classification of disease, and he saw the
patient as representative of a class of people with the same type of disease. The
patient's biological individuality was not given great weight.
Oslerian philosophy is still influential today and, couched in mathematical terms,
means that individual patients are regarded as being "randomly drawn" from
large homogenous populations with the same diagnostic characteristics or
properties. Following this interpretation, it does indeed make sense to quantify
the conditional "probability" that a patient in question has the disease, given his
or her test result. Without accepting such a stochastic nature for the individual
disease status, however, probability computations would be meaningless. In fact,
the question whether a probability can retrospectively be ascribed to a single,
factual event lies at the heart of the debate between Bayesian and non-Bayesian
statisticians.
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In 1998, the Max-Planck-Institute for Human Development, Berlin, performed a
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systematic study on how low-risk clients were counselled for AIDS in Germany
[Gigerenzer G, Hoffrage U, Ebert A, 1998, AIDS counselling for low-risk clients,
Aids Care 10: 197-211]. One of the researchers visited 20 public health centres in
the country as a client to take 20 counselling sessions and HIV tests. A majority
of the counsellors explained that false positives do not occur, and half of the
counsellors told the client that, if he tests positive, it is 100% certain that he is
infected with the virus. In reality, however, the probability of a test-positive
person being infected is only 1.95% if that person belongs to a low-risk
population.
The false positive rate of the latest ELISA test for HIV infection is of the order of
0.5%. The ELISA test detects antibody against a single HIV antigen, and the
influenza vaccine has been reported to sometimes result in a false positive test.
Other causes include pregnancy, recent blood transfusion and autoimmune
diseases such as lupus. However, the most common reason for a false positive
ELISA is laboratory error! Therefore, the false positive rate of this diagnostic test
can be reduced by independent re-testing of individuals with a positive result
(among whom the prevalence, i.e. 1.95% compared to 0.01%, is much higher
than in the general population).
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The most practically relevant aspect of Bayes' Theorem in a diagnostic context is
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that it yields the posterior probability of disease, given the test result. This is the
number that eventually has to be communicated to the client or patient. Note
that the posterior probability of disease depends critically upon the prior
probability of disease (i.e. the prevalence) and is therefore a hallmark of the
population in which the test is applied. Since the posterior probability of disease
also reflects the accuracy with which a disease status can be derived from a test
result, it is often called the 'predictive value' of the test.
The probability with which a test outcome is positive for diseased individuals is a
natural measure of its 'sensitivity'. If the test is not capable of detecting many
affected people, its sensitivity is poor. In contrast, the probability with which the
test shows a negative result among clients without the disease in question
indicates the 'specificity' of the test. Sensitivity and specificity are characteristics
of the technical, medical or biochemical conditions of the test procedure. With all
relevant factors unchanged, the test should in principle perform equally in any
high-risk or low-risk group.
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Both the positive and negative predictive value of a diagnostic test should ideally be
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large. However, in a population with a low prevalence, the posterior probability of
disease will be small irrespective of the test result, unless the specificity of the
test is very close to 100%. Since the specificity of the ELISA test is "only" 99.5%,
i.e. there is a 0.5% chance of false positive results, it is not surprising that the
positive predictive value is still small for realistic prevalence figures (i.e. of the
order of 0.1% or less).
On the other hand, the sensitivity of the ELISA is so high (99.5%) that the
negative predictive value is sufficiently high for virtually all practically relevant
prevalence figures. Thus, even among intra-venous drug abusers where the
prevalence of HIV infection is as high as 15%, a negative test result implies that
the proband is almost certainly (>99.9%) not infected.
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Brain atrophy is obviously a poor diagnostic marker for schizophrenia. At a typical
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prevalence of 1.5%, only 18.6% of positive test results would be "true positives"
in the sense that the atrophic individual would also be schizophrenic. Unlike the
ELISA test for HIV, this diagnostic test cannot be repeated independently in order
to exclude false positives from the first round of testing.
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The slope of the positive predictive value of brain atrophy for schizophrenia is
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comparatively steep, owing to its high specificity (only 2% of non-schizophrenics
show signs of atrophy), but the PPV is still small for the actual prevalence of the
disease (i.e. 1.5%). The negative predictive value, in contrast, is poor for the
wide range of prevalence figures characteristic of high-risk groups, simply
because the low sensitivity of brain atrophy (only 30% of schizophrenics are
atrophic) implies that negative findings are no reliable indicators of the absence
of disease.
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A more succinct version of Bayes' Theorem can be obtained when likelihood ratios
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instead of conditional probabilities are taken into account.
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The odds-based presentation of Bayes' Theorem has two advantages. First, it is
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easier to remember ("posterior odds equal prior odds times likelihood ratio").
Second, the information provided by different, conditionally independent tests can
be accumulated successively, using the posterior odds upon one test as prior odds
for the next test. Here, "conditionally" independent means that the outcome of
one test is independent of the outcome of the other test only among probands
with the same disease status (i.e. affected or not affected).
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The data on this slide were taken from a 1967 study by A.F. Smith, published in the
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Lancet [July 22; 2(7508): 178-182]. Approximately two third (230/360) of male
patients with a suspected myocardial infarction (MI) actually had an MI (prior
odds: 1.77). Of these, 215 had an increased serum creatine kinase level (≥80
U/l), which corresponds to a sensitivity of 93.5% for the test. Only 16 of the 130
patients without a confirmed MI (i.e. 12.3%) showed an increased enzyme level,
so that the positive likelihood ratio was 7.60. This implies that it was 1.77⋅7.60 =
13.45 times more likely than not for a patient with an increased creatine kinase
level in that study to have had an MI (posterior odds: 13.45).
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The literature on diagnostic testing often provides definitions of test validity that are
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related to sensitivity and specificity. According to this view, a test is 'valid' if it
detects most people with the target disorder (i.e. has a high sensitivity), excludes
most people without the disorder (i.e. has a high specificity), and if a positive test
usually indicates that the disorder is present (i.e. has a high positive predictive
value). Other sources would define such tests as "performing well" whilst giving
'validity' a more technical meaning. A diagnostic test is valid in that sense if, by
design, it measures what it is intended to measure (e.g. an enzyme activity or an
antibody titre).
The 'reliability' of a diagnostic test is high when any variation between tested
individuals represents true differences. A common measure of reliability is the
'test-retest reliability' where one and the same diagnostic test is administered to
the same set of examinees on two separate occasions. The reliability of the test is
then expressed by the correlation between the two sets of results.
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A 'gold standard' is a procedure that is often either slower, less convenient, or more
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expensive than the diagnostic test under consideration, but which ideally gives a
definitive answer as to the disease status of a proband. Thus, the gold standard
hypothetically provides 100% sensitivity and 100% specificity, i.e. it never yields
any false negative or false positive diagnoses. In reality, however, even decisions
based upon the gold standard may be incorrect, and the results of a diagnostic
study must therefore be interpreted in the context of current scientific
knowledge. The gold standard may change over time in response to medical and
scientific progress and, what is more, medical experts may hold differing views as
to what the gold standard actually is.
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With a high sensitivity, most clients that have the disease or disease predisposition
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yield a positive test result. Therefore, a negative test result provides reassurance
as to the absence of disease.
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With a high specificity, most clients that do not have the disease or disease
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predisposition yield a negative test result. For a disease of sufficiently high
prevalence, a positive test result therefore provides reassurance as to the
presence of disease. That this is not necessarily the case if the prevalence is too
low has been exemplified already by the HIV test provided to low-risk individuals.
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The two lists of criteria shown on this slide should be interpreted as so-called
'positive lists', which means that the different criteria are linked by the logical
expression "or". The more criteria on one of the two list are met, the more
important is the maximisation of the corresponding measure of test performance
(i.e. sensitivity or specificity).
The 'Guthrie test', also known as the 'Guthrie bacterial inhibition assay', is a
diagnostic test performed on newborn infants to detect phenylketonuria (PKU), an
inborn error of amino acid metabolism. PKU is a genetic disorder in which the body
lacks phenylalanine hydroxylase, the enzyme necessary to metabolize phenylalanine
to tyrosine. If left untreated, the disorder can cause brain damage and progressive
mental retardation as a result of the accumulation of phenylalanine and its
breakdown products. If the condition is diagnosed early enough, however, an
affected child can grow up with normal brain development, by eating a special diet
low in phenylalanine. This requires severely restricting or eliminating foods high in
protein.
The Guthrie test has been widely used throughout North America and Europe as one
of the core newborn screening tests since the late 1960s. In recent years, it is
gradually being replaced in many areas by newer techniques, such as tandem mass
spectrometry, that can detect a wider variety of congenital diseases.
The Guthrie test is named after Robert Guthrie, an American bacteriologist and
physician, who devised it in 1962.
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William John Youden (1900-1971) was born in Townsville, Australia, but came to the
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United States at an early age. He studied chemical engineering at the University
of Rochester, then chemistry at Columbia University. For many years, he worked
at an institute for plant research but eventually joined the National Bureau of
Standards. He always considered himself a chemist, but, for the last 40 years of
his life, he surely was a statistician as well. During World War II, Youden served
with the US Air Force in Europe and the Pacific and showed exceptional skill in
inventing novel statistical tools of experimental design to cope with problems of
bombing accuracy. He was awarded the Medal of Freedom for this contribution to
the allied victory.
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At first glance, brain atrophy as a diagnostic marker for schizophrenia appears to
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violate the 'SpPIn rule' whereby a high specificity should result in a high positive
predictive value (PPV). However, the present specificity of 98% is still too low to
compensate for the low sensitivity of 30% and the low prevalence of 1.5%. Even
with a specificity of 99.9%, the PPV would only be 82.0%.
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It is not uncommon for medical researchers in general to artificially dichotomize
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continuous variables. Examples of this are "pass–fail" on a test or task, "young–
old" in reference to age, and "rich–poor" in lieu of yearly income. Perhaps the
most common argument to justify this practice is that clinicians have to make
dichotomous decisions to treat or not to treat, so it makes sense to have a binary
outcome. However, dichotomization usually entails a loss of information since
measurements close to the cut-off are treated in the same way as measurements
far away from it. In such situations, those developing or applying a diagnostic test
have to decide whether caseness or the original measurement value is the reality,
and then decide whether dichotomization is indeed appropriate.
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In this study, all participants underwent a 50 g glucose challenge test between the
24th and 28th gestational week. Within one week, they were also administered a
diagnostic 3 hour 100 g oral glucose tolerance test (representing the gold-standard),
following a 12 hours period of fasting and 3 days of 150-200 g (minimum)
carbohydrate diet. Gestational diabetes was diagnosed if two or more values of the
tolerance test equalled or exceeded the cut-offs proposed and adopted by the 4th
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International Workshop Conference on Gestational Diabetes.
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'Receiver Operating Characteristic (ROC)' curves plot the sensitivity of a diagnostic
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test, based upon a dichotomized continuous measurement, against its false
positive rate (i.e. 1-specificity) for various cut-offs. It therefore shows the tradeoff between sensitivity and specificity (any increase in sensitivity will be
accompanied by a decrease in specificity). The name 'ROC' comes from signal
detection theory, developed during World War II for the analysis of radar images.
Radar operators had to decide whether a blip on the screen represented an
enemy target, a friendly ship, or just noise. Their ability to make these important
distinctions was called the 'Receiver Operating Characteristics'. ROC curves were
developed in the 1950's as a by-product of research into making sense of radio
signals contaminated by noise. It was not until the 1970's, however, that signal
detection theory was also recognized as useful for interpreting medical test
results.
The performance of a diagnostic test depends on how well the test separates the
group being tested into those with and without the disease in question. The
potential of a particular laboratory procedure or other diagnostic measure to
provide a well-performing test can be quantified by the area under the ROC
curve. An area of 1, corresponding to a rectangular ROC curve, represents a
perfect test. An area of 0.5, as obtained from the 45° line associated with cointossing, indicates a worthless test.
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All possible combinations of true and false positive rate are located on the ROC
curve. In order to find an optimal threshold, one has to decide on the relative costs
of false positive and false negative results. If both were regarded as equally costly,
and if the prevalence of the disease in question were 50%, then the best
combination of true and false positive rate would be the one that maximizes
Youden's index. This combination can be found simply by drawing a line parallel to
the 45° line that is tangential to the ROC curve. If false positives and false negatives
have different costs, or if the prevalence of the disease differs from 50%, then the
slope of the corresponding tangent would have to be proportional to both the cost
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ratio and the prior odds of the disease.
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In the present example, gestational diabetes was diagnosed in 53 (10%) of the 520
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women tested. The area under the ROC curve was 0.897 for fasting plasma
glucose concentration and 0.815 for the glucose challenge test, indicating that the
former test may be generally more useful than the latter.
The ROC curves also indicate that the best cut-off value for fasting plasma
glucose concentration is 4.8 mmol/l. Universal screening for gestational diabetes
using this threshold would have a sensitivity of 81% and a specificity of 76%.
This implies that approximately 30% (0.10⋅0.81+0.90⋅0.24=0.297) of women
would still have to proceed with a tolerance test.
A generally accepted cut-off value for the 50 g glucose challenge test is
7.8 mmol/l, yielding a sensitivity of 59% and a specificity of 91%. Although only
14% of women would be administered a diagnostic glucose tolerance test with
this screening method, more than twice as many cases of diabetes (41%
compared to 19%) would be missed. Obviously, as is indicated by the ROC curve,
there are better cut-off values for the glucose challenge test (7.0 mmol/l or lower)
than the one in current use.
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Bayes' Theorem can in principle also be used as a tool for differential diagnostics.
Instead of only two possible causes for the presence of a diagnostic marker, such as
"disease" and "no disease", any number of causes Kj can be taken into account.
It should be noted, however, that for Bayes' Theorem to yield the correct result, the
events Kj must be exhaustive and mutually exclusive. Both assumptions may not be
fulfilled in reality, for example, if the Kj denote potentially co-occurring disease
states. However, exhaustiveness can always be achieved by invoking an additional
"unrelated" cause, and if the individual events Kj are sufficiently rare, the probability
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of their co-occurrence is at least approximately equal to zero.
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The classification of pathological findings, by definition, usually provides a
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decomposition of the "universe" into exhaustive and mutually exclusive events,
suitable for inclusion into Bayes' Theorem. The non-invasive molecular marker
described here is helpful in two respects. First, the presence of mutational and
epigenetic variation at the respective genes reduces the chance of a tumour
being benign by 50% in smoking patients. Second, it nearly doubles the risk of a
tumour being of non-small cell type in non-smokers. Thus, the presence of the
marker is likely to have important consequences for the subsequent management
of the disease in both groups of patients.
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