Statistical Methods in Medical Research 4ed

56 Probability
converted to a ratio of posterior probabilities by multiplication by the likelihood
ratio.
A third form of Bayes' theorem concerns the odds in favour of hypothesis Hi:
p ijy
1 p ijy
ˆ pi
1 pi
l yji
P…yjnot Hi† : …3:5†
In (3.5), the posterior odds are derived from the prior odds by multiplication
by a ratio called the Bayes factor. The denominator P(yjnot Hi) is not a
pure likelihood, since it involves the prior probabilities for hypotheses other
than Hi.
In some examples, the outcomes will be continuous variables such as weight
or blood pressure, in which case the likelihoods take the form of probability
densities, to be described in the next section. The hypotheses Hi may form a
continuous set (for example, Hi may specify that the mean of a population is
some specific quantity that can take any value over a wide range, so that there is
an infinite number of hypotheses). In that case the summation in the denominator
of (3.3) must be replaced by an integral. But Bayes' theorem always takes
the same basic form: prior probabilities are converted to posterior probabilities
by multiplication in proportion to likelihoods.
The example provides an indication of the way in which Bayes' theorem may
be used as an aid to diagnosis. In practice there are severe problems in estimating
the probabilities appropriate for the population of patients under treatment; for
example, the distribution of diseases observed in a particular centre is likely to
vary with time. The determination of the likelihoods will involve extensive and
carefully planned surveys and the definition of the outcome categories may be
difficult.
One of the earliest applications of Bayes' theorem to medical diagnosis was
that of Warner et al. (1961). They examined data from a large number of patients
with congenital heart disease. For each of 33 different diagnoses they estimated
the prior probability, pi, and the probabilities l yji of various combinations of
symptoms. Altogether 50 symptoms, signs and other variables were measured on
each individual. Even if all these had been dichotomies there would have been 2 50
possible values of y, and it would clearly be impossible to get reliable estimates of
all the l yji. Warner et al. overcame this problem by making an assumption which
has often been made by later workers in this field, namely that the symptoms and
other variables are statistically independent. The probability of any particular
combination of symptoms, y, can then be obtained by multiplying together the
separate, or marginal, probabilities of each. In this study firm diagnoses for
certain patients could be made by intensive investigation and these were compared
with the diagnoses given by Bayes' theorem and also with those made by
experienced cardiologists using the same information. Bayes' theorem seems to
emerge well from the comparison. Nevertheless, the assumption of independence