Bayes's theorem First mathematical formulation, by the Reverend Thomas Bayes (1702-1761), of the principle of conditional probabilities allowing calculation of the probability that an event will occur or that an affirmation will be correct under certain conditions (e.g. that a test is positive or
Dictionary of Pharmacoepidemiology Author: Bernard Begaud Copyright © 2000 John Wiley & Sons Ltd ISBNs: 0-471-80361-8 (Hardback); 0-470-84254-7 (Electronic)
negative, or that a symptom is present or absent).
For diagnostic tests, Bayes' theorem can be expressed as follows:
p (T+ | D) x p (D) + p (T+ | D) x p (D)' In this notation:
• p (D | T+) is the probability that the subject is diseased if the test is positive (otherwise known as the positive predictive value of the test).
• p (T+ | D) is the probability that the test will be positive if the disease is present (otherwise known as the sensitivity of the test).
• p (D) is the a priori probability that the subject is diseased before knowing the result of the test (otherwise known as the risk in the population from which the subject is drawn).
• p (T+ | D) is the probability that the test will be positive if the disease is absent, that is, the complement of the test's specificity (1 — specificity).
• p (D) is the a priori probability that the subject is not diseased; p (D) thus equals 1 — p (D).
Bayes' theorem may more easily be expressed as follows:
p (D | T+) = pjD) p (T+ | D) p (D | T+) p (D) p (T+ | D).
In this formulation, the odds of presenting the disease given a positive test (posterior odds) is equal to the odds of presenting the disease estimated before analysis (prior odds) multiplied by the odds that the test will be positive if the disease is present. The multiplier of the prior odds is called the likelihood ratio. For diagnostic tests, we have seen above that this can be calculated by:
sensitivity 1 — specificity'
Bayes's theorem has numerous applications in clinical and epidemiological research. Among other uses, it has been applied to causality assessment in individual cases. The principle consists of fixing, before analysis, the prior odds that the drug of interest is responsible for the adverse event observed in a given patient. These odds are 1 if there is no reason to favour the hypothesis of the responsibility or non-responsibility of the drug for the adverse event. In the most favourable cases, prior odds can be estimated from the relative risk (RR) or odds ratio quantified by a previous epidemiological study (cf. aetiologic fraction of the risk in the exposed, EFRe):
The prior odds are then multiplied by one or more likelihood ratio(s) corresponding to available information, signs or criteria relevant to the causality analysis. Each likelihood ratio is calculated by dividing the probability of the criterion being present, if the drug is responsible for the event, by the probability of the criterion being present if the drug is not responsible.
Example: If 67% of patients presenting a given adverse effect are female when the proportion of females among all users of a drug of interest is 42%, the likelihood ratio for gender will be 0.67/0.42 = 1.6 if the subject is female and 0.33/0.58 = 0.57 if the subject is male.
The multiplication of prior odds by all the likelihood ratios
Prior odds =
relevant for the causality analysis gives the final or posterior odds that the drug is responsible for the adverse event of interest. These odds can easily be transformed into a posterior probability:
For example, an odds of 5.6 corresponds to a probability of 5.6/(5.6 + 1) = 0.85.
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