Likelihood ratio interpretation of the relative risk

Suhail A R Doi et al.

Interpreting diagnostic test results in medicine The likelihood ratio (LR) is today commonly used in medicine for diagnostic inference. Historically, it was preceded by introduction, in 1966, of the predictive value of a diagnostic test in Medicine1 and within a decade of the latter, it was realised that the true-positive to false-positive ratio2 3 also then called the likelihood value4 was the main driver of the change from prior probabilities to posterior predictive values. The latter were also called posttest likelihoods and this ratio became known as the LR in Medicine. The change from prior probabilities to posterior predictive values was formulated using Bayes’ theorem5 and represented a more versatile approach to predictive values. The reason this is considered more versatile is that Bayes’ theorem allows a physician to compute the predictive value (probability) of a diagnosis conditional on a specific test result. For example, if we denote test status as +ve (positive) and –ve (negative) and the gold standard (eg, underlying diagnosis) as D (diagnosed) and nD (not diagnosed), respectively, then from Bayes’ theorem,5 the posterior (after the test result) probability (expressed in odds form) of the diagnosis can be derived from test sensitivity and specificity as follows: Pr( D|+ve) 1−Pr( D|+ve) = Pr( +ve|D ) Pr( +ve|nD) × Pr( D ) 1−Pr( D )  In the preceding expression, Pr ( D| + ve) is the predictive value and ‘Pr’ is probability, | is ‘conditional on’ and thus read as probability of D (diagnosis) conditional on test result = +ve). The first and last terms in  are odds and the middle term is the LR and, thus posttest odds+ve = LR+ve × pretest odds  The expression for the likelihood ratio (LR+ve) within expressions  and  is LR+ve = Pr( +ve|D ) Pr( +ve|nD) = Pr( true positive) Pr( false positive) = sensitivity 1−specificity ≡ posttest odds+ve pretest odds  with the ratio being the ratio of sensitivity (numerator) to 1—specificity but the ratio equivalent to a ratio of two odds from Bayes’ theorem as indicated in expression .

Key messages What is already known on this topic ⇒ The risk ratio (relative risk) is a ratio of two risks that is interpreted as connecting the intervention conditional risks in a clinical trial. What this study adds ⇒ It is demonstrated that the conventional interpretation of the risk ratio is in conflict with Bayes’ theorem. ⇒ The interpretation of the risk ratio as a likelihood ratio connecting prior (unconditional) intervention risk to outcome conditional intervention risk is required to avoid conflict with Bayes’ theorem. How this study might affect research, practice or policy ⇒ The interpretation of the risk ratio as an effect measure in a clinical trial is naïve and better replaced by its interpretation as a likelihood ratio. ⇒ The ratio of the complementary risk ratio’s (or likelihood ratio’s) is what should actually be interpreted as an effect measure connecting the intervention conditional risks in a clinical trial.

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