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Performance Metrics: Positive Likelihood Ratio

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What is the Positive Likelihood Ratio?

The Positive Likelihood Ratio (LR+, +LR, likelihood ratio positive or likelihood ratio for positive results) gives the change in odds of the true value being positive when the predicted value is positive. This is expressed as a ratio. An LR+ of 5 indicates a 5-fold increase in the odds of the true value being the same as the predicted positive value. The larger this value, the more informative this ratio is.

The LR+ is calculated as follows:

\text{LR}+ = \frac{\text{sensitivity}}{1 - \text{specificity}}

Nevertheless, this is a likelihood ratio. Within this context, using the conditional probabilities in the numerator and denominator makes more sense:

 \text{LR}+ = \frac{\Pr({T+}\mid D+)}{\Pr({T+}\mid D-)}

An LR+ of 1 means that the model is completely useless. In this case, the odds of a positive true value before and after knowing the predicted value haven’t changed — i.e. the change in odds is 1.

Finally, it is worth noting that the LR+ lacks interpretability. Within medicine, they are one of the best measures of diagnostic accuracy. However, they are seldom used because it can be hard for laymen to convert the odds to probabilities.

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