## What is the Negative Likelihood Ratio?

The **Negative** **Likelihood Ratio** (LR-, -LR, likelihood ratio negative or likelihood ratio for negative results) gives the change in odds of the true value being positive when the predicted value is negative. This is expressed as a ratio. It is analogous to the Positive Likelihood Ratio. An LR- of 6 indicates a 6-fold increase in the odds of the true value being positive, when the predicted value is negative. The larger the ratio, the more informative it is. Furthermore, the LR- is independent of the prevalence. I.e. this performance metric is resistant to class imbalance.

## Calculating the Negative Likelihood Ratio

The LR- is calculated as follows:

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

However, this is a likelihood ratio. Given this, 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 negative true value before and after knowing the predicted value haven’t changed — i.e. the change in odds is 1.

Further reading:

- With likelihood ratios, Bayes’ Theorem is never far away.
- Risks, Odds & ROC Curves