Positive Likelihood Ratio (LR+)

The positive likelihood ratio asks how much more often a positive result turns up in people who have the condition than in people who do not. It is $\mathrm{Se}/(1-\mathrm{Sp})$: the true-positive rate over the false-positive rate. A ratio of 18 means a positive result is eighteen times as likely to have come from someone with the condition as from someone without.

This one has no originator, and it is worth saying so plainly rather than inventing an attribution. The likelihood ratio is Bayes' theorem restated on a 2×2 table — the ratio of two conditional probabilities that the theorem already contains — and it entered clinical epidemiology through teaching materials and evidence-based-medicine curricula in the 1970s and 1980s rather than through any founding paper. AHRQ's methods guidance for systematic reviews of diagnostic tests describes it in exactly those terms, as a derived quantity rather than a discovery. Where a textbook attributes it to a named author, that author was a teacher of it, not its source.

Its virtue over sensitivity and specificity is that it is a single number that does arithmetic. Sensitivity and specificity have to be carried around as a pair and combined by hand with prevalence; a likelihood ratio multiplies straight into the pre-test odds and gives the post-test odds. The rough calibration usually taught: above 10 is a large and often conclusive shift, 5 to 10 is moderate, 2 to 5 is small, and below 2 is barely worth the phlebotomy.

Two cautions. The first is that a likelihood ratio, unlike a predictive value, is stable across populations — but only as far as sensitivity and specificity are, and those drift with spectrum. A test validated on advanced disease performs worse on the early disease a screening programme finds, because the early cases are the harder ones. The second is that dichotomising a continuous result into "positive" and "negative" throws away information: a result far past the threshold carries a much larger likelihood ratio than one just over it, and stratum-specific ratios keep that difference instead of discarding it.