Post-Test Odds (Odds × Likelihood Ratio)

This is Bayes' theorem with the arithmetic taken out of it. In probabilities, updating on evidence means computing a denominator that sums over every hypothesis. In odds, that denominator cancels, and the entire update is one multiplication: pre-test odds × likelihood ratio = post-test odds. It is the same theorem, and it is the reason odds survive in clinical epidemiology long after they have been abandoned elsewhere.

Three conversions do all the bookkeeping. A probability becomes odds by $O = p/(1-p)$; odds become a probability by $p = O/(1+O)$; and prevalence is the pre-test probability when you have nothing else to go on. Work the clinic table through: prevalence 20 % gives pre-test odds of $0.20/0.80 = 0.25$. A positive result on a test with LR+ of 18 gives post-test odds of $0.25 \times 18 = 4.5$. Convert back: $4.5/5.5 = 0.818$. That is the positive predictive value the same table gives directly, reached without building the table at all.

The practical value is that it chains. A history that raises the odds, then an examination finding, then a test, then a second test — each is one multiplication, applied in any order, and the order does not change the answer. This is what the Fagan nomogram draws with a straight edge, and what makes likelihood ratios teachable at the bedside in a way that a table of predictive values is not.

The catch, and it is a real one, is that chaining assumes the pieces of evidence are conditionally independent given the disease state. Clinical findings rarely are. Two tests that measure related biology — two inflammatory markers, an examination sign and the symptom that produced it — carry overlapping information, and multiplying their ratios double-counts it and overstates the final confidence. The multiplication is exact; the independence is an assumption, and it is the assumption that fails.