How do prior odds and a likelihood ratio combine to interpret a positive workplace drug test?

## Odds, and why they are the right currency Probability `p` and odds `p/(1-p)` carry the same information, but odds turn Bayes' theorem into a multiplication. Write it for a hypothesis H (this employee uses the drug) and evidence E (the test came back positive): ``` P(H|E) / P(not H|E) = [ P(H) / P(not H) ] x [ P(E|H) / P(E|not H) ] posterior odds = prior odds x likelihood ratio ``` The awkward normalising denominator in the usual statement of Bayes' rule cancels, because it is identical in the numerator and denominator of the odds. Conversions you should be able to do instantly: odds `a:b` means probability `a/(a+b)`; probability `p` means odds `p:(1-p)`. So 1:49 is 2%, and 19:49 is 19/68 = 27.9%. ## The worked example Assume 2% of the workforce actually uses the substance, and the assay has 95% sensitivity and 95% specificity. - **Prior odds**: `0.02 : 0.98 = 1 : 49`. Forty-nine clean employees for every user. - **Positive likelihood ratio**: `LR+ = P(+ | user) / P(+ | non-user) = sensitivity / (1 - specificity) = 0.95 / 0.05 = 19`. A positive is 19 times more likely to come from a user than from a non-user. - **Posterior odds**: `1:49 x 19 = 19:49`. - **Posterior probability**: `19 / (19 + 49) = 0.279`, about 28%. A nineteen-fold multiplication of the evidence still leaves the accused employee more likely innocent than guilty, because 19 is not enough to overcome 49-to-1 starting odds. That is the base-rate effect stated in a single arithmetic line. ## The negative likelihood ratio The symmetric quantity for a negative result is `LR- = P(- | user) / P(- | non-user) = (1 - sensitivity) / specificity = 0.05 / 0.95 = 1/19`. Applying it: `1:49 x 1/19 = 1:931`, about 0.1%. A negative on this test is far more decisive than a positive, for the same reason the positive is weak -- the pool of non-users is huge. Some useful landmarks for likelihood ratios: - `LR = 1` means the result is worthless; it leaves the odds exactly where it found them. - `LR` around 2 to 5 is a nudge; 10 to 20 is moderately strong; above 100 is decisive for anything but an extremely low prior. - `LR` below 1 is evidence **against** the hypothesis. Notice that the likelihood ratio is a property of the test alone. It travels between populations. The prior odds are a property of who you are testing. Reporting the two separately is more honest than quoting a single posterior, because a reader in a different setting can substitute their own prior. ## Chaining evidence Because the update is multiplicative, independent pieces of evidence chain: ``` posterior odds = prior odds x LR_1 x LR_2 x ... ``` A confirmatory second test with `LR = 19` would take the odds from 19:49 to 361:49, about 88%. This is the standard two-stage design: a cheap sensitive screen followed by a different, highly specific confirmation. The critical caveat is the word *independent*. The multiplication is valid only if the two results are **conditionally independent given the person's true status**. Repeating the same assay on the same sample usually violates this badly: whatever cross-reacting substance or handling error produced the first false positive is still there and will very likely produce a second one. A genuine confirmation has to use a different mechanism, or you are multiplying by a likelihood ratio the second test does not really have. ## Common errors - Multiplying probabilities instead of odds. `0.02 x 19 = 0.38` is not the posterior; the odds form is only valid on odds. - Using `1 - specificity` in the numerator, or otherwise inverting the ratio. The numerator is always the probability of the observed result under the hypothesis. - Quoting the likelihood ratio as though it were the answer. An LR of 19 is not a 19-to-1 chance of guilt; it multiplies whatever the prior odds were. - Forgetting that a policy that acts on a single positive is acting mostly on innocent people when the base rate is low. Twenty-eight percent is a reason to confirm, never a reason to conclude. ## How to answer in the room Write the three-term identity, convert the prevalence to odds out loud, compute the likelihood ratio from sensitivity and specificity, multiply, convert back. Then say the sentence that shows you understand the result rather than just the arithmetic: the test is strong evidence, and the person is still probably clean, because strong evidence times long odds is short of certainty.