Likelihood ratio

A likelihood ratio is the probability of observed evidence under one hypothesis divided by its probability under another. It quantifies how strongly the evidence favors one hypothesis over the other.

Likelihood function: A function that gives the probability of observed data under each possible parameter value or hypothesis. Each hypothesis supplies a likelihood in the ratio’s numerator or denominator.
Likelihood-ratio test: A statistical test that compares maximized likelihoods under nested hypotheses. This test uses a likelihood ratio to assess whether a more flexible model improves fit.
Conditional probability: The probability of an event given that another event has occurred. Each likelihood is the probability of the evidence conditional on a hypothesis.
P-value: The probability, under a specified null hypothesis, of a test statistic at least as extreme as the observed one. A p-value measures tail-area extremity, not the relative probability of evidence under two hypotheses.
Bayes’ theorem: A rule for updating probabilities using evidence and conditional probabilities. Bayes’ theorem turns prior odds and a likelihood ratio into posterior odds.
Diagnostic likelihood ratio: A measure comparing the probability of a test result in people with a condition to its probability in people without it. In diagnosis, positive and negative test results have distinct likelihood ratios for updating disease odds.
Statistical hypothesis: A claim about a probability distribution or parameter that can be assessed using data. The two hypotheses specify the competing conditions under which evidence is evaluated.
Maximum likelihood estimation: A method that estimates model parameters by maximizing the probability of observed data. Estimation finds a best-fitting parameter; a likelihood ratio compares hypotheses or models.
Bayes factor: The ratio of the marginal likelihoods of observed data under two statistical models. It is the model-comparison counterpart of a likelihood ratio, averaging over parameter uncertainty.
Sequential probability ratio test: A sequential hypothesis test that accumulates evidence until a likelihood-ratio boundary is crossed. It applies the ratio repeatedly as data arrive, allowing early decisions.

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