🔬 Likelihood Ratio (LR)

Likelihood Ratio (LR)#

The ratio of data likelihoods under two hypotheses; central to many sequential and diagnostic tests.

Important

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What it is#

The likelihood ratio (LR) measures how much more likely the observed data is under one hypothesis than another, by dividing their likelihoods:

\[\Lambda = \frac{L(\text{data} \mid H_1)}{L(\text{data} \mid H_0)}.\]

\(\Lambda = 1\) means the data is equally likely under both; \(\Lambda > 1\) favours \(H_1\); \(\Lambda < 1\) favours \(H_0\).

Two roles in testing#

Likelihood-ratio test (LRT) — the generalised statistic \(\Lambda = \sup_{\theta \in \Theta_0} L(\theta) / \sup_{\theta \in \Theta} L(\theta)\) compares the best fit under the null to the best fit overall; a small \(\Lambda\) is strong evidence against \(H_0\). (By the Neyman–Pearson lemma, the LR is the most powerful test for simple hypotheses.)
Sequential probability ratio test (SPRT) — accumulate the ratio as data arrives, \(\Lambda_n = L(\text{data}_{1:n} \mid H_1)/L(\text{data}_{1:n} \mid H_0)\), and stop when it crosses an upper bound \(A\) (accept \(H_1\)) or lower bound \(B\) (accept \(H_0\)), else continue.

Worked example#

Seven heads in 10 tosses, with \(H_0: p = 0.5\) vs \(H_1: p = 0.7\):

\[L(H_0) = \binom{10}{7}(0.5)^{10} \approx 0.117, \quad L(H_1) = \binom{10}{7}(0.7)^7(0.3)^3 \approx 0.266, \quad \Lambda = \frac{0.266}{0.117} \approx 2.27,\]

so the data is about 2.3 times more likely under the biased hypothesis. Rough reading: \(\Lambda \approx 1\) no evidence, \(> 3\) moderate, \(> 10\) strong.

Where it shows up#

The LR is the backbone of likelihood-based inference: the LRT and SPRT, nested model comparison, and — closely related — the Bayes factor, which is a likelihood ratio when the hypotheses are simple (and an evidence ratio of marginal likelihoods when they are composite). In medicine, diagnostic LR+ and LR− update the odds of disease from a test result.

Theme: Model Evaluation & Uncertainty · All terminology

Hint

Mind map — connected ideas

Sequential Probability Ratio Test (SPRT) · Bayesian Sequential Testing · Marginal Likelihood (also called The Model Evidence or Integrated Likelihood) · Binomial Likelihood · Frequentist · Posterior Probability

Hint

More in Model Evaluation & Uncertainty

Average Absolute Error (AAE) · Baseline Heuristics · Bootstrap · Bootstrap Confidence Intervals (CIs) · Coverage · Cramér’s V · DeLong’s Test · KS Statistic (Kolmogorov–Smirnov Statistic) · Mann–Whitney U Test (also called the Wilcoxon rank-sum test) · MASE (Mean Absolute Scaled Error) · Mean Absolute Error (MAE) · Mean Absolute Percentage Error (MAPE) · Mean Squared Error (MSE) · Relative accuracy