Likelihood Ratio

Likelihood Ratio: Concept Architecture

Compare how well two explanations predict the same evidence

A likelihood ratio compares the likelihood of the same observed result under two specified conditions or models. In diagnostic testing, it compares the probability of a test result among people with a condition with that among people without it, allowing prior odds to be updated. In statistical inference, a likelihood ratio compares how well two parameter values or fitted models account for observed data. The underlying idea is shared, but the numerator, denominator, and interpretation must be named every time.

A likelihood is viewed as a function of a hypothesis or parameter for fixed observed data. It is not, by itself, the probability that the hypothesis is true. A ratio of 8 means the specified result is eight times as likely under the numerator condition as under the denominator condition; it does not mean there is an 80% chance the numerator hypothesis is correct. Clinical decisions also require baseline risk, consequences, and uncertainty.

Diagnostic likelihood ratios

For a binary test, sensitivity is $P(+\mid D)$ and specificity is $P(-\mid\overline{D})$, where $D$ denotes the target condition. The positive and negative likelihood ratios are:

$$\mathrm{LR}_{+}=\frac{P(+\mid D)}{P(+\mid\overline{D})} =\frac{\mathrm{Se}}{1-\mathrm{Sp}},\qquad \mathrm{LR}_{-}=\frac{P(-\mid D)}{P(-\mid\overline{D})} =\frac{1-\mathrm{Se}}{\mathrm{Sp}}.$$

A useful positive test has $\mathrm{LR}_{+}>1$; a useful negative test has $\mathrm{LR}_{-}<1$. A value of 1 provides no discrimination for that particular result. These are conditional on disease status, and so they are mathematically distinct from positive or negative predictive value, which asks for disease probability given the result. The likelihood ratios need an accurate reference standard and a threshold defining positive and negative test results.

Given a relevant pretest probability $p$, convert to odds $p/(1-p)$, multiply by the ratio for the observed result, and convert back to probability:

$$O_{\mathrm{post}}=O_{\mathrm{pre}}\times\mathrm{LR},\qquad P_{\mathrm{post}}=\frac{O_{\mathrm{post}}}{1+O_{\mathrm{post}}}.$$

The pretest probability should reflect the actual clinical population and information available before the test. A likelihood ratio estimated in a highly selected study may not transport unchanged to primary care: disease severity, alternative diagnoses, test administration, and verification practices can alter sensitivity and specificity. Predictive values change with prevalence even if sensitivity and specificity happened to remain constant.

A diagnostic example with a visible 2-by-2 table

Suppose a hypothetical cohort has 1,000 people, 100 with the condition and 900 without it. A test has sensitivity 80% and specificity 90% in this illustrative population. That yields 80 true positives, 20 false negatives, 90 false positives, and 810 true negatives.

The positive ratio is $0.80/(1-0.90)=8$ and the negative ratio is $(1-0.80)/0.90=2/9\approx0.222$. Before testing, odds of disease are $100/900=1/9$. After a positive result, odds become $(1/9)\times8=8/9$, giving disease probability $8/17\approx47.06\%$—the same as $80/170$ in the table. After a negative result, odds become $(1/9)\times(2/9)=2/81$, giving disease probability $2/83\approx2.41\%$—the same as $20/830$.

The positive LR of 8 is therefore not a post-test probability of 80% or 8%. The probability after a positive result is below one half here because the starting prevalence is only 10% and false positives among the larger disease-free group matter. The 2-by-2 table also makes the distinction between a negative result's disease probability, $20/830$, and the negative predictive value, $810/830$, explicit. All values are illustrative point estimates without confidence intervals.

A result can have more than two levels

For a continuous or ordinal test, a threshold throws away some information. A result-specific likelihood ratio compares densities or probabilities at an observed result or within a defined result interval: $\mathrm{LR}(r)=f(r\mid D)/f(r\mid\overline{D})$ where the densities are comparable. Different ranges can have different ratios; a result just above a positive threshold may be less informative than a very high value. State how ranges were chosen, estimate uncertainty, and avoid assuming a published binary LR applies identically to every positive value.

Combining two results by multiplying their individual likelihood ratios requires conditional independence given both disease and non-disease, or a justified joint model. Two correlated biomarkers or a score that already includes one test can otherwise count the same evidence twice. An indeterminate result needs its own management and cannot be quietly assigned to whichever group improves reported accuracy.

Likelihood ratios for statistical models

For observed data $y$ and two fully specified parameter values $\theta_1$ and $\theta_0$, a simple likelihood ratio is $L(\theta_1;y)/L(\theta_0;y)$. For example, suppose eight successes are observed in ten independent Bernoulli trials. Comparing a fixed success probability of 0.8 with a fixed probability of 0.5 gives:

$$\frac{L(0.8;y)}{L(0.5;y)} =\frac{0.8^8\,0.2^2}{0.5^{10}}\approx6.872.$$

The same binomial coefficient would appear in numerator and denominator and cancels. The observed sequence or count is about 6.87 times as likely under the specified 0.8 model. That ratio alone does not give a probability that the success probability equals 0.8, nor is it a frequentist p-value. It depends on both particular models and the observed data.

For nested fitted models, the conventional likelihood-ratio test compares the maximum likelihood under a restricted model with that under the larger model:

$$\Lambda=\frac{\sup_{\theta\in\Theta_0}L(\theta;y)} {\sup_{\theta\in\Theta_1}L(\theta;y)},\qquad G^2=-2\log\Lambda=2(\ell_1-\ell_0),\qquad \Theta_0\subseteq\Theta_1.$$

Here $\ell_0$ and $\ell_1$ are maximized log-likelihoods on the same observed data under compatible likelihood definitions. The ratio $\Lambda$ is at most 1 when the models are truly nested and optimized correctly; larger $G^2$ indicates more improvement from relaxing the restriction. Under standard large-sample regularity conditions, $G^2$ has an approximate chi-square null distribution with degrees of freedom equal to the difference in freely estimated parameters. Boundary parameters, unidentified mixtures, small samples, different observations or likelihood bases, and post-selection testing can invalidate that simple reference distribution. Verify the comparison and use an appropriate alternative calibration when its assumptions fail.

A likelihood-ratio test asks about evidence for a particular model restriction. It is not a Bayes factor: the latter compares marginal likelihoods integrated over parameter priors. Nor does rejection prove that a more complex model predicts well or represents a causal mechanism. Inspect fit, assumptions, clinical relevance, and out-of-sample performance in addition to a model-comparison statistic.

Use in health economic decisions

Diagnostic likelihood ratios can update disease probabilities at a test-and-treat decision node. Those post-test probabilities influence expected true and false classifications, follow-up tests, treatment harms and benefits, resource use, costs, and QALYs. Carry uncertainty in test sensitivity, specificity, prevalence, and downstream outcomes rather than treating a single LR as an entire cost-effectiveness analysis. A test can be informative yet have little decision value if it rarely changes management, while a modest LR may matter near a treatment threshold.

Likelihood-ratio tests can compare nested statistical models used to estimate treatment effects, costs, survival, or calibration parameters. A statistically significant extra parameter does not ensure a better economic decision model; assess effect size, predictive validation, structural assumptions, and the uncertainty that propagates to incremental results. State explicitly whether “likelihood ratio” refers to a diagnostic result, a fixed-model evidence ratio, or a test statistic.

Report disease and reference-standard definitions, study population and setting, threshold or test range, sensitivity and specificity with uncertainty, the relevant pretest probability, and posterior odds or probabilities for diagnostic use. For model testing, report the models and parameters, identical data and likelihood basis, maximized log-likelihoods, $G^2$, degrees of freedom, calibration method, and substantive implication. The ratio is useful only when its two sides and its decision context are visible.

Sources and further reading

• Cochrane Clinical Answers: Diagnostic test accuracy reviews. Primary evidence-synthesis explanation of positive and negative likelihood ratios.
• Usher-Smith JA, Sharp SJ, Griffin SJ. The spectrum effect in tests for risk prediction, screening, and diagnosis. BMJ (2016). Explains why test accuracy can differ across patient populations and settings.
• R stats documentation: Analysis of deviance for generalized linear models. Primary software documentation for likelihood-ratio comparisons of nested models.
• statsmodels documentation: Likelihood-ratio comparison of regression models. Primary software documentation on the nested-model requirement.