Assume that we are given some data for which we have a statistical model with parameter . Suppose that the
maximum likelihood estimate for is \hat{\theta}. Relative plausibilities of other values may be found by comparing the likelihoods of those other values with the likelihood of \hat{\theta}. The
relative likelihood of is defined to be \frac{~\mathcal{L}(\theta \mid x)~}{~\mathcal{L}(\hat{\theta} \mid x)~}, where \mathcal{L}(\theta \mid x) denotes the
likelihood function. Thus, the relative likelihood is the
likelihood ratio with fixed denominator \mathcal{L}(\hat{\theta} \mid x). The function \theta \mapsto \frac{~\mathcal{L}(\theta \mid x)~}{~\mathcal{L}(\hat{\theta} \mid x)~} is the
relative likelihood function.
Likelihood region A
likelihood region is the set of all values of whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a
% likelihood region for is defined to be. \left\{\theta : \frac{\mathcal{L}(\theta \mid x)}{\mathcal{L}(\hat{\theta\,} \mid x)} \ge \frac p {100} \right\}. If is a single real parameter, a % likelihood region will usually comprise an
interval of real values. If the region does comprise an interval, then it is called a
likelihood interval. Likelihood intervals, and more generally likelihood regions, are used for
interval estimation within likelihood-based statistics ("likelihoodist" statistics): They are similar to
confidence intervals in frequentist statistics and
credible intervals in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of
coverage probability (frequentism) or
posterior probability (Bayesianism). Given a model, likelihood intervals can be compared to confidence intervals. If is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for will be the same as a 95% confidence interval (19/20 coverage probability). In a slightly different formulation suited to the use of log-likelihoods (see
Wilks' theorem), the
test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a
chi-squared distribution with degrees-of-freedom (df) equal to the difference in df-s between the two models (therefore, the −2 likelihood interval is the same as the 0.954 confidence interval; assuming difference in df-s to be 1). ==Relative likelihood of models==