statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, when selecting a

statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repre ...

for given data, the relative likelihood compares the relative plausibilities of different candidate models or of different values of a

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

of a single model.

Relative likelihood of parameter values

Assume that we are given some data for which we have a statistical model with parameter . Suppose that the

maximum likelihood estimate In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...

for is

\hat

. Relative plausibilities of other values may be found by comparing the likelihoods of those other values with the likelihood of

\hat

. The ''relative likelihood'' of is defined to be :

\frac,

where

\mathcal(\theta \mid x)

denotes the

likelihood function A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...

. Thus, the relative likelihood is the likelihood ratio with fixed denominator

\mathcal(\hat \mid x)

. The function :

\theta \mapsto \frac

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\.

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 In statistics, interval estimation is the use of sample data to estimate an '' interval'' of possible values of a parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interval es ...

within likelihood-based statistics ("likelihoodist" statistics): They are similar to confidence intervals in frequentist statistics and

credible interval In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter value has a particular probability \gamma to fall within it. For example, in an experime ...

s in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of

coverage probability In statistical estimation theory, the coverage probability, or coverage for short, is the probability that a confidence interval or confidence region will include the true value (parameter) of interest. It can be defined as the proportion of i ...

(frequentism) or

posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posteri ...

(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 In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...

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

The definition of relative likelihood can be generalized to compare different

s. This generalization is based on AIC (Akaike information criterion), or sometimes

AICc The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to e ...

(Akaike Information Criterion with correction). Suppose that for some given data we have two statistical models, and . Also suppose that . Then the ''relative likelihood'' of with respect to is defined as follows. ::

\exp \left( \frac \right)

To see that this is a generalization of the earlier definition, suppose that we have some model with a (possibly multivariate) parameter . Then for any , set , and also set

\hat\theta

. The general definition now gives the same result as the earlier definition.

Notes

{{reflist Likelihood Statistical models

Relative likelihood of parameter values

Likelihood region

Relative likelihood of models

See also

Notes