Concept:
In statistics, a likelihood
ratio test is a statistical test used to compare the fit of two models, one of which (the null model)
is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data
are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a p-value, or compared to a critical value to decide whether to reject the null model in favour of the
alternative model.
In statistics, a likelihood
ratio test is used to compare the fit of
two models, one of which is nested within the other. This often occurs when
testing whether a simplifying assumption for a model is valid, as when two or
more model parameters are assumed to be related.
Both models are fitted to the data and their log-likelihood recorded. The test
statistic (usually denoted D) is twice
the difference in these log-likelihoods:
model with more parameters will always fit at
least as well (have a greater log-likelihood). Whether it fits significantly
better and should thus be preferred can be determined by deriving the
probability or p-value of the obtained
difference D. In many cases, the probability distribution of the test statistic can be approximated by a chi-square distribution with (df1 − df2) degrees of freedom, where df1 and df2 are
the degrees of freedom of models 1 and 2 respectively.
The test requires nested models, that is, models in which the more
complex one can be transformed into the simpler model by imposing a set of
linear constraints on the parameters.
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