Likelihood-ratio Test - Use

Use

Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by D) is twice the difference in these log-likelihoods:

The 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 is determined by deriving the probability or p-value of the difference D. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to df2 − df1 . Symbols df1 and df2 represent the number of free parameters of models 1 and 2, the null model and the alternative model, 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 constraints on the parameters.

For example: if the null model has 1 free parameter and a log-likelihood of −8024 and the alternative model has 3 degrees of freedom and a LL of −8012, then the probability of this difference is that of chi-squared value of +2·(8024 − 8012) = 24 with 3 − 1 = 2 degrees of freedom. Certain assumptions must be met for the statistic to follow a chi-squared distribution and often empirical p-values are computed.