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The likelihood ratio statistic (LR statistic) is a statistical measure used in hypothesis testing and model selection. It is based on the likelihood ratio, which assesses the relative fit of two competing statistical models.
To calculate the LR statistic, two models are compared: the null model and the alternative or extended model. The null model represents the null hypothesis, while the alternative model represents the alternative hypothesis. The Null Model is usually a simplified model that assumes that certain parameters or relationships do not exist. The Alternative Model includes additional parameters or provides an alternative representation of the data.
The first step in calculating LR statistics is to maximize the likelihood function for each model. The likelihood function measures the probability that the observed data will occur under the given model assumptions. The maximum likelihood is achieved by choosing the parameter values that provide the greatest probability for the observed data.
The LR statistic is then calculated by taking the logarithm of the quotient of the maximum likelihoods of the two models. In formal terms:
LR statistic = 2 * (log-likelihood of the alternative model - log-likelihood of the null model)
The LR statistic usually follows a chi-square distribution if the sample size is large enough and certain assumptions are met. The LR statistic can be used to perform hypothesis testing by setting critical thresholds for the LR statistic. If the calculated LR statistic exceeds the critical threshold, the null hypothesis can be rejected, and there is evidence that the Alternative Model provides a better fit to the data.
The LR statistic is also used in model selection to decide between different competing models. In this case, the model with the larger LR statistic is considered the better model because it provides a better fit to the data.
It is important to note that the use of the LR statistic depends on certain assumptions and preconditions, particularly the validity of the asymptotic distribution properties. In addition, the LR statistic should not be considered in isolation, but rather in conjunction with other information and considerations when interpreting the results.