as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected. An example The following example is adapted and abridged from . Suppose that we have a random sample, of size , from a population that is normally-distributed. Both the mean, , and the standard deviation, , of the population are unknown. We want to test whether the mean is equal to a given value, . Thus, our null hypothesis is and our alternative hypothesis is . The likelihood function is With some calculation (omitted here), it can then be shown that where is the -statistic with degrees of freedom. Hence we may use the known exact distribution of to draw inferences. Asymptotic distribution: Wilks’ theorem If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine. Assuming is true, there is a fundamental result by Samuel S. Wilks: As the sample size approaches , the test statistic defined above will be asymptotically chi-squared distributed () with degrees of freedom equal to the difference in dimensionality of and . This implies that for a great variety of hypotheses, we can calculate the likelihood ratio for the data and then compare the observed to the value corresponding to a desired statistical significance as an approximate statistical test. Other extensions exist. See also Akaike information criterion Bayes factor Johansen test Model selection Vuong's closeness test Sup-LR test Error exponents in hypothesis testing References Further reading External links Practical application of likelihood ratio test described R Package: Wald's Sequential Probability Ratio Test Richard Lowry's Predictive Values and Likelihood Ratios Online Clinical Calculator Statistical ratios Statistical tests