How to perform a likelihood ratio test?

To perform a likelihood ratio test, calculate the likelihoods of two competing hypotheses and compare them using a test statistic.

A likelihood ratio test is used to compare two competing hypotheses, where one is a null hypothesis and the other is an alternative hypothesis. The null hypothesis is usually the simpler of the two, and the alternative hypothesis is more complex. The test involves calculating the likelihoods of both hypotheses and comparing them using a test statistic.

To perform the test, first calculate the likelihood of the null hypothesis, denoted by L(θ0), and the likelihood of the alternative hypothesis, denoted by L(θ1). The test statistic is then calculated as the ratio of these two likelihoods:

λ = L(θ1) / L(θ0)

Under the null hypothesis, the test statistic follows a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters between the two hypotheses. The p-value can then be calculated using this distribution, and if it is less than the chosen significance level, the null hypothesis is rejected in favour of the alternative hypothesis.

For example, suppose we have a sample of data and we want to test whether it comes from a normal distribution with mean μ and variance σ^2, or from a normal distribution with mean μ and variance 2σ^2. The null hypothesis is that the variance is σ^2, and the alternative hypothesis is that the variance is 2σ^2. We calculate the likelihoods of both hypotheses using the sample data, and find that L(θ0) = f(x|μ,σ^2) and L(θ1) = f(x|μ,2σ^2). We then calculate the test statistic λ = L(θ1) / L(θ0), and compare it to the chi-squared distribution with one degree of freedom. If the p-value is less than the chosen significance level, we reject the null hypothesis in favour of the alternative hypothesis.