Likelihood-ratio test

The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.

Definition

General

Suppose that we have a statistical model with parameter space ${\displaystyle \mathrm{\Theta }}$. A null hypothesis is often stated by saying that the parameter ${\displaystyle \theta }$ lies in a specified subset ${\displaystyle {\mathrm{\Theta }}_0}$ of ${\displaystyle \mathrm{\Theta }}$. The alternative hypothesis is thus that ${\displaystyle \theta }$ lies in the complement of ${\displaystyle {\mathrm{\Theta }}_0}$, i.e. in ${\displaystyle \mathrm{\Theta }\mathrm{\setminus }{\mathrm{\Theta }}_0}$, which is denoted by ${\displaystyle {\mathrm{\Theta }}_0^{\text{c}}}$. The likelihood ratio test statistic for the null hypothesis ${\displaystyle H_0:\theta \in {\mathrm{\Theta }}_0}$ is given by:

$${\displaystyle {\lambda }_{\text{LR}}=-2\ln \left[\frac{\underset{\theta \in {\mathrm{\Theta }}_0}{sup}\mathcal{L}(\theta )}{\underset{\theta \in \mathrm{\Theta }}{sup}\mathcal{L}(\theta )}\right]}$$

where the quantity inside the brackets is called the likelihood ratio. Here, the ${\displaystyle sup}$ notation refers to the supremum. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one and the likelihood ratio test statistic between 0 and infinity.

Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods $${\displaystyle {\lambda }_{\text{LR}}=-2\left[\ell ({\theta }_0)-\ell (\hat{\theta })\right]}$$ where $${\displaystyle \ell (\hat{\theta })\equiv \ln \left[\underset{\theta \in \mathrm{\Theta }}{sup}\mathcal{L}(\theta )\right]}$$ is the logarithm of the maximized likelihood function ${\displaystyle \mathcal{L}}$, and ${\displaystyle \ell ({\theta }_0)}$ is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes ${\displaystyle \mathcal{L}}$ for the sampled data) and $${\displaystyle {\theta }_0\in {\mathrm{\Theta }}_0\text{ and }\hat{\theta }\in \mathrm{\Theta }}$$ denote the respective arguments of the maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) ${\displaystyle {\lambda }_{\text{LR}}}$ converges asymptotically to being χ²-distributed if the null hypothesis happens to be true. The finite-sample distributions of likelihood-ratio statistics are generally unknown.

The likelihood-ratio test requires that the models be nested – i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below.

If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood.

Case of simple hypotheses

A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter ${\displaystyle \theta }$:

$${\displaystyle \begin{array}{ccc}{H}_0& :& \theta ={\theta }_0,\\ H_1& :& \theta ={\theta }_1.\end{array}}$$ In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:

$${\displaystyle \mathrm{\Lambda }(x)=\frac{\mathcal{L}({\theta }_0\mid x)}{\mathcal{L}({\theta }_1\mid x)}.}$$

Some older references may use the reciprocal of the function above as the definition. Thus, the likelihood ratio is small if the alternative model is better than the null model.

The likelihood-ratio test provides the decision rule as follows:

• If ${\displaystyle \mathrm{\Lambda }>c}$, do not reject ${\displaystyle H_0}$;
• If ${\displaystyle \mathrm{\Lambda }<c}$, reject ${\displaystyle H_0}$;
• If ${\displaystyle \mathrm{\Lambda }=c}$, reject ${\displaystyle H_0}$ with probability ${\displaystyle q}$.

The values ${\displaystyle c}$ and ${\displaystyle q}$ are usually chosen to obtain a specified significance level ${\displaystyle \alpha }$, via the relation $${\displaystyle qPr(\mathrm{\Lambda }=c\mid H_0)+Pr(\mathrm{\Lambda }<c\mid H_0)=\alpha .}$$ The Neyman–Pearson lemma states that this likelihood-ratio test is the most powerful among all level ${\displaystyle \alpha }$ tests for this case.