Here I introduce three classical approaches to hypotheses testing in the frequentists’s approach of statistics: the likelihood-ratio test, the Lagrange multiplier test (or the score test), and the Wald test.
- The likelihood-ratio test
- The score test
- The Wald test
- The relationship between the three methods and their applications
- Inference and model selection in Bayesian statistics
The likelihood-ratio test
The oldest is the likelihood-ratio test, sometimes abbreviated as ‘LQT’. It compares the goodness of fit of two statistical models being considered by the ratio of their likelihoods, which is found either by maximization over the entire parameter space, or either by imposing certain constraints. The constraints, for instance, can be expressed as the null hypothesis. If the constraints are supported by the data, the two likelihoods should not differ by more than sampling error. Therefore, the likelihood-ratio test tests whether the ratio of two likelihood values is significant from one.
The Neyman-Pearson lemma states that when we compare two models each of which has no unknown parameters, the likelihood-ratio test is the most powerful test among all statistical tests.
The likelihood-ratio test is a standard statistical test for comparing nested models. Two models are nested if one model contains all the terms of the other, and at least one additional term. By simulation, it is also possible to use the test for non-nested models. For details, see Lewis, Butler, and Gilbert, 2010.
Besides the likelihood-ratio test, the other two approaches of hypothesis testing are the score test (also known as ‘Lagrange multiplier test’) and the Wald test, which is named after the Hungarian mathematician Abraham Wald, who discovered the survivorship bias.
The score test
The score test evaluates constraints on the parameter to be estimated based on the gradient of the likelihood function with respect to the parameter, which is known as the score, evaluated at the hypothesized parameter value under the null hypothesis. If the estimator is near the maximum of the likelihood function, then the score should not differ from zero by more than sampling error.
The Wald test
The Wald test in essence is based on the weighted distance between the estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate, namely the reciprocal of the variance. So the form resembles closely that of a t-test. Indeed, when testing a single parameter, the square root of the Wald statistic can be understood as a (pseudo) t-ratio, which is, however, not actually t-distributed except for the special case of linear regression.
The relationship between the three methods and their applications
The three approaches are asymptotically equivalent (all approaching the chi-square distribution), though in the case of finite samples, the results can differ strongly between the methods.
In bioinformatics and computational biology, these methods are used in many settings, for instance estimating differential gene expression in bulk and single-cell RNA sequencing studies.
Inference and model selection in Bayesian statistics
In Bayesian statistics, suppose that we have a prior distribution of a parameter, which can be our intuition or prior knowledge, some data associated with that parameter, and we wish to estimate (infer) the value of a parameter given the prior distribution and the data. One can use the mode of the posterior distribution, which is distribution of the parameter updated from the prior model by the data, as the best guess of the parameter. The mode of the posterior distribution is known as the Maximum a posteriori probability estimate, or simply the MAP estimate. Hypothesis testing is then addressed by quantifying the confidence interval of the MAP estimate with respect to the hypothesis.
If we have two hypotheses, we can choose the one with the highest posterior probability. This is known as the MAP hypothesis test. See the course’s website of Introduction to Probability, Statistics and Random Processes for mathematical details and examples of MAP hypothesis test. And see the blog post by Jonny Brooks-Bartlett Probability concepts explained: Bayesian inference for parameter estimation for graphical examples and more explanations.
But what happens that we have two competing hypotheses with strong difference in complexity? Then we need to select among the models, following the principle of Occam’s Razor: Use fewer things unless necessary.
The mostly used Bayesian versions of model selection are Akaike information criterion (AIC) and Bayesian information criterion (BIC). They both reward parameters and models that maximize likelihood, while penalizing complex models.
In the frequentist’s approach of statistics, one can use the likelihood-ratio test, the score test, or the Wald test to test a hypotheses against another. The likelihood-ratio test is used for model selection, either directly for nested models, or via simulation for non-nested models.
In the Bayesian approach, we often use MAP estimates and confidence intervals to test a hypothesis. AIC and BIC uses likelihood and penalties of model complexity to select the parsimonious model that fit the data at best.