Maximum likelihood methods have a long history of use for point and interval estimation in statistics. In contrast, likelihood principles have only more gradually emerged as a foundation for an alternative to traditional hypothesis testing via frequentist test statistics. The alternative framework stresses the use of likelihood and information theory as the basis for parameterizing and selecting among competing models, or in the simplest case, among competing point estimates of a parameter of a model.
In contrast to traditional approaches, in which the statistical models are often constrained by the choice of a particular test statistic, a likelihood framework stresses the specification of both the "scientific" model that embodies the hypotheses and relationships to be tested, and the appropriate "probability" model that characterizes the statistical properties of the data and the error structure.
There are 4 general steps involved in a likelihood analysis:
- model specification, including both alternate scientific models and appropriate error structures
- maximum likelihood parameter estimation, using optimization methods
- model comparison, using information theory, and
- model evaluation, using a variety of metrics of precision, bias, and goodness of fit.