A number of different approaches exist for comparing and selecting statistical models, each with its own assumptions, emphases and definitions on what model selection actually means. This talk gives first a review of the most important of such approaches: significance tests, lack-of-fit indices, Bayesian model selection, AIC and related criteria, and cross-validation and other predictive criteria. The role of simplicity of models in some of the approaches is then explored in more detail. Several model selection quantities (such as the well-known BIC and AIC statistics) are expressed as penalized criteria which appear to express a trade-off between the fit of a model and its complexity. An interpretation of the penalty term in such criteria can provide insight into the concept of simplicity of models and why simple models are generally desirable. It also helps to understand differences between different penalized criteria. The approaches are illustrated using models for comparisons of patterns of social mobility across nations.
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