The Estimation Problem

The vast increase in computing power over recent decades has led to the emergence of multilevel models and its equivalents as practical and powerful analysis tools.1–6 This approach involves specifying two or more levels, or stages, of relationships among study variables and parameters. These levels are arranged in a hierarchy; hence the approach is also commonly known as hierarchical modelling or hierarchical regression. Ordinary regression techniques represent a special case in which there is only one level of the hierarchy. Multilevel modelling has appeared and reappeared in many forms over the last 50 years. For example, the following techniques are all special cases of or equivalent to hierarchical regression: Bayesian, empirical-Bayes (EB), Stein, penalized likelihood, mixed-model, ridge, and random-coefficient regression, and variance-components analysis.1–6 Conventional estimation methods, such as least squares and maximum likelihood, are also special cases in which there is only one level in the underlying model. The hierarchical approach unifies these methods and clarifies their meaning. Most importantly, it unifies the seemingly disparate methods of frequentist (‘classical’) and Bayesian analysis. This unification is a major benefit of the approach, for it leads to methods that are superior to both classical frequentist and Bayesian methods. I will here barely touch on the diverse applications, forms, and examples of multilevel modelling. My reason is that a more basic conceptual introduction seems necessary. Despite several decades of published examples and methodological studies demonstrating the superiority of the multilevel (hierarchical) perspective, and its widespread acceptance in the social sciences, it is not widely understood, taught, or employed in the health sciences. Lack of affordable and user-friendly software has no doubt been an obstacle to use, although as discussed below many software options are now available for multilevel modelling. Another obstacle, however, is that most presentations move into technical and subject-matter details too rapidly for healthscience readers, and few discuss the general multilevel perspective. I will develop that perspective here, although to limit the scope I will focus on estimation error, which is just one of several major problems addressed by multilevel modelling. I will explain how one by-product of multilevel modelling— hierarchical estimation—can help address that problem. The examples used here are artificial and too simple to warrant even an ordinary regression analysis. Their purpose is only to provide a basis for understanding the much more complex real examples that can be found in the references.