Rank-Based Analysis of Linear Models and Beyond: A Review

In the 1940s Wilcoxon, Mann and Whitney, and others began the development of rank based methods for basic one and two sample models. Over the years a multitude of papers have been written extending the use of ranks to more and more complex models. In the late 60s and early 70s Jureckova and Jaeckel along with others provided the necessary asymptotic machinery to develop rank based estimates in the linear model. Geometrically Jaeckel's fit of linear model is the minimization of the distance between the vector of responses and the column space of the design matrix where the norm is not the squared-Euclidean norm but a norm that leads to robust fitting. Beginning with his 1975 thesis, Joe McKean has worked with many students and coauthors to develop a unified approach to data analysis (model fitting, inference, diagnostics, and computing) based on ranks. This approach includes the linear model and various extensions, for example multivariate models and models with dependent error structure such as mixed models, time series models, and longitudinal data models. Moreover, McKean and Kloke have developed R libraries to implement this methodology. This paper reviews the development of this methodology. Along the way we will illustrate the surprising ubiquity of ranks throughout statistics.