This contribution is an educational plea to put the Mahalanobis distance ( \(d_M\) ) more in the spotlight when teaching linear models. When compared to statistical multivariate analysis, we learn that the role of the Mahalanobis distance in inference for linear models is underrated. The Mahalanobis distance is in fact the key ingredient to arrive at confidence ellipsoids and F-tests for estimable parameters. We therefore want to stress, in this educational note, the usefulness and the obviousness of the Mahalanobis distance in linear models (regression models and multi-factor models), by showing the role of the Mahalanobis distance not only in regression diagnostics but also—even more importantly—in testing general linear hypotheses. Although the results in this note are not intrinsically new, the didactic message is that the Mahalanobis distance provides an overarching way to treat statistical hypotheses testing in linear models in a unified way. We will in fact demonstrate that testing is (knowing basic matrix algebra) easy to explain and to teach when looking at it from a Mahalanobis distance perspective.

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Mahalanobis Distance and General Linear Hypotheses in Linear Models

  • Paul Janssen,
  • Luc Duchateau,
  • Noël Veraverbeke

摘要

This contribution is an educational plea to put the Mahalanobis distance ( \(d_M\) ) more in the spotlight when teaching linear models. When compared to statistical multivariate analysis, we learn that the role of the Mahalanobis distance in inference for linear models is underrated. The Mahalanobis distance is in fact the key ingredient to arrive at confidence ellipsoids and F-tests for estimable parameters. We therefore want to stress, in this educational note, the usefulness and the obviousness of the Mahalanobis distance in linear models (regression models and multi-factor models), by showing the role of the Mahalanobis distance not only in regression diagnostics but also—even more importantly—in testing general linear hypotheses. Although the results in this note are not intrinsically new, the didactic message is that the Mahalanobis distance provides an overarching way to treat statistical hypotheses testing in linear models in a unified way. We will in fact demonstrate that testing is (knowing basic matrix algebra) easy to explain and to teach when looking at it from a Mahalanobis distance perspective.