Regression and ANOVA

  • Optimal Design Characterizing designs that allow both efficient estimation of a regression model as well as powerful testing of likely deviations from the model has been an interest of our group. The additional demand that deviations from the model must be readily detectable, leads to new optimality criteria. The resulting optimal designs are quite different from the usual ADE-optimal ones.

  • Two-Way Plots Two-way plots are graphical representations of models for two-way ANOVA tables in which the row and column structure is maintained and visible. The fitted values are shown by intersecting line segments, one segment per row and per column. We have explored the construction of such plots and the fits they can represent. Generalizations, where the geometric objects representing rows and columns are not necessarily lines and where all or some of the residuals are added to the plots have also been studied.

  • L_1 Regression Minimizing the sum of absolute residuals is an old idea for fitting linear models. It results in a robust method whose asymptotic properties are well understood. The method does not, however, have positive breakdown point if one is allowed to change the design. We have studied the breakdown point under the restriction that only outlying observations can be created whereas the design remains fixed.

  • Residuals from L_1 Fits The residuals obtained by L^1 fitting exhibit several weaknesses. First of all they are ambiguous in the sense that there are a multitude of L^1 fits, sometimes quite far apart. Second, typical algorithms produce as many exact zero residuals as there are contrasts fitted. As a result, the non-zero residuals do not give an accurate reflection of the errors that occurred during the experimental runs.