Regression Diagnostics, Transformations and Constructed Variables

Summary Two diagnostic displays for outlying and influential observations in multiple regression are reviewed. In some examples apparent outliers vanish after transformation of the data. Interpretation of the score statistic for transformations as regression on a constructed variable leads to partial residual plots for transformations and makes the two diagnostic displays available for detection of the influence of individual observations on the evidence for a transformation. Two forms of the partial residual plot are compared. A quick estimate of the transformation parameter is derived. This is shown to have poor properties and a superior sequence of estimates is found. The plots and estimates are studied for several examples using the power transformation for non-negative observations. The corresponding theory is developed for a “folded” transformation for proportions and percentages and an example given. The relationship with other constructed variables is explored, especially Tukey's one degeee of freedom for non-additivity. Pregibon's link test for generalized linear models is interpreted in terms of constructed variables and plots given for an example. This leads to the choice between two separate models for which a graphical Monte-Carlo procedure is suggested.