Binary ROC analysis has solid decision-theoretic foundations and a close relationship to linear discriminant analysis (LDA). In particular, for the case of Gaussian equal covariance input data, the area under the ROC curve (AUC) value has a direct relationship to the Hotelling trace. Many attempts have been made to extend binary classification methods to multi-class. For example, Fukunaga extended binary LDA to obtain multi-class LDA, which uses the multi-class Hotelling trace as a figure-of-merit, and we have previously developed a three-class ROC analysis method. This work explores the relationship between conventional multi-class LDA and three-class ROC analysis. First, we developed a linear observer, the three-class Hotelling observer (3-HO). For Gaussian equal covariance data, the 3-HO provides equivalent performance to the three-class ideal observer and, under less strict conditions, maximizes the signal to noise ratio for classification of all pairs of the three classes simultaneously. The 3-HO templates are not the eigenvectors obtained from multi-class LDA. Second, we show that the three-class Hotelling trace, which is the figure-of-merit in the conventional three-class extension of LDA, has significant limitations. Third, we demonstrate that, under certain conditions, there is a linear relationship between the eigenvectors obtained from multi-class LDA and 3-HO templates. We conclude that the 3-HO based on decision theory has advantages both in its decision theoretic background and in the usefulness of its figure-of-merit. Additionally, there exists the possibility of interpreting the two linear features extracted by the conventional extension of LDA from a decision theoretic point of view.