Do unbalanced data have a negative effect on LDA?

For two-class discrimination, Ref. [1] claimed that, when covariance matrices of the two classes were unequal, a (class) unbalanced dataset had a negative effect on the performance of linear discriminant analysis (LDA). Through re-balancing 10 realworld datasets, Ref. [1] provided empirical evidence to support the claim using AUC (Area Under the receiver operating characteristic Curve) as the performance metric. We suggest that such a claim is vague if not misleading, there is no solid theoretical analysis presented in [1], and AUC can lead to a quite different conclusion from that led to by misclassification error rate (ER) on the discrimination performance of LDA for unbalanced datasets. Our empirical and simulation studies suggest that, for LDA, the increase of the median of AUC (and thus the improvement of performance of LDA) from re-balancing is relatively small, while, in contrast, the increase of the median of ER (and thus the decline in performance of LDA) from re-balancing is relatively large. Therefore, from our study, there is no reliable empirical evidence to support the claim that a (class) unbalanced data set has a negative effect on the performance of LDA. In addition, re-balancing affects the performance of LDA for datasets with either equal or unequal covariance matrices, indicating that having unequal covariance matrices is not a key reason for the difference in performance between original and re-balanced data.