# Under which circumstances does LDA achieve a higher classification accuracy than QDA?

Since some weeks, I pursue the question "Under which circumstances will LDA achieve a higher classification accuracy than QDA using the same training and test set as well as the same prior probabilities?".

Unfortunately, the answers do not satisfy me completely.

Therefore, I would like to ask this question again in the hope of comprehensive responses consisting of descriptive examples.

• Have you considered a case where the data generating process corresponds to LDA? The true model should beat any competitors in large samples and it should beat more complex competitors (such as QDA) in small samples, too. – Richard Hardy Oct 6 '19 at 19:14
• I have, and the QDA accuracy was always higher than the LDA accuracy. – Daniel Oct 6 '19 at 19:33
• That is weird... – Richard Hardy Oct 7 '19 at 6:14
• @RichardHardy do you maybe have any other idea? – Daniel Nov 16 '19 at 21:07
• Thank you for your answer! I will check your answer and notify as soon as I have finished writing my paper. – Daniel Dec 23 '19 at 13:35

Scenario 2: Details are as in Scenario 1, except that within each class, the two predictors had a correlation of $$−0.5$$. The center panel of Figure 4.10 indicates little change in the relative performances of the methods as compared to the previous scenario.
Scenario 3: We generated $$X_1$$ and $$X_2$$ from the $$t$$-distribution, with 50 observations per class. The $$t$$-distribution has a similar shape to the normal distribution, but it has a tendency to yield more extreme points — that is, more points that are far from the mean. In this setting, the decision boundary was still linear, and so fit into the logistic regression framework. The set-up violated the assumptions of LDA, since the observations were not drawn from a normal distribution. The right-hand panel of Figure 4.10 shows that logistic regression outperformed LDA, though both methods were superior to the other approaches. In particular, the QDA results deteriorated considerably as a consequence of non-normality.