Is it appropriate to do an ANOVA on a feature selected via inspecting PCA results? I've been given a dataset consisting of 8 dimensional feature vectors for 4 classes of objects. I was asked to find the features that best distinguish the classes, and write up a short report.
My first instinct was to do a PCA on the data. This yielded the following plot, among others.

From visual inspection, it seems that the squares have higher 8000 Hz values than the other symbols.
My next instinct, on seeing this trend, was to do a one-way ANOVA on the 8 kHz feature, followed by a Tukey HSD test, to reassure myself that the difference between classes was significant. These test came to mind not because I know them to be appropriate, but rather because they're the only tests I'm familiar with. I have no idea whether or not they're well suited to this context.
My questions: is an ANOVA, followed by a planned comparison, the right course of action here? Or would some other test be more appropriate? 
 A: You only have 8 features in your data, it is not so many. The simplest approach is to conduct 8 ANOVAs, one for each feature, and correct for multiple comparisons with e.g. Bonferroni or Holm-Bonferroni correction.
If you had 8000 features, then this would probably result in no feature being significant, and one would need to think about more suitable approaches, but with 8 features you can safely start with simple ANOVAs.
Crucially, your PCA analysis can not be considered as a way out of the multiple comparisons problem. If you had decided to test your 8000Hz feature a priori, then you would not have needed to use a Bonferroni correction. But if you look at the PCA plots first, it is by definition not a priori anymore. You looked at the features and tried to select the "best" one; whether you did it with PCA or with conducting all possible ANOVAs is immaterial -- you have already run into the multiple comparisons territory.
A: Again, my main question here is which statistical test to use to figure out which features best distinguish my classes.
@StatsStudent suggests that I do a MANOVA, since I'm investigating the effect of one independent variable - class - on multiple dependent variables. This seems reasonable but I'm unfamiliar with the procedure and don't have time to read up on it.
@amoeba suggests doing one ANOVA on each feature followed by planned comparisons. He points out that my visual analysis of the PCA is, in effect, conducting a set of multiple comparisons. If I'm going to test for effects that are visually suggested by the PCA, I'll need to be careful to control my experiment-wide $ \alpha $.
I'm going to follow @ameoba's advice and do a one way anova on the 8 kHz feature, followed by a Tukey HSD test. To keep my experiment-wide error, $ \bar{\alpha} $, below 0.05, I'll use Bonferroni correction.
Bonferroni correction requires setting $ \alpha_{per comparison} $ to $\bar{\alpha} / k $, where  $k$ is the number of comparisons. My PCA is doing 6 comparisons (4 classes, pairwise) on 8 features, for a total of 48 comparisons.  So $ \alpha_{per comparison} = \bar{\alpha} / k = 0.05 / 48 = 0.001 $.
