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Is it appropriate to do nonparametric and parametric significance testing on clustered data?

More precisely, I performed clustering on PCA dimension reduced data, and created cluster labels for data points using the usual procedure with standardization, parameter sampling, elbow... The clusters were created over the PCA reduced data (call it dataset 'X-hat')---reduced from a couple hundred features (call the original dataset 'X') to around 30 projected features. Is it fair to apply the usual statistical analysis techniques like anova, logistic regression, pairwise t-tests... over dataset X?

I'm not sure how else to explain these clusters. All of my coursework never discussed this, and idk what books/papers to take a look at.

Addendum: I'm hoping to achieve some inferential explainability as to why these clusters are formed particularly in the preimage of the PCA process where we have all the features defined in their "original condition" directly corresponding to the distribution of each feature.

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  • $\begingroup$ Can you say more about what your overall goal is, and describe the data in a bit more detail? $\endgroup$
    – mkt
    Jul 14 at 21:09
  • $\begingroup$ I'm doing unsupervised clustering on a dataset about people over features corresponding to demographic and behavioral information. I have clusters grouping these people, each representing a data point profiling each individual. I'd like to explain and characterize the statistical significance of each cluster based off of the demographic and behavioral features. $\endgroup$ Jul 14 at 21:56
  • $\begingroup$ Can you expand on that? What are you hoping to achieve by defining these clusters? Also, it's better to edit the question directly and not add details in the comments, because people often don't read comments. $\endgroup$
    – mkt
    Jul 14 at 22:01
  • $\begingroup$ I edited the post. $\endgroup$ Jul 15 at 15:22
  • $\begingroup$ What is/are the null hypotheses you want to test? I don't understand what exactly you want to test. $\endgroup$ Jul 15 at 23:19

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