I understand that in observational studies, if we want to use regression for analysis, and if some variables are supposed to be a confounding variable, we should control for them in the regression model. However, I am a bit confused about adding covariates in ANOVA kind of analysis. Here is the scenario:
Scenario: In a randomised experiment in which participants are randomly assigned into 3 different treatment groups, and then we measure their performance (e.g., response times) on some task as our only DV. However, participants' IQ and EQ are supposed to be covariates, so they were measured prior to the treatment.
Question1: In general, should we run ANOVA or ANCOVA?
Question2: If we indeed found no significant difference in participants' IQ, shall we treat IQ as covariate in our model?
Question2: If we unfortunately find that the 3 groups have different levels of EQ even though we used random assignment, should we run ANOVA or ANCOVA?
Edit 1: Assume that the observations are independent of each other by random sampling. Also assume that the distribution of DV is normal.