Normal probability is not the goal, identifying sources of variation is. In industrial engineering, we use statistics for process improvement, not necessarily for making great models. You will likely know that better than most of us - you haven't even said what academic area you're in. If you're worried about your defense, you have to anticipate what they will expect, not what a statistician would do. Alternatively, I might use a Bayesian approach to simultaneously probabilistically identify (and suitably use or adjust for) potential outliers with the rest of the identification and estimation. If I anticipate some contamination by some other process that would produce outliers not from my process of interest, I'd usually try to choose a procedure robust to them. If my model is not suitable, I had a problem with model choice - which should come before data collection. That is, both your options are things I would usually try to avoid. I tend to avoid removing points in any arbitrary fashion if at all possible. on the behavior of p-values) can sometimes be dramatic. It's about your choices and their consequences - and the consequences of data-based model choice and then inference on the same data (e.g. It's not like there's a police force of model choice. I am mystified by the intent of "can" and "should" here.
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