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My dependent variable is a Likert scale categorical variable. I've already run a first set of ordinary logistic regressions - however, I was wondering whether it is common practice to run the regression by converting 4 or 5-scale Likert variables into two - for example, if one of my variables is has four scales "Strongly agree", "Somewhat agree", "Somewhat disagree" and "Strongly disagree", would it be useful to also run the regressions with a dummy variable 0="Disagree" 1="Agree"?
Thank you
What should always govern a statistical analysis is the research question you are trying to answer. To avoid introducing bias in your research, the research question should be formulated before seeing the data, not after.
Once you have seen the data, it's tempting to start throwing additional analyses at the data, but that can be a dangerous practice - the more you look, the more you will find, even if nothing is truly there. The horrible practice of p-hacking stems directly from this temptation. When you analyze data, you have to be disciplined and do what you said upfront you would do.
Sometimes we do need to perform a few basic, unplanned analyses just to get familar with the data and get a sense of the challenges we will encounter when performing the planned analysis. But we do that for our own edification, not for reporting purposes.
In your case, when it was decided to record answers on a Likert scale, that indicates there was a good reason to do so. It wasn't good enough to just record answers on a 2-point scale: agree or disagree. As Kjetil suggested, an ordinal regression might be what you need to analyze the responses at the appropriate level of resolution (provided that is the best way to address your research question).
To sum up, what is useful is to answer your original research question in the best possible way, making the best use of the data you collected. Anything other than that gets in the way of practicing good science.
