I'm running a moderation analysis, and I was taught that you should center all continuous independent variables. In my department, professors usually treat variables that use Likert scales as continuous variables. My 1st IV is 1-item question asking participants to rate their health; it's on a 4-point Likert scale ranging from 1(poor) to 4 (excellent). My moderator is a sum of 4 items that asks how approachable parents/guardians are; it's on a 5-point Likert scale ranging from 1 (very difficult) to 5 (very easy). Since the response options are scale options, can I refrain from centering before creating the interaction term for the moderation analysis?
Centering would be important, in this case. If you just multiply these values to get an interaction term, this term will be scaled from 1 to 20 with 1 meaning low health and difficult and 20 meaning good health and easy. The scale points, however, will not be evenly distributed as 1 point difference in health means 5 scale points difference if the guardians are approachable an only 1 point in difference if they are difficult. That means, the upper end of the scales has a much stronger impact on the interaction term than their lower end. This might make sense if it's ratings that go from low/not present to high/intense and you may expect a skewed distribution with a mode around 1-2 but not in your case, where the lower end of the scale holds information and should have an impact.
By centering the values, you get an interaction term that ranges from -5 to 5 and is sensitive to both very low health/difficult and very good health/easy. Theoretically, this makes much more sense in your case.
This is the main difference mean centering makes. It changes the impact of low and high values on the interaction.