I have consulted the HLM literature on scaling issues (raw scores, grand mean centering, group mean centering), experimented with the data you sent, and talked with my friend and colleague Paul Bliese, who is an expert on this topic. At this stage, I have reached the following conclusions.
- The main argument I have found for grand mean centering is that it gives the intercept an intuitive interpretation, i.e., the estimated value of Y when X is at its mean. However, this estimate can also be obtained by using the coefficients from a regression equation using raw scores to compute Y-hat at the mean value of X. Moreover, a confidence interval around any estimated value of Y can be easily obtained because these values are weighted linear combinations of the intercept and slope for X. As such, I don’t find this rationale for using grand mean centering to be very compelling.
- Compared to using raw scores, grand mean centering does not affect the coefficient on X, so the interpretation of the slope relating X to Y is unaffected.
- With group mean centering, the mean for each group is naturally zero, and so is the overall mean of X. As such, the intercept again equals the estimated value of Y when X is at the overall mean as well as the mean for each group. Although perhaps convenient, this result does not constitute a compelling reason for group mean centering, because again the estimated value of Y at any value of X can be found by computing Y-hat at that value of X, and that value could be the mean of X, the mean of a group, or any other value of interest.
- Group mean centering affects the coefficient relating X to Y because that part of this relationship attributable to group differences is eliminated by centering each group at its mean the X variable. Whether this is a desirable property is a matter of judgment.
- For polynomial regression models in which Z is a function of X and Y, the means of X and Y will generally differ, as will the means for the various groups nested within the X and Y variables. Centering X and Y at different values, whether these values are grand means or group means, transforms the X and Y scales such that they are no longer commensurate (i.e., they no longer share the same scale). Without commensurate measures, it is impossible to locate the relative standing of cases on X and Y, and congruence cannot be assessed or empirically studied no matter how the data are analyzed.
In my work, I have recommended scale-centering because it locates the point X = 0, Y = 0 at the center of the possible ranges of X and Y. The formulas that Mark Parry and I derived for response surface methodology were based on this type of scaling, but it is a matter of convenience rather than a strict requirement. I could reproduce a complete response surface analysis from scale-centered variables using X and Y in their original metrics, but I doubt that most people would have the desire or patience to tackle this task (which is unnecessary in the first place if scale-centered variables are used).
Incidentally, if you are using HLM merely to take into account the clustering of followers within leaders, the primary issue is the lack of independence of the residuals within group, which can be handled using clustered or robust standard errors without resorting to HLM (for a relevant discussion, see McNeish, D., Stapleton, L. M., & Silverman, R. D. (2017). On the unnecessary ubiquity of hierarchical linear modeling. Psychological Methods, 22, 114-140).