I am trying to estimate the mediation effect of a continuous mediator m on a dichotomous variable y. The independent variable x is dichotomous as well. While there are methods to estimate the indirect effect when the dependent variable is dichotomous, I am struggling to find a way to do this in a multilevel setting. This setting applies to my data because I have 660 people and for each person I have 60 observations for x, m and y each. So all observations are nested (clustered) within a person, i.e. it is a 1-1-1 design. The variable x is randomised over time, the mediator and the outcome variables are not.

The interclass correlation (ICC) of the dependent variable y is 8% (95%CI: 0.06; 0.11). The ICC of the mediator variable m is in the same range. Can this be considered as strong? If not, I plan to use
Code:
khb logit y x || m, vce(cluster id)
If the estimated ICC should be considered as strong, I was thinking of two ways to take it into account:

1. To use the individual means of x and m to center these variables and use the centred variables, e.g. by using
Code:
khb logit y centered_x || centered_m
(Source: Kohler, U., Karlson, K.B., Holm, A. (2011). Comparing Coefficients of Nested Nonlinear Probability Models. The Stata Journal 11:420-438).

2. To use fixed effects:
Code:
khb xtlogit y x || m, fe
Does any of this approaches solve the problem? Do you have a better idea how I should proceed?

Any help is greatly appreciated!