I am doing the research, using Multilevel modeling, with limited dependent variable number of days- it is limited downward (0) and upward (30). Is it necessary to use Multilevel logit model? Or is it acceptable to use Multilevel linear modeling and Winsorize the dependent variable (for example at ninety-ninth percentile). Does it give the same output or do you think the result will be biased using the second option?
It sounds like your dependent variable is a count (number of days). That should probably be accounted for in your analysis. Winsorizing this would not make sense, nor does logistic; you should use some sort of count model, such as Poisson or negative binomial regression.
Whether you need multilevel modeling depends not on the dependent variable, but on whether the errors are independent. Assuming that they are not, then a nonlinear MLM may be needed. In SAS, PROC GLIMMIX allows this. In R see this thread.
You could do this as a fractional logit model by dividing all your variables by the max, i.e., do it as a percentage of days of the month. Google "fractional logit" or see Papke, Leslie E., and Jeffrey M. Wooldridge. "Econometric methods for fractional response variables with an application to 401 (k) plan participation rates." Journal of Applied Econometrics 11.6 (1996): 619-632.