Is there a way to run a fractional logit model in Mplus? (See “Econometric Methods for Fractional Response Variables with an Application to 401 (K) Plan Participation Rates” Leslie E. Papke and Jeffrey M. Wooldridge, Journal of Applied Econometrics, Vol. 11, No. 6 (Nov. - Dec., 1996), pp. 619-632., Also see http://www.ats.ucla.edu/stat/stata/faq/proportion.htm
It's not super clear which method you are interested in. The most simple one that is mentioned there is nothing more than define: y=log(y/1-y); model: y on x; You can further constrain the variance of y somehow using the technique from User's guide example 5.23.
I would recommend running the data as a two-level model where the proportion is recorded via the original binary variable (in cluster format) then use a model (modifying User's guide example 9.3) MODEL: %WITHIN% %BETWEEN% u ON w; For example 0.33 proportion from 3 observations for person 1 would be recorded as cluster 1 u c 1 1 0 1 0 1 etc.
There are some other possibilities such as using count distributions / negative binomial and exposure predictor.
We do not have any of the quasi-likelihood methods at this time.