We need to estimate the effect of a treatment (1= treated; 0=else) and other covariates on a dependent variable Yi which is a proportion, thus limited in the interval [0,1]. We observe a few zeros and a few ones. For theoretical purposes, we need to apply a threshold, c=0.333. After applying this threshold, we seek to estimate the effect of the treatment on three aspects:
  1. The effect of treatment on the probability that the proportion lies below the threshold. The dependent variable here is dichotomous: =0 if Yi<c; =1 if Yi≥c.
  2. The effect of treatment on the mean of the dependent variable for cases at or above the threshold. In this case our dependent variable is truncated from below (for values Yi<c), and we need to model a proportion in the interval [0.333,1].
  3. The effect of the treatment over the mean of the censored proportion, where all values lying below the threshold are coded as 0. In this case, the dependent variable is 0 if Yi<c, and a proportion in the interval [0.333,1] if Yi≥c. An inflation in 0 results from the censoring of values below the threshold.
We would appreciate suggestions on how to estimate each of the three effects, particularly the second one. Thank you in advance!