Non-traditional Difference in Difference: first voluntary treatment then compulsory I have the following scenario. I have two groups. With the first one nothing happens during the whole time span. However, for the second one situation changes. First, Reform 1 happens and allows individuals from the second group to voluntarily be selected into the treatment(basically they have an option for half of the year to try the treatment). Then after half of the year, all the individuals of the second group are treated. I suppose that I can't in this situation just use a standard Difference in Difference model which assumes two time periods (pre and post) and just one treatment. But how can I approach the problem with the period where the second group can choose whether it wants to be treated or not?
 A: Paired tests within 2nd group. I'd look at the second group first. You have two types
of subjects in there. Ones who volunteered to start
treatment early and those who didn't. 
For those who stared early, use a paired t test 
to see if there is a difference in differences
between the no treatment phase and the treatment
phase.
For those who started later (if you gave tests timed 
to do it), do a similar paired test to see if there
is a difference between 'full' treatment and 'half'.
Two-sample tests between two main groups.
Between the two groups (no treatment at all and
at least some treatment), do a Welch 2-sample t test to
see if there is a difference in differences.
(Post - Pre in both groups).
Depending on findings above, the timing of various tests,
in the second group, and the number of subjects in each
subgroup of the second group, other comparisons between the two main groups might
be worthwhile investigating.
Plan analysis before taking data. As always, if perhaps only in retrospect, one appreciates the advantages
of planning the methods of analysis before beginning
the study.
