In some instances, it may also be reasonable to assume something like

there is no effect on the outcome in strata where the intermediate outcome is

unaffected by treatment.

Now, in general, more assumptions are going to be required to identify

causal effects within principal strata when M takes on more than two values.

So as an example, if M were to take 3 ordered values, 0,1 and

2, we have 3 principal strata with M(1 ) less than M(0).

So we might go ahead and assume that's not possible.

So that means that the probabilities in those principal strata is 0.

Then if we assume monotonicity, we're going to end up with 5 of the 9

principal stratum probabilities being identified.

But without at least one additional restriction,

the remaining probabilities are not.

In other words, monotonicity alone isn't going to get us

identification of all the principal stratum probabilities for

the case where M takes 3 ordered values, okay.

Now, furthermore, even if a restriction were imposed to get us identification on

the principle strata probabilities, without further restrictions on the values

of the principle strata effects, these would not be identified.

So, for example, suppose we now knew the principle strata probabilities.

Then by the monotonicity condition, if you see that M1 is 0,

that tells you that M0 must have been 0.

So, the principal stratum effect in this stratum is identified.

And similarly, if you see that M0 is 2, that means that M1 is 2, so

that's the probability that M0 is 2, so

the principle stratum effect is identified in this stratum.

However, you can show that the remaining principle stratum effects

are not identified without imposing further restrictions.

Now if M takes on more than 3 values,

identification will require even stronger assumptions.

So as before for continuous outcomes,

one might use some kind of parametric mixture model with covariates.

So for some examples of how one might go about this, there is a paper by Jin and

Rubin in 2008 in a journal of The American Statistical Association.

And also there's a paper by Joffe, Small and Hsu in 2007 in Statistical Science.

Both of these are pretty nice papers.

All right, I'm going to switch topics now, I'm going to take up regression

discontinuity designs which are related to instrumental variables,

in case it turns out.

So, the regression discontinuity design

arises when treatment depends on thresholding a continuous score.

So, that the reader may see the similarities and differences between this

design and its analysis and the previous material on mediation.

Instrumental variables and complaints, I will denote the score Zi, treatment Mi and

the outcome Yi.