How to deal with within-treatment variability that is of substantive interest?

Question

In my area of research (conflict research), it is not uncommon (increasingly becoming to norm) to examine the variability within treatment and how this variability impacts the outcome. I'll demonstrate some examples:

United Nations Peacekeeping Operations (UN PKOs)

• Dummy Treatment: Whether or not a country received a UN PKO
• More Complex Treatment: The number of armed military personnel within a UN PKO (as opposed to to number of police personnel or non-armed personnel)

Peace agreements

• Dummy Treatment: Whether or not a peace agreement was reached
• More Complex Treatment: Whether a peace agreement includes power sharing provisions

Humanitarian intervention

• Dummy Treatment: Whether or not a conflict received a humanitarian intervention
• More Complex Treatment: Whether a humanitarian intervention is biased in favor of the rebels (as opposed to being neutral or in favor of the government)

How should one account for situations such as these in a causal inference framework? This becomes difficult given the consistency assumption. A simple dummy measure of treatment - while convenient - masks a lot of potential variation within the values that take on a value of "1" for the dummy variable. In these scenarios, consider that I am more interested in the more complex iterations of treatment. I see a couple of options, but I am unsure which is correct:

• Conceptualize treatment only as the more complex version of treatment. While I can see how this helps satisfy consistency of treatment, this can be problematic because values that take on "0" (not receiving treatment) creates a wildly varying reference category. For example, if I decide that I am interested in biased humanitarian interventions, any biased intervention is coded as "1", and any unbiased intervention is coded as "0". However, scenarios where there is no intervention at all are also coded as 0. This feels problematic to me.

• Alternatively, (continuing with the humanitarian intervention example) I could only use observations where a humanitarian intervention occurs so that the reference category is not as varied. I feel like this approach could also be problematic.

• I could include dummies for each type of humanitarian intervention in a model, omitting one to serve as a reference category. I see this approach a lot, but it seems like this would be problematic to model given what I've learned about DAGs and the logic of causally-informed control variables. This approach feels like I'm controlling for treatment itself and I wonder the damage it does for common support if within-treatment variability does not overlap (if differing types of treatment are never applied to the same units)

• I could interact the dummy treatment with the more complex treatment of interest (Humanitarian intervention x biased intervention).

Overall, I don't know what the correct approach is in situations such as these. I feel like inconsistency within treatment is a common issue. As demonstrated with this line of research, inconsistencies within treatment are often of theoretical interest. So I think that it is important to understand how to deal with such scenarios.

Answer 1

Binary treatments can be seen as special case of a much broader class of causal models

The dummy treatment variable and the measures it gives rise to such as ATE average treatment effect are very convenient and some of the most frequently used ideas from causality in practice. This is also the setting that is most often used to introduce the potential outcomes framework [Rubin 1974]. They can, however, be seen as a special case of (linear) structural causal models (SCM) [Pearl 2000] which allow for all kinds of causal relationships (see also this question).

Imagine for example the following causal model $X\to Y$ with exogenous noise terms $N$:

$X = N_X;\quad Y = X + N_Y$.

If $X$ only takes the values 0 and 1, it is very useful to define the ATE $E(Y\mid X=1) - E(Y\mid X=0)$. In your application, $X=1$ could indicate the presence of UNPKO, and $X=0$ its absence.

If you suspect that there is a relationship between the number of deployed troops and outcome $Y$, we can redefine $X$ to be the number of deployed troops (e.g. in 10,000), and estimate the functional dependency between $X$ and $Y$, for example using a linear model:

$Y = \beta X + N_Y$,

where $\beta$ is now the target parameter of our estimation. How to correctly estimate $\beta$ in more complex causal models is one of the primary questions in causal inference (e.g. by finding appropriate adjustment sets). Such a setup is standard practice in disciplines such es econometrics and economic history. Looking at some of those papers (e.g. this classic by Acemoglu et al.) might be a good source of inspiration and also of the difficulties that arise in these kind of statistical designs.

In principal, such models would not have to be linear, and the noise need not be additive, etc.. The statistical theory of structural causal models is very well developed for more general model classes (see e.g. Section 3 in the Elements of Causal Inference). In practice however, simplifying assumptions like linearity or additive and homogenous noise are common.