SUTVA vs. independence In the Wikipedia article on the Rubin causal model I stumbled upon the following quote:

We require that "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units" (Cox 1958, §2.4). This is called the Stable Unit Treatment Value Assumption (SUTVA), which goes beyond the concept of independence.

(https://en.wikipedia.org/wiki/Rubin_causal_model)
Please, can anyone tell me in what sense the SUTVA 'goes beyond the concept of independence'? I don't get the difference between the two.
 A: I think SUTVA violations come in two flavors, which are not always distinct:

*

*"spillovers/interferences" that arise from contact across individuals in social, commodity, or physical space (independence flavor)

*dilution/concentration of treatment effects that stem from changes in the prevalence of treatment (what economists call general equilibrium effects or failure of the ceteris paribus assumption flavor)

Consider a job training program that teaches a handful of people how to knit and sell their output on Etsy (a small program in a large market). If you have treated trainees that teach control group people how to crochet, or more knitting takes place when you treat groups of friends (knitting is often a social activity), you have an example of (1). Two real world examples of this are patients in early AIDS drug trials sharing their medication or irrigation/rain causing fertilizer runoff from treated to control plots.
If you a have a mandatory job training program that teaches knitting and selling at a local farmer's market (large program in a small market), you might expect the prices of scarves and socks to plummet, with the pecuniary benefit of the knitting knowledge declining with the number of people treated. A real world example is the effect of charter schools on academic achievement, which might change if you had a large influx of public school students into the other sector, or a program that teaches farmers to all grow a particular type of crop. You can think of this dilution as either a dosage change or as a kind of treatment effect change.
I think it is frequently hard to make these two fully distinct, and (2) often operates through (1)-type channel: the inrush of public school students is only problematic because of rival resource constraints or peer effects. However, (2) is more subtle than spillover/interference, so I think it does go "beyond independence" in some sense.
I think (1) is often more harmful, since it undermines internal validity of an estimate, though we can sometimes redefine the unit of analysis to be the community within which individuals interact rather than the individuals themselves.
I think of (2) as circumscribing the external validity, since when trials are small, we can think of the estimated partial equilibrium effects as a kind of bound on the general equilibrium effects that would be seen if the program was scaled up and prices and inputs or "dosage" change. This limits what you can claim, but if the costs of the small trial program already exceed the benefits, and we anticipate the benefits to decline if the program is scaled up, that is still useful information. Alternatively, SUTVA may only hold for some part of our data, and the analysis can proceed once the rest is discarded. This makes (2) less pernicious.
Here's a slightly more rigorous way to think about this. We can write the treatment effect for person $i$ as a function of the $(N-1) \times 1$ indicator vector $\mathbf{t}$ that gives you treatment assignments in the remaining population: $$\Delta_i(\mathbf{t})=y^1_i(\mathbf{t})-y^0_i(\mathbf{t})$$
We can think about how $\Delta_i$ varies as we change $\mathbf{t}$ in particular ways.
Let $T=\vert \mathbf{t} \vert$, the $L_1$ norm of the treatment assignment vector. This tells you how many people got treated in a particular treatment configuration. If $\Delta_i$ depends on where the ones are in $\mathbf{t}$, holding $T$ fixed, you have SUTVA violation of type (1). This means it matters whether people "connected" to person $i$ are treated or not, a kind of dependence.
If $\Delta_i$ only changes with $T$, but is the same for all pairs $\mathbf{t}'$ and $\mathbf{t}$ where $\vert \mathbf{t'} \vert= \vert \mathbf{t} \vert,$ you have a type 2 violation.
If $\Delta_i(\mathbf{t})=y^1_i-y^0_i,$
SUTVA is fully satisfied since the potential outcomes do not depend on how the treatment is rolled out.
To summarize all this, there are two types of SUTVA violations which are not fully conceptually distinct, but have differing implications, which makes it useful to emphasize their differences.
