Thank you for giving us an insight into the difficulties faced by SEM researchers trying to reconcile “potential outcome” idioms with traditional SEM framework. But, for the life of me, I still do not understand what drove you into this painful journey. As I said in my post above, SEM researchers do not need to ask whether (Y(1),Y(0)) are independent on other variables, nor to see these counterfactuals explicitly in their model.

Quoting :

“On the other extreme we have students of SEM, for whom counterfactuals are but byproducts of the structural model (as the First Law dictates); so, they too do not need to see counterfactuals explicitly in their graphs. For these researchers, policy intervention questions do not require counterfactuals, because those can be answered directly from the SEM-graph, in which the nodes are observed variables. The same applies to most counterfactual questions, for example, the effect of treatment on the treated (ETT) and mediation problems; graphical criteria have been developed to determine their identification conditions, as well as their resulting estimands (see here and here).

So, who needs to see counterfactual variables explicitly in the graph?

Can you describe to us what research problem drove you into this journey? Here, I mean a genuine research question (e.g., needed to estimate XXX, from data YYY, given assumptions ZZZ) not merely an intellectual curiosity to understand what Wang and Sobel are saying in their article.(on which Bollen and I commented elsewhere, see R-393).

Now, assuming that it was merely an intellectual curiosity to understand the new talk in town, “potential outcomes”, “ignorability” etc etc. Did you find anything missing from my translation: “(Y(0),Y(1)) are none others but what SEMers

call “error terms” or “missing factors”, i.e., the factors that cause Y to vary when X is held constant (at zero or at one).” Nothing more to it.

In posting after posting I have been trying to tell SEMers: counterfactuals grow organically in you own garden, they are simple to understand and simple to analyze with conventional SEM language, much much simpler than what you would find

in the ignorability-speaking literature. (This is why I posted the First Law and its flowers).

From your comment, I gather that I was not very convincing, and that there is perhaps something I left out.

I am eager to find out what it is, so that I can post another flower, for fun and insight.

Judea

Representing counterfactuals in a manner that could make them ‘visible’ for even laymen out there is on my agenda too, of course. This comes from arriving in the statistics field through the SEM backdoor (and a chunk of physics college work), which added some challenges to grasping common statistical entities, but empowered me with a visual understanding mechanism that still benefits me.

One such SEM object that can be stretched so it can cover many common entities is the latent variable of course, which I have thrown upon things that were not commonly considered LVs as such, like propensity scores (existing ‘out there’ but never fully accessible), and now even counterfactuals. I know you wrote about such a representation option too, for me it became obvious when trying to ‘see’ these variables with my own eyes, in a data format, like a worksheet: where are these two Y(x0) and Y(x1) variables (under say two treatment conditions 0 and 1), and what makes them ‘disappear’ when nature (or researchers) opt for one vs. the other one? And where do they go, are they gone forever, or can we resurrect them for some autopsies, when needed?

With 2 such options, as Conrad pointed out, ‘seeing’ them is much easier than with a continuous X range of possibilities, of course. But one can literally see them, I’ve shown this to myself and others from the MMM conference (thanks for coming over there, of course, that put me on this counterfactuals adventure path!) http://scholar.google.com/scholar?cluster=6396546620311161616&hl=en&oi=scholarr

I benefited from the Wang & Sobel 2013 chapter too, where they posed a challenge to SEMers, which I will detail on SEMNET for some resolution soon, they also used the visual display of the ‘variables’: seeing Y(x0) and Y(x1) and Y [observed] as 3 such ‘variables’ makes things much easier to grasp, especially when assigning meaning to statements about ‘ignorability’ or such: for SEMers one can now think of relations between 100% observed (or observable) variables, 50% observable ones by nature (like Y(x0) and Y(x1), in an experiment with a 0 and a 1 condition, both are missing 1/2 of the ‘values’ by design), or 100% unobservable by design, like the Y(M1(0)), or Y when X=0, but for M under X=1.

I was able to ‘see’ then that some claims are made at the “all Y’s” level, where one sees 3 such Ys, or even more, when mediator is considered, like ignorability, whereas others are at the observed/observable variables level only, which is a subset of the ‘many Y’s’ set. And that one simply needs to keep things straight in their mind when talking about causality assumptions and claims, and be aware of skipping from one level to another (and back), like when explaining why ignorability cannot be tested with observed data: we talk about correlations between variables that are missing 1/2 of their values, but complementary such halfs unfortunately.

I am trying to bring such insights (still developing) into the LV representations in SEM, like a Y being literally a composite of such many Y latents (the two in my example above only), although for now I see only an imagery/visual-grasp benefit to it. Anyways, good that you are pointing to the different ‘traditions’ out there, as I am working my way through Morgan and Winship, and Shpitser, and others. I’ll try to provide some translational advice to SEMers on a blog soon, I need a bit more critical mass I guess.

Nice flowers! ]]>

You are absolutely right. No one would load the graph with ALL counterfactuals; their

number is super exponential. Rather, the needed counterfactual is generated upon demands, for example, when we wish to

test whether a given counterfactual is independent on another give some variable. We then generate

the nodes, test for independence, and remove them, only to wait for another query.

The example you brought up illustrates that, indeed,

the counterfactual node created needs to be a conjunction of all counterfactual variables {Y(x1), Y(x2),….,Y(xn)}

If this joint variable satisfies the independence queried, then any subset satisfies it. Conversely, when we say

that the omitted factor, or “disturbance term” is independent of something, we mean that each of its instantiation is

independent of that something.

It is for this reason that I find it more meaningful to judge independence of “omitted factors” than independence

of conjunctions of counterfactuals like {Y(x1), Y(x2),….,Y(xn)}. Some researchers think the latter is more

scientific, and only by writing down the latter you show that “you know what you are talking about”.

I am for flowers.

Judea ]]>

Perhaps this is just me, but I feel that adding counterfactual nodes make the graph too cumbersome.

I’m trying to think of a situation where you have 4 levels of X (0,1,2,3). Y(0)=Y(1)=Y(2) (i.e. no arrow from X to Y when X=0,1,2), and Y(0) is not equal to Y(3) (i.e. an arrow from X to Y when X=3). Putting all this in a single graph, to my view, makes it too complicated.

-Conrad.

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