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Note: I have rewritten this question quite a lot, because pzivich's answer made me realize that I had not formulated it accurately enough . In order to give the original context of pzivich's answer, I have left my original question below the updated one.

Updated Version

Description:

Consider a treatment $T$ that can take on $M$ values $T\in \{1,...,M\}$. Let $Z$ be some set of observed properties of the treatments. Say, for instance, that the population of interest were refugees, who received residence permit in a given country, and the treatment variable $T$ was which area within the country they were assigned to settle down in. Then $Z$ could e.g. be the settlement area's population density, overall employment rate and non-native population share. Let $Y^d$ denote the potential outcome of interest that is realized under treatment $d$. I would like to estimate the quantity:

$\delta(l,m,r,q) = E[Y^l - Y^m\ |\ Z_l=r,Z_r=q]$

where $Z_d$ denote the values of $Z$ for treatment $d$. Continuing the example with the impact of settlement areas on refugees, let the outcome $Y$ be whether the refugee finds employment within the first two years of residence. Then $\delta(l,m,r,q)$ is supposed to be the difference in employment probability caused by assigning a refugee to a settlement area with characteristics $r$ instead of a settlement area with characteristics $q$.

Assume that the treatment assignment is random.

Questions:

I have two questions.

Q1: Do you know if this kind of causal inference problem has a name? I know many examples of studies estimating the conditional average treatment effect $E[Y^l - Y^m\ |\ X=x]$, where $X$ is some set of observed properties of the individuals in the population, but I don't know of any estimating $E[Y^l - Y^m\ |\ Z_l=r, Z_r=q]$. I guess the reason is that usually the treatment is binary or a single continuous value, but in my case the treatment is characterized by a set of properties $Z$.

Q2: What is the proper causal interpretation of the estimated relationship between the $Z$ variables and the outcome? My intuition is the following. Continuing with the example from above, say e.g. that we estimate that refugees, who are assigned to settlement areas with a higher population density, have a greater employment probability. Since the settlement areas are randomly assigned, we can conclude that being assigned to a settlement area with higher population density causes a refugee to have a higher employment probability. However, we cannot conclude that higher population density causes refugees to have a higher employment probability, since the relationship between population density and employment could e.g. be confounded by unobserved labor market conditions. In other words, we can conclude that assigning a refugee to an area with high population density increases the refugee's employment probability, but we cannot conclude that increasing the population density of an area will increase the employment probability of refugees being assigned to the area. Do you agree with this interpretation? If so, do you have any ideas for how to formalize and prove this intuition?


Original Version

Consider a treatment $T$ that can take on $M$ values $T\in \{1,...,M\}$. Let $X$ be some set of observed covariates of the individuals in the population of interest (this could e.g. be the individuals' age, gender and ethnicity). Let $Z$ be some set of observed covariates of the treatments. Say, for instance, that the population of interest were refugees, who received residence permit in a given country, and the treatment variable $T$ was which area within the country they were assigned to settle down in. Then $Z$ could e.g. be the area's population density, overall employment rate and non-native population share. Let $Y^d$ denote the potential outcome of interest that is realized under treatment $d$. I would like to estimate the quantity:

$\delta(l,m,x,r,q) = E[Y^l - Y^m\ |\ X=x,Z_l=r,Z_r=q]$

where $Z_d$ denote the values of $Z$ for treatment $d$. Continuing the example with the impact of settlement areas on refugees, let the outcome $Y$ be whether the refugee finds employment within the first two years of residence. Then $\delta(l,m,x,r,q)$ is supposed to be the difference in employment probability caused by assigning a refugee with characteristics $x$ to a settlement area with characteristics $r$ instead of a settlement area with characteristics $q$. One potential benefit of knowing $\delta$ could be to direct the assignment of a refugee towards the type of settlement area, where he/she has the highest employment probability.

Assume that the treatment assignment is random.

Q1: Do you know if this kind of causal inference problem has a name? I know that it falls under the general themes of heterogeneous treatment effects and multiple treatments. However, I have not been able to find theoretical or applied studies, where the treatment variable is characterized by a set of observed covariates.

Q2: How would you approach estimating $\delta$, given the assumption that treatment is randomly assigned?

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How the problem is described (i.e., $T$ is the area assigned and $Z$ is the characteristics of that area), it sounds like the features of $Z$ are already implied by $T$. For a discussion of how treatment is being defined and the subsequent implications, I would recommend Hernan 2016 for an introduction to the problem. Defining treatments is also referred to as 'causal consistency' or 'well-defined interventions'.

In the case I am misreading the problem (and $Z$ can vary withing $T$), there are several approaches. Similar to the Hernan 2016 paper, you can define a new $T^*$ that is a function of both $T$ and $Z$ and estimate the effect. Note that this could either increase the dimension of $T^*$ or reduce it compared to $T$. Based on the language provided in the question, I would write the estimand as $$\delta(t^*, t^{*'}) = E[Y^{t^*}-Y^{t^{*'}}]$$ with the original $T$, $$\delta(t, t', r, q) = E[Y^{t, r}-Y^{t',q}]$$ Note that the potential outcomes are defined by both the values of $T$ and $Z$.

As for the interpretation, I would interpret $E[Y^{t^*}-Y^{t^{*'}}]$ as if all migrants had been assigned to $t^*$ compared to all migrants had been assigned $t^{*'}$ would have resulted in ___ more (less) unemployed migrants by two years. So I agree with you interpretation. You are correct when you say we cannot make a claim regarding increasing the population density itself. The only claims supported are on assigning migrants a location.

I will say that this problem assumes that there is no interference (i.e. the assignment of migrant $i$ does not effect the employment of migrant $j$). This is likely an issue (if all migrants were sent to the same location, the labor market may be saturated and result in lower net employment than estimated).


Response to original questions:

The heterogeneous treatment effect (HTE) problem has been referred to as a few different names. Here are some others I have heard previously: conditional average treatment effect, and variable importance. A related problem is the optimal treatment regime. This sets about searching through the space of HTE to find which plan maximally reduces (increases) the outcome. Optimal treatment effects look through HTE, but the estimation of those HTE is often done in similar ways.

Ideally, even though $T$ is randomized, I would choose nonparametric approaches. For how to apply this, I would recommend starting with Kennedy arXiv 2020. This approach makes weaker assumptions regarding the structure of the $Z$ variables. If $Z$ only consists of a few categorical variables or you are willing to make parametric assumptions, van der Laan 2006 talks about how to proceed under that approach. The 'pseudo-outcome' algorithm that Kennedy describes comes from van der Laan. I think both of these papers would be good to start with and either are referenced by, or reference the larger literature.

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    $\begingroup$ Thanks for the answer, @pzivich. Reading your answer, I think that my question was not formulated accurately enough with respect to exactly what about my specific HTE problem I am confused about. I would be very grateful, if you have time to read the updated question and see if you can help me with that. $\endgroup$ Sep 16, 2020 at 11:44
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    $\begingroup$ I have updated my answer. Hope it helps! $\endgroup$
    – pzivich
    Sep 16, 2020 at 17:45
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    $\begingroup$ I agree that a no interference assumption is quite problematic in this case, but I wanted to think about the multidimensional treatment aspect of the problem first :-) $\endgroup$ Sep 17, 2020 at 12:27
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    $\begingroup$ There isn't a particular reason using $Z$ as treatment but there are two issues. First treatment can become very high dimensional. Second you make the assumption that only the differences in $Z$ are important differences for $T$. There can be other differences ($U$) but $U$ is unimportant for $T$ on $Y$. I am not familiar with this topic area, so I can't comment on how reasonable this assumption is in your context. I think it might be easier to collapse $T$ into a smaller number of versions (e.g. from $K$ to 2) by merging $T$'s that are similar in their values of $Z$ $\endgroup$
    – pzivich
    Sep 17, 2020 at 14:41
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    $\begingroup$ The later does still assume that any $U$'s are unimportant for $T$ on $Y$ though (it is a similar concept as discussed in the above linked Hernan paper). But yes, interference makes things much more complicated! (only wanted to make sure it wasn't lost) $\endgroup$
    – pzivich
    Sep 17, 2020 at 14:44

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