Policy relevance of causal quantities in networks

Abstract

In settings where units' outcomes are affected by others' treatments, there has been a proliferation of ways to quantify effects of treatments on outcomes, including via indirect exposure to other units' treatments. Here we consider two properties we might want estimands to have: being interpretable as summaries of unit-level effects, and being relevant to choice of a policy governing treatment assignment. We characterize many estimands as involving one of two orders of averaging over units in a population and over treatment assignments under a policy. The more common representation often results in quantities that are insufficient for optimal policy choice. This occurs because these quantities summarize outcomes under homogeneous exposure to treatment, but even homogeneous policies often lead to heterogeneous exposures. The other representation often yields quantities that lack an interpretation as summaries of unit-level effects. We argue that, among various estimands, the expected average outcome, which averages over units and treatment assignments in either order, deserves further attention from researchers. This estimand, or contrasts among these estimands under different policies, is both a summary of unit-level effects and is sufficient for optimal policy choice with utilitarian welfare.

Figures (4)

• Figure 1: The complete set of potential outcome functions $y_i(\boldsymbol{z})$ for every unit $i\in[n]$ is encoded in the unknown "science" table (in blue), where each row is a unit $i$, and each column is a set of treated units $\left\{j\in[n] \, \middle\vert \, Z_j=1\right\}$. We only observe one column (in red) of the table: $Y_i \triangleq y_i(\boldsymbol{Z})$ for the realized treatment vector $\boldsymbol{Z}\in\{0,1\}^n,\boldsymbol{Z}\sim\pi$, $V\triangleq\left\{j\in[n] \, \middle\vert \, Z_j=1\right\}$. We can "compress" this table into interpretable estimands by moving in one of two directions: (a) integrating over the columns, i.e. marginalizing over the treatments under unit-specific policies $\pi_i$ (move right), or (b) integrating over the rows, i.e. averaging the outcomes over treatment-specific node sets (move down). These averages can be executed in succession --- a form of "double averaging" --- and the resulting estimands will be typically sensitive to their ordering. In other words, the averaging operations generally do not commute; see Proposition \ref{['thm:commuting_fixedcardinality']}. Here, the averaging is done with respect to the "exposure map" $D_i \triangleq d_i(\boldsymbol{Z})$ of the number of treated neighbors a unit has.
• Figure 2: Different kinds of dose--response curves. (a) Average focal expected outcomes (AFEOs) for each unique exposure evaluated when units are treated with probability $\pi = 0.5$. Here, $d$ is the number of treated neighbors of each unit. (b) The same AFEOs as in panel (a) but as a function of the policy $\pi$. The circles correspond to the AFEOs shown in panel (a). That these AFEOs are not constant in $\pi$ is a consequence of the exposure map being misspecified. (c) The expected average outcome (EAO) as a function of the homogeneous Bernoulli policy; several such curves are all compatible with the AFEOs in panels (a, b). (d) Expected focal average outcomes (EFAOs) as a function of the policy $\pi$ corresponding to the solid EAO curve in the center. Comparisons of the EFAOs for a given $\pi$ characterize the outcomes of units with different numbers of treated neighbors under that policy, but these comparisons are not summaries of unit-level causal effects. Notes: Looking at the AFEO curve in panel (a) one might be tempted to conclude that the average outcome increases as more units are treated, but that question is answered by the EAO curves in (c). However, quite different EAO curves are compatible with the same AFEOs, some of which are not even monotonic --- as shown by the solid, dashed, dotted, dash-dotted, long-dotted EAO curves. In other words, the EAO is generally not identified by the AFEO. Furthermore, looking at the AFEOs in panel (a), one might be tempted to conclude that, under a given design $\pi$, the expected average outcome of exposure $2+$ units is more than that of exposure $1$ units, but that question is answered by the curves in panel (d). In fact, there exists no i.i.d. Bernoulli design for which the expected average outcome of exposure $2+$ units is larger than that of exposure $1$ units.
• Figure 3: (a) A biclique $K_{2,3}$. (b) Directed acyclic graph representing the causal relationships in a setting with interference according to a correctly specified exposure map. The policy $\pi$ produces the treatment assignment vector $\boldsymbol{Z}$, which determines, in a unit-specific way, each unit's exposure $D_i$ alongside the full network structure encoded in the adjacency matrix $\mathbf{A}$. If the exposure mapping is correctly specified, then $\boldsymbol{Z}$ only affects $Y_i$ via $D_i$. The decision maker is interested in some aggregate quantity (e.g., welfare) given by $W$. (c) There are no non-trivial (heterogeneous) Bernoulli policies with varied probabilities of treatment for degree-2 and degree-3 units that induce equal probability of exposure for the exposure mapping of the number of treated neighbors $\{0, 1, 2+\}$, except for the all-or-none-treated policies; that is, there is no other common points where the these curves meet in all three panels. So there is no such non-trivial policy for which the average outcomes by exposure are jointly relevant. (d) When probabilities of exposure are not homogeneous across units, we can only partially identify the expected average outcomes from the average outcomes by exposure --- even when the exposure map is correctly specified; see Corollary \ref{['thm:correctly_specified_offpolicy']} --- as shown here using data from cai2015insurance. AFEOs by exposure (left) only partially identify the EAO curve (right). Lines indicate bounds on the EAO, with red lines being the bounds when outcomes are assumed to be monotonic under exposure levels. Error bars and bands are 95% confidence intervals.
• Figure 4: For a correctly specified exposure map, panel (b) shows that the AOE (Definition \ref{['def:avg_exposure_outcome']}), i.e. AFEO by exposure, is not a function of the treatment policy $\pi$, as suggested by Proposition \ref{['thm:correctly_specified_exposure']}. See the caption of Figure \ref{['fig:aoe_vs_eao']} for further details on these dose--response curves.

Theorems & Definitions (88)

• Definition 1: Potential outcome
• Remark
• Definition 2: Policy
• Definition 3: Focal map
• Remark
• Definition 4: Deterministic focal map
• Definition 5: Positive policy w.r.t. focal map
• Remark
• Definition 6: Homogeneous focal map
• Proposition 1