Limits of Approximating the Median Treatment Effect
Raghavendra Addanki Siddharth Bhandari
Average Treatment Effect (ATE) estimation is a well-studied problem in causal
inference. However, it does not necessarily capture the heterogeneity in the
data, and several approaches have been proposed to tackle the issue, including
estimating the Quantile Treatment Effects. In the finite population setting
containing $n$ individuals, with treatment and control values denoted by the
potential outcome vectors $\mathbf{a}, \mathbf{b}$, much of the prior work
focused on estimating median$(\mathbf{a}) -$ median$(\mathbf{b})$, where
median($\mathbf x$) denotes the median value in the sorted ordering of all the
values in vector $\mathbf x$. It is known that estimating the difference of
medians is easier than the desired estimand of median$(\mathbf{a-b})$, called
the Median Treatment Effect (MTE). The fundamental problem of causal inference
-- for every individual $i$, we can only observe one of the potential outcome
values, i.e., either the value $a_i$ or $b_i$, but not both, makes estimating
MTE particularly challenging. In this work, we argue that MTE is not estimable
and detail a novel notion of approximation that relies on the sorted order of
the values in $\mathbf{a-b}$. Next, we identify a quantity called variability
that exactly captures the complexity of MTE estimation. By drawing connections
to instance-optimality studied in theoretical computer science, we show that
every algorithm for estimating the MTE obtains an approximation error that is
no better than the error of an algorithm that computes variability. Finally, we
provide a simple linear time algorithm for computing the variability exactly.
Unlike much prior work, a particular highlight of our work is that we make no
assumptions about how the potential outcome vectors are generated or how they
are correlated, except that the potential outcome values are $k$-ary, i.e.,
take one of $k$ discrete values.