Many research questions which aid in decision making are the question of “how does T affect Y, controlling for X”. this is such a common and useful question to ask that it is occasionally mistaken to be the entire enterprise of causal inference
this problem has a familiar causal diagram, the confounding triangle (show it)
in my career, some examples I’ve seen over and over: how does using product T affect customer outcome Y (revenue, usually) when controlling for the mix of user types
more formally, we are usually looking to estimate the treatment effect; the average result of being moved from control to treatment
$\underbrace{\Delta}\textrm{Treatment effect} = \underbrace{\mathbb{E}[y \mid T = 1]}\textrm{Average outcome when treated} - \underbrace{\mathbb{E}[y \mid T = 0]}_\textrm{Average outcome when control} $
or the conditional treatment effect, which is the treatment effect given some specific information $X$ which we know, the conditional treatment effect:
$\underbrace{\delta(X)}\textrm{Conditional Treatment effect} = \underbrace{\mathbb{E}[y \mid T = 1, X]}\textrm{Average outcome when treated} - \underbrace{\mathbb{E}[y \mid T = 0, X]}_\textrm{Average outcome when control} $
stratification is basically always the first step for me when doing an analysis like this, even though there are lots of directions it might take after that
1.1: A helpful reference, from which a number of these ideas were adapted directly, are the lecture notes for Stanford’s Data Mining: 36-462/36-662 with Rob Tibshirani. These notes and these notes in particular are useful, as is section 7.10 of ESL.