Collider in RCT Subgroup Analysis

arthur_albuquerque:

I noticed that stratifying by subgroup and including the T \times S parameter could open a backdoor path in both interaction or effect modifier frameworks if an unobserved variable influences both T \times S and the outcome:

I’m still a bit confused.

People who are part of the subgroup also have a higher/lower risk for the outcome than people not part of this subgroup, even if they are not given the treatment. That’s @Pavlos_Msaouel EGFR example. People from the subgroup also have a different relative treatment effect (on the scale of interest) when given treatment than people not in the subgroup.

The second DAG:

Being part of the subgroup tells us nothing about the risk of the outcome if no treatment is given. Under treatment it does.

This DAG is confusing me :

Why is there an arrow from \text{Unobserved} \rightarrow \text{T x S} ? We are not modeling the \text{T x S} Interaction using U? I think this corresponds to this graph from Pavlos publication. But the above DAG would somehow lead me to think that people who are part of U and \text{subgroup} have a different relative treatment effect than people who are part of \text{subgroup} and not U?

Quote from Epi paper which helps a bit:

It is possible that there are factors that influence the likelihood of both expo- sures occurring concurrently (marked by an arrow into the interaction E G node) and also influence disease risk (marked by an arrow into the disease D node); such a fac- tor would create a back-door path and this would be ex- plicitly visualized in the DAG (see Figure 2c). Continuing with our example of smoking (E), asbestos (G) and lung cancer (D), a potential confounder (C) would be a factor that increases the likelihood of both smoking and asbesto- sis exposure, such as socio-economic status. Although this back-door path could also be captured by arrows from C to both E and G if the E G node were omitted, the inter- action node prompts the researcher to think about factors that affect both exposures simultaneously.

I’ll try to simulate some data to wrap my head around this.