A bit late, but you might find this interesting, osf.io/preprints/ps.... I think we have the same graph about Lord’s paradox.

This leads to an embarrassing thought: what I draw in my DAGs might itself be the result of a collider in some meta-DAG of the universe. I drew Sex → Weight and was so sure of the structure. But in a higher-order universe, this might itself be the result of collider conditioning.

What does “unconditional” really mean? P(data) seems unconditional, and P(data | boys) conditional. But imagine an alien landing on Earth and seeing P(data). It says, “Oh, so you’re conditioning on humans, not tigers.” Every “unconditional” is just conditional on a world we take for granted.

We’re too obsessed with decomposing direct and indirect effects in mediation. "mediation should not be understood in terms of decomposition...Once the priority of research questions is established, the practical irrelevance of statistical effect decomposition directly follows" osf.io/preprints/ps...

a fun part is, these two approaches might give conflicting results about the effect of T. I think this can be another version of Lord's paradox.

I think your approach is ok. You just defined your question as the effect of T on Y/X, and there’s nothing wrong with it. But it might be good to think about why you're using Y/X. If you want to account for the role of X, another option is Y~T+X, which gives the effect of T on Y holding X constant.

Card & Krueger's (1994) minimum wage study may be such an extreme case of confounding: "State" (NJ vs. PA), a confounder, perfectly correlates with the causal variable "minimum wage." Their interest was in the effect of minimum wage on employment, not the effect of restaurants' state location.

Looking for a tool to more easily draw your DAGs and reason on them? Try PV-dagger (pvverse.github.io/pv_dagger/). Specifically designed by @fusarolimichele.bsky.social to deal with the complex DAGs involved in pharmacovigilance, helps positioning and color-coding confounds, measurement errors, etc

A key insight is the equivalence btw suppressors and instrumental variables. Yes, DAGs are useful for understanding why S is zero-related with Y, yet can increase the overall prediction.

Card & Krueger’s minimum wage study may be a real example of a positivity violation. Their DiD addresses positivity, not unconfoundedness.
osf.io/preprints/ps...

This sounds like the same error I blogged about a few years ago, the common error of trying to control for population (or body size or many etc) by dividing the outcome variable by it. Props to the authors for seeking review and taking the issue seriously. Role models for us all.

Why HIGHER? If not, A² also be part of the Y model, implying A² → Y, which violates the exclusion restriction. This shows why the DAG representation suggested in shorturl.at/Tj8am is useful. A² = A × A can be described in DAGs, offering intuition for analysis mechanics.

Clear from the DAG, A² acts as an instrumental variable (conditional on A), enabling the identification of the M → Y effect even with U. This is what shorturl.at/1TgCm showed: mediation analysis can be valid (even with U) if the M model has a higher order of A than the Y model.

Very happy to share this final version with you. Thank you! ;-)

Easy to see why the cor btw the first-order and interaction terms (indicating collinearity) after centering becomes zero (though this is not the reason for centering); why centering X1 only (not X2) change the coef on X2​ while leaving the coefs on X1 and the (centered) interaction term unchanged.

DAGs (causal graphs) can be used to understand the mechanics of linear interaction analysis. See more here: bpspsychub.onlinelibrary.wiley.com/doi/10.1111/...