Collapsibility Explained with Clinical Logic: When Crude Equals Adjusted (and When It Doesn’t)

🧩 Collapsible = “Crude ≈ Adjusted”

If you don’t adjust for age/sex and then do adjust, the result doesn’t change much.

💊 Example: You're comparing depression scores (BDI) between two groups (treatment vs control).

regress BDI6mo treat
regress BDI6mo treat BDI0 age sex

If both give almost the same β for treat, the effect is collapsible.Why? Because mean difference (MD) is a simple subtraction — adjusting doesn't twist the result.

🧠 Think: "Same story, whether or not I look at age/sex."

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🔄 Non-Collapsible = “Crude ≠ Adjusted”

Now, imagine you use odds ratio (OR):

logistic outcome treat
logistic outcome treat BDI0 age sex

Now OR changes even if no confounding!

Why? Because OR is like taking a curvy road (logarithm) — once you adjust, the curve shifts.

🧠 Think: "Just adjusting age/sex can change the story — not because of bias, but because of math."

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🧮 Marginalisation = “Make conditional result reflect the full group”

Suppose you did logistic regression with age and sex.

Now you want to know:❓“What would the OR or risk be for the whole population, not just adjusted?”

You use:

margins r.treat, predict(pr)   // risk for each group
margins r.treat, predict(or)   // marginal OR

This tells you the big-picture effect, like making the adjusted result “speak for the average patient”.

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🎯 Final Cheat Table

Concept	Translation in real talk
Collapsible	Crude = Adjusted (e.g. mean diff)
Non-collapsible	Crude ≠ Adjusted (e.g. odds ratio, hazard ratio)
Marginalisation	Adjusted result → generalized for full population

Let me know if you want this as a diagram or animated logic. Would you like a visual version?