What Is Collapsibility in Clinical Statistics?

Collapsibility is a property of effect measures that determines whether the measure changes when you (statistically) adjust or don’t adjust for covariates, even in the absence of confounding.

✅ Collapsible Measure

Example: Risk Difference (RD) or Mean Difference (MD)
Property:
- If no confounding, the crude (unadjusted) effect estimate equals the adjusted effect estimate.
- That is:

Risk Difference Note

\( RD_{\text{adjusted}} = RD_{\text{crude}} \quad \text{(if no confounding)} \)

Why? Because RD and MD are linear, and averaging over subgroups does not distort the overall effect.

❌ Non-Collapsible Measure

Example: Odds Ratio (OR) or Hazard Ratio (HR)
Property:
- Even if there’s no confounding, adjusting for covariates changes the effect estimate.
- That is:

Odds Ratio Note

\( OR_{\text{adjusted}} \ne OR_{\text{crude}} \quad \text{(even if no confounding)} \)

Why? These are non-linear transformations of the data (log-odds, log-hazard) and inherently distort when marginalizing.

🧪 Illustrative Stata Examples

Crude vs Adjusted (Collapsible case)

regress BDI6mo treat
regress BDI6mo treat BDI0 age sex

If treat is your intervention and BDI6mo your outcome (e.g., depression score),
The β for treat changes little: ➡️ Mean Difference is collapsible

Crude vs Adjusted (Non-Collapsible case)

logistic outcome treat
logistic outcome treat BDI0 age sex

The ORs differ: ➡️ Odds Ratio is non-collapsible, even if covariates aren't confounders.

🔁 What Is Marginalisation?

Marginalisation is the process of recovering population-average (marginal) effects from conditional (adjusted) regression models.

In other words:

“Given I fitted a logistic model adjusting for covariates, what is the overall effect of treat averaged over the covariate distribution?”

📦 In Stata:

logistic outcome treat age sex
margins r.treat, predict(pr)   // Predicted risk
margins r.treat, predict(or)   // Marginal OR
marginsplot                    // Visualize marginal effects

This tells you: “On average, what is the risk/OR if everyone received the treatment vs. control?”

🔑 Summary Table

Concept	Meaning
Collapsible	Effect estimate doesn't change with covariate adjustment (e.g., RD)
Non-collapsible	Effect changes due to mathematical form, not confounding (e.g., OR)
Marginalisation	Transforming model-based (conditional) effects into population averages

🧠 Secret Insight from CECS Logic

Always distinguish change due to bias (confounding) vs change due to non-collapsibility.
Marginal effects are policy-relevant; conditional effects are decision-support relevant.
Collapsibility is mathematical. Confounding is causal. Don’t confuse them.