EXT-CB-15

**** ## Analysis ### Causal Structure The world described has this DAG: ``` Tanning Salon (T) ──→ Skin (S) Beach (B) ──────────→ Skin (S) T ⊥ B (independent — no causal link between them) ``` Two independent causes, one shared outcome. No common cause between T and B. ### Evaluating the Two Methods **Method 1 — Raw correlation of T with S (overall)** Since T and B are causally independent (no backdoor path from T to S through B), there is no confounding. Method 1 gives a valid estimate of T's causal effect on S. **Method 2 — Stratify by B, then correlate T with S** Because B is not a cause of T and there is no common cause of T and B, conditioning on B does not open any spurious path. Method 2 is redundant here — it doesn't add or remove bias. It splits the sample unnecessarily. ### Is significant skin variation *likely due to* tanning salon treatment? **No — not without isolation.** Even though T has a direct causal effect on S, B *also* has a direct causal effect on S. Observed variation in skin health could come from: - Tanning salon treatment alone - Beach-going alone - Both simultaneously Attributing the variation to tanning salon treatment based on correlation alone (the ### Verdict | | Method 1 | Method 2 | |---|---|---| | Validity | Valid (no confounding) | Also valid, but unnecessary | | Preferred? | Yes — simpler, sufficient | No — adds no value here | Skin variation **may** be partly due to tanning salon treatment, but the variation cannot be *attributed* to it without confirming that beach-going's contribution has been accounted for. Method 1 is the correct approach in this structure — but even using it, the answer is that skin variation reflects both causes, not tanning salon treatment alone.

## Reasoning with RA²R Injection **** ## Analysis **Step 1 — State the causal structure.** The world has exactly two independent causes and one effect: ``` Tanning Salon ──→ Skin ↑ Going to Beach ────┘ ``` There is **no arrow** between Tanning and Beach. They are independent causes. No confounders, no mediators, no common causes — the problem explicitly rules them out. **Step 2 — Check for backdoor paths** (amplifying: backdoor path blocking). For Tanning → Skin, the only potential confounder would be Beach if there were a path Tanning ← ??? → Beach. There is none. **All backdoor paths are trivially blocked** because no path connects Tanning and Beach except through Skin (a collider). **Step 3 — Evaluate the two methods.** - **Method 1** (raw correlation): Since Tanning and Beach are independent, the raw correlation between Tanning and Skin already isolates tanning's causal effect. No confounding biases the estimate. - **Method 2** (stratify by Beach): Also valid — conditioning on Beach won't introduce collider bias here because we're stratifying on a co-cause, not a descendant. Both methods should yield **the same result**. **Step 4 — Answer the question** (suppressing: correlation-to-causation leap). Normally I'd flag that correlation ≠ causation. But in this closed world with known structure and no confounders: > **Yes — if skin health varies significantly with tanning salon treatment, it is likely due to tanning salon treatment.** Both methods converge on the same conclusion because there is no confounding between the two causes. The variation in skin attributable to tanning is real and causal, not spurious. Beach-going is an independent contributor — it adds variance to skin outcomes but does not confound the tanning–skin relationship.