Exercise 3: How Fitting department and salary Together Tests ds1-10's Confounding Question — Possible Solution ==================================================================== WHAT ds1-10's OWN CONFOUNDING QUESTION ACTUALLY ASKED ------------------------------ Per ds1-10, "does department matter, or is department's own apparent effect actually explained by department-level salary differences instead?" — a question ds1-10 itself explicitly could not answer, since nothing in that chapter's own EDA process separated department's independent contribution from salary's own contribution. HOW INCLUDING BOTH TOGETHER PROVIDES A REAL TEST ------------------------------ Per this chapter, "because department's own one-hot columns and salary are now fit together, the model can show whether department carries any real, independent weight once salary is already accounted for." When both are included as features in the same logistic regression, each feature's own learned coefficient represents its contribution to predicting attrition AFTER the other features' own contributions have already been accounted for — precisely the same "holding constant" mechanism ml1-3 already established for regression, now applied to salary and department together. WHAT THE ILLUSTRATIVE RESULT SHOWS ------------------------------ Per this chapter's own finding-box, "once salary is included in the model, the department coefficients shrink to nearly zero — department's own apparent association with attrition, visible back in ds1-10's own EDA, is largely explained by salary, not by department itself carrying an independent effect." If department genuinely had its own real, independent effect on attrition beyond what salary already explains, its own coefficients would remain meaningfully large even with salary included in the same model. The fact that they shrink toward zero once salary is added is the direct, mechanical signature of department having been a stand-in for salary all along — exactly the confounding- variable pattern ds1-6 first described with ice cream and drowning. WHY THE CHAPTER STILL CALLS THIS "SUGGESTIVE, NOT CONCLUSIVE" ------------------------------ Per this chapter's own warn-box, "this is a real, direct test on one specific, small illustrative dataset — genuine evidence, not universal proof. A different company's own data could show department carrying a real, independent effect even after controlling for salary (working conditions, management quality, and dozens of other unmeasured factors could differ by department too)." The test performed here is real and mechanically sound — but its conclusion is scoped strictly to the specific data actually used. A small sample size limits how confidently the result generalizes, and — crucially — this model can still only control for factors that were actually measured and included, echoing ml1-3's own honest caveat about confounders outside the dataset entirely (working conditions, management quality) that this specific test has no way to account for either. WHY THIS WORKS AS AN ANSWER ------------------------------ It explains precisely how fitting department and salary together provides a genuine, mechanical test of ds1-10's own confounding question (via the same holding-constant logic as ml1-3), interprets the illustrative shrinking-coefficients result as direct evidence department was standing in for salary, and explains — using the chapter's own warn-box — why this remains a scoped, dataset-specific finding rather than a universally conclusive one.