SUBMISSION: Project 4 — Multivariable Regression with Multicollinearity Diagnostics (VIF) & Interaction Terms

# SUBMISSION: Project 4 — Multivariable Regression with Multicollinearity Diagnostics (VIF) & Interaction Terms > 🎯 **Hands-on Portfolio Deliverable.** Regression is the most-used model in DS — and the most misread. This project proves you check assumptions instead of quoting R². ## 🏢 Business Scenario & Executive Prompt **CloudNine SaaS** (50,000 accounts, 4.8% quarterly churn) wants to know *why* customers leave. The PM's draft model regresses churn on 12 features and reports R² = 0.31 with *"all p-values significant."* You spot `sessions_per_week`, `logins_per_week`, and `active_days_per_week` in the same model — plus a suspicion that tenure effects differ by plan tier. The VP Sales asks: *"Which three levers actually move churn, and how much do you trust each number?"* ## 📋 Source Data & Input Schemas **`cloudnine_churn.csv`** — 50,000 rows (mock shape; scaffold generates it): | column | dtype | description | |---|---|---| | `acct_id` | str | unique account | | `churned` | 0/1 | churned this quarter (4.8% base rate) | | `tenure_months` | int | account age | | `plan` | category | `basic \| pro \| enterprise` | | `sessions_pw` | float | sessions/week (collinear trio #1) | | `logins_pw` | float | logins/week (collinear trio #2) | | `active_days_pw` | float | active days/week (collinear trio #3) | | `support_tickets` | int | tickets this quarter | | `contract_value_inr` | float | ARR in ₹ | ## 🎯 Required Deliverables 1. **`churn_regression.ipynb`** — diagnostics → VIF triage → interactions → validation notebook. 2. **`churn_model.py`** — final OLS specification (formula + fit + `get_summary()` printing VIF table). 3. **`vif_report.csv`** — `feature, vif_before, vif_after, action (keep \| drop \| combine)`. 4. **`regression_memo.md`** — three levers with per-lever effect sizes + CIs, assumption verdicts, and what you would NOT claim. ## 🛠️ Step-by-Step Implementation Guide 1. **Baseline + residual diagnostics.** Fit OLS on all 12 features (statsmodels). Residual-vs-fitted, Q-Q, scale-location plots. Name each violated assumption in plain language. 2. **VIF triage.** Compute VIF for all features; the engagement trio will print VIF > 10. Decide per feature: drop, combine (e.g., `engagement_index = z-mean`), or keep-with-HC-SEs. Record in `vif_report.csv`; refit and show coefficient stability (before/after table). 3. **Interactions.** Test `tenure_months × plan` (enterprise tenure behaves differently). F-test for the interaction block; interpret one interaction coefficient as *"for enterprise, each extra month changes churn log-odds by…"*. 4. **Partial-effect interpretation.** Every reported coefficient stated as *"holding others constant"* with 95% CI. One paragraph on why the PM's *"all p-values significant"* claim is wrong under multicollinearity. 5. **Memo.** Three levers ranked by standardized effect; honest limits (observational, no causation). ## 💻 Production Starter Scaffold ```python import numpy as np, pandas as pd import statsmodels.api as sm from statsmodels.stats.outliers_influence import variance_inflation_factor rng = np.random.default_rng(21) N = 50_000 tenure = rng.integers(1, 60, N) plan = rng.choice(["basic","pro","enterprise"], N, p=[.55,.3,.15]) sess = rng.gamma(4, 1.2, N) + (plan=="enterprise")*2 df = pd.DataFrame({ "acct_id": [f"CN-{i:05d}" for i in range(N)], "tenure_months": tenure, "plan": plan, "sessions_pw": sess, "logins_pw": sess*1.4 + rng.normal(0,.5,N), # collinear by construction "active_days_pw": (sess/2).clip(0,7) + rng.normal(0,.3,N), "support_tickets": rng.poisson(1.2, N), "contract_value_inr": np.round(rng.lognormal(10.5,.8,N),2), }) logit = -3 + -0.02*tenure + (plan=="pro")*-0.4 + (plan=="enterprise")*-0.9 \ + 0.15*df.support_tickets - 0.1*sess + (plan=="enterprise")*tenure*-0.01 df["churned"] = (rng.uniform(size=N) < 1/(1+np.exp(-logit))).astype(int) print("churn rate:", df.churned.mean().round(4)) # --- VIF triage --- X = pd.get_dummies(df[["tenure_months","sessions_pw","logins_pw","active_days_pw", "support_tickets","contract_value_inr","plan"]], drop_first=True).astype(float) vif = pd.DataFrame({"feature": X.columns, "vif": [variance_inflation_factor(X.to_numpy(), i) for i in range(X.shape[1])]}) print(vif.sort_values("vif", ascending=False).head(8)) ``` ## 🧪 Acceptance Criteria & Verification Checks - [ ] `vif_report.csv` lists all features with before/after VIF and a justified action per feature. - [ ] Final model has max VIF < 5 and a before/after coefficient-stability table in the notebook. - [ ] `tenure × plan` interaction tested with an F-test; one interaction coefficient interpreted correctly. - [ ] Residual diagnostic trio (residual-vs-fitted, Q-Q, scale-location) present with verdicts. - [ ] Memo names three levers with CIs and explicitly disclaims causal reading. --- ## 📊 Evaluation Rubric (100 points — passing ≥ 80) | # | Criterion | Points | What earns full mark