Evaluate channels and allocate budget

You have a daily aggregated marketing dataset broken down by channel, with columns for spend, visits, form_starts, form_completes, loan_funded, and revenue.

Tasks:

1. Construct a funnel showing conversion rates between stages and CAC at each meaningful milestone.
2. Calculate ROAS and payback using both last-click and 7-day first-touch attribution, and explain how you would re-attribute if only raw click/impression logs were available.
3. Recommend channel-level budget shifts for the next month using a simple response curve such as a log or Hill function, and estimate the marginal ROAS.
4. Design an incrementality test using a geo or time-based holdout to validate the recommendations, covering the randomization unit, sample size, contamination risks, and success criteria.
5. Flag channels that appear effective but are likely non-incremental because of retargeting or brand spillovers, and suggest diagnostics to detect this.

Give concrete formulas, state any assumptions you need, and describe how you would communicate the recommendation with risk bounds.

Overview: This question assesses marketing analytics and experimentation skills for a Data Scientist role, spanning funnel analysis, conversion and CAC metrics, attribution modeling, response-curve-based budget allocation, and incrementality test design using channel-level spend and outcome data.