Compare first-score vs all-scores estimators

You have two candidate estimators for survey quality based on the score column over 2025-08-26 to 2025-09-01: - E_first: For each user×survey pair, take the first score in-window; then average across pairs. - E_all: Average across all in-window scores (users with more responses contribute more weight). Answer precisely: 1) Define the target estimand and write E_first and E_all formally. Show that E_all is a weighted average of per-user means with weights proportional to each user’s in-window response count. Under what conditions are E_first and E_all equal in expectation? 2) Discuss bias risks for E_all when the number of responses per user correlates with their latent satisfaction. Provide a concrete example where E_all over- or under-estimates the estimand. 3) Derive large-sample variance estimators for both approaches. For E_all, propose user-cluster-robust standard errors; for E_first, standard IID SEs over user×survey pairs. Explain when cluster-robust SEs are required for E_first. 4) Suppose repeated scores within a user×survey pair follow an exchangeable correlation with intracluster correlation ρ and average cluster size m. Show how the design effect DE = 1 + (m−1)ρ affects effective sample size and confidence intervals for E_all. 5) Recommend which estimator to report by default and when to prefer the alternative. Justify using bias–variance tradeoffs and interpretability.