Kish Unequal-Weight Effective Sample Size (n_eff) Formula

Why raking or propensity weights shrink precision from nominal N to Kish n_eff

Even in an unclustered sample, post-stratification raking or inverse-propensity weighting assigns unequal weights w_i across respondents, inflating sampling variance whenever a few observations carry huge weights. Kish (1965, 1992) showed that under homoskedastic outcomes uncorrelated with weights, variance inflates by DEFF_w = 1 + CV_w^2 = (N * SUM(w_i^2)) / (SUM w_i)^2, reducing effective sample size to n_eff = (SUM w_i)^2 / SUM(w_i^2)!

Audit N = 6 weighted respondents B2:B7 = {1, 1, 1, 1, 2, 6}: total weight is SUM(w) = 12 ((SUM w)^2 = 144), while squared weights sum to 1+1+1+1+4+36 = 44. Kish effective sample size drops to n_eff = 144 / 44 = 3.2727 (weighting efficiency 3.2727 / 6 = 0.5455, or 45.45% precision loss!) with DEFF_w = 6 / 3.2727 = 1.8333. Trimming or equalizing weights to {1,1,1,1,1,1} restores full n_eff = 6.0000 and DEFF_w = 1.0000.

Kish Unequal-Weight Effective Sample Size n_eff = (SUM w)^2 / SUM(w^2) Formula

Notice that Kish n_eff is completely scale-invariant: multiplying every weight by 1,000 leaves (SUM w)^2 / SUM(w^2) unchanged! Guard against negative weights or zero total weight.

=LET(w_col, B2:B7, sum_w, SUM(w_col), IF(OR(MIN(w_col) < 0, sum_w <= 0), "Invalid Survey Weights", ROUND((sum_w^2) / SUMPRODUCT(w_col, w_col), 4)))

Unequal-Weighting Design Effect DEFF_w = (N * SUM(w^2)) / (SUM w)^2 Formula

Monitor DEFF_w during survey raking: when DEFF_w exceeds 2.0 (efficiency < 50%), analysts routinely cap extreme weights at the 95th or 99th percentile and check bias-variance trade-offs.

=LET(w_col, B2:B7, n_nom, ROWS(w_col), sum_w, SUM(w_col), IF(OR(MIN(w_col) < 0, sum_w <= 0), "Invalid Survey Weights", ROUND((n_nom * SUMPRODUCT(w_col, w_col)) / (sum_w^2), 4)))

Further reading

• Arte Sheets Lesson — Kish Cluster Design Effect (DEFF) and Intracluster Correlation
• Arte Sheets Lesson — Horvitz-Thompson vs Hajek Inverse Probability Weighting (IPW) ATE