Statistical Analysis
Kish Design Effect (DEFF), Cluster ICC & Effective Sample Size Formula
Deflate nominal survey and cluster-randomized sample size N using Leslie Kish’s (1965) cluster DEFF_c = 1 + (m - 1)*rho and unequal-weighting DEFF_w = 1 + CV(w)^2.
Why 1,000 clustered, weighted respondents can carry the statistical precision of only 400
Treating 1,000 observations sampled across 40 schools, clinics, or retail stores as 1,000 independent simple-random draws severely understates standard errors! Even a tiny intracluster correlation rho = 0.04 across average cluster size m = 26 inflates variance by Leslie Kish’s (1965) cluster design effect DEFF_c = 1 + (m - 1) * rho = 1 + 25 * 0.04 = 2.00.
When post-stratification or non-response weights w_i vary with coefficient of variation CV_w = SD(w) / Mean(w) = 0.50, Kish’s unequal-weighting factor adds a multiplicative penalty DEFF_w = 1 + CV_w^2 = 1.25. Combined DEFF = 2.00 * 1.25 = 2.50 cuts effective independent sample size to N_eff = 1,000 / 2.50 = 400 and widens confidence-interval half-widths by DEFT = SQRT(2.50) = 1.5811 (+58.11%).
| Survey Design Diagnostic | Kish Formula | Sample (N=1,000, m=26, rho=0.04, CVw=0.50) |
|---|---|---|
| Cluster Design Effect DEFF_c | 1 + (m - 1) * rho | 2.0000 |
| Weighting Design Effect DEFF_w | 1 + CV_w ^ 2 | 1.2500 |
| Combined DEFF & Root DEFT | DEFF_c * DEFF_w | SQRT(DEFF) | 2.5000 | 1.5811x SE multiplier |
| Effective Sample Size N_eff | N_nominal / DEFF_total | 400.00 independent equivalents |
Combined Cluster & Weighting Design Effect (DEFF) Formula
Multiply cluster variance inflation (1 + MAX(0, m - 1) * rho) by Kish unequal-weight inflation (1 + CV_w^2).
=ROUND((1 + MAX(0, B3 - 1) * B4) * (1 + (B5 ^ 2)), 4)Effective Sample Size N_eff & Standard-Error Multiplier DEFT
Divide nominal sample size B2 by combined DEFF for N_eff, and take SQRT(DEFF) for confidence-interval multiplier DEFT.
=ROUND(B2 / ((1 + MAX(0, B3 - 1) * B4) * (1 + (B5 ^ 2))), 2)Try the example
Use the original values or change them to test your understanding.