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State the covariance model
Suppose n measurements each have variance sigma² and every distinct pair has covariance rho × sigma². This is an equicorrelation illustration, not a description of every cluster or time series. The admissible correlation range must make the covariance matrix valid.
Include all pairwise covariance terms
The variance of the mean is sigma²[1 + (n − 1)rho]/n. For n = 10 and rho = 0.2, the bracket is 2.8. The independent formula sigma²/10 would omit those positive covariance contributions.
Define a limited equivalent count
Matching this variance to sigma²/n_eff gives n_eff = n/[1 + (n − 1)rho]. In the stated example, n_eff ≈ 3.5714. It is a variance-equivalent count for this estimator and model, not a new number of observed rows or a universal degrees-of-freedom formula.
| rho | Variance multiplier versus independence | n_eff |
|---|---|---|
| 0 | 1 | 10 |
| 0.2 | 2.8 | 3.5714286 |
| 1 | 10 | 1 |
Use the actual design for a real uncertainty estimate
Within-branch relationships, unequal group sizes and temporal dependence can require a more specific covariance calculation. Do not substitute a guessed rho or this effective count into every test. The illustration explains why collecting many related records can add less information than independent records.