Prerequisites
Variance and covariance, independent experimental units, and random-intercept models.
In y_ij=mu+u_i+e_ij, observations share u_i within group i. Independent group effects with variance a and independent residuals with variance b give Var(grand mean)=a/n+b/(nm) for n equal groups of m observations. More observations within one group do not create new independent groups.
Worked example
For n=3, m=5, a=6, b=10, the grand-mean variance is 6/3+10/15=8/3. Treating all 15 values as independent would incorrectly omit their within-group covariance.
Model and Assumptions
Fictional plant-height study: four independently selected trays, ten plants per tray. A schedule would be assigned at tray level. Assume u_i and e_ij are mutually independent zero-mean terms, with between-tray variance a=9 and residual variance b=16 in squared height units.
Further inquiry
Derive the grand-mean variance by summing all covariance terms. Relate it to the design effect 1+(m-1)rho, then discuss why unequal tray sizes, shared tray histories, or estimated variance components require further analysis.
Review criteria
- Count within-tray covariances rather than assuming nm independent values.
- Recover design effect 4.24 for m=10 and rho=0.36.
- Separate the stated variance calculation from a justified treatment comparison.