Optimality of Equal vs. Unequal Cluster Sizes in Multilevel Intervention Studies: A Monte Carlo Study for Small Sample Sizes

Abstract

Optimality of equal versus unequal cluster sizes in the context of multilevel intervention studies is examined. A Monte Carlo study is done to examine to what degree asymptotic results on the optimality hold for realistic sample sizes and for different estimation methods. The relative D-criterion, comparing equal versus unequal cluster sizes, almost always exceeded 85%, implying that loss of information due to unequal cluster sizes can be compensated for by increasing the number of clusters by 18%. The simulation results are in line with asymptotic results, showing that, for realistic sample sizes and various estimation methods, the asymptotic results can be used in planning multilevel intervention studies.

Keywords:

• D-optimality
• D _s -optimality
• Mean squared error
• Multilevel intervention studies
• Relative efficiency
• (Restricted) maximum likelihood
• Unequal cluster sizes

Mathematics Subject Classification:

• Primary 62K05
• Secondary 65C05

Notes

Note 1. A more detailed description of the different distributions can be found in Table 2.

Note 2. Since a numerical investigation based on asymptotic results showed that a variance ratio of 1 yielded the minimum relative efficiency in terms of D _s (fixed), the Monte Carlo simulations for small samples were limited to a variance ratio of 1.

Note 1: f _a  = number of clusters of size g _a (small), f _b  = number of clusters of size g _b (medium), f _c  = number of clusters of size g _c (large)

Note 2: Cluster sizes and cluster frequencies are chosen such that the total number of clusters is equal to 42, the average cluster size is 24, the difference between the largest and smallest cluster size is equal to 36 (except for the negatively skewed distribution, where the range is 28 to prevent g _a  < 0) and all cluster size frequencies and cluster sizes are even.