Neyman Allocation

Stratified samples are commonly used when supplementary information is available to help with sample design. The precision of a stratified design is influenced by how the sample elements are allocated to strata. Neyman allocation is a method used to allocate sample to strata based on the strata variances and similar sampling costs in the strata. A Neyman allocation scheme provides the most precision for estimating a population mean given a fixed total sample size.

For stratified random sampling, the population is divided into H mutually exclusive strata. In each stratum, a simple random sample is drawn without replacement. Neyman allocation assigns sample units within each stratum proportional to the product of the population stratum size (Nh) and the within-stratum standard deviation (Sh), so that minimum variance for a population mean estimator can be achieved. The equation for Neyman allocation is

where nh is the sample size for stratum h and n is the fixed total sample size. The effect of Neyman allocation is to sample more heavily from a stratum when (a) the population size of the stratum is large; (b) the variability within the stratum is large, so that the heterogeneity needs to be compensated.

Of note, Neyman allocation is a special case of optimal allocation whose objective in sample allocation is to minimize variance of an estimator for a population mean for a given total cost. It is employed when the costs of obtaining sampling units are assumed to be approximately equal across all the strata. If the variances are uniform across all the strata as well, Neyman allocation reduces to proportional allocation where the number of sampled units in each stratum is proportional to the population size of the stratum. When the variances within a stratum are different and are specified correctly, Neyman allocation will give an estimator with smaller variance than proportional allocation.

The major barrier to the application of Neyman allocation is lack of knowledge of the population variances of the study variable within each stratum. In some situations, historical estimates of strata variances can be used to provide good approximation to Neyman allocation for the current survey sample. For example, the Medical Expenditure Panel Survey Insurance Component (MEPS IC) is an annual survey of establishments that collects information about employer-sponsored health insurance offerings. To implement Neyman allocation, stratum variance estimates were obtained from the 1993 National Employer Health Insurance Survey for the initial MEPS IC 1996 and later from prior MEPS IC surveys.

In situations where estimated population variances within each stratum are not easily available, an alternative is to find a surrogate variable (a proxy) that is closely related to the variable of interest and use its variances to conduct a Neyman allocation. For example, the U.S. Government Accountability Office conducted a survey in 2004–2005 to estimate the average and median purchase prices of specified covered outpatient drugs (SCODs) in a population of 3,450 hospitals. Since a direct measure of purchase prices for SCODs was not available at the time of sample selection, the total hospital outpatient SCOD charges to Medicare was used as a proxy to carry out the Neyman allocation.

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