4. Simulation Study

Benmei Liu, Partha Lahiri and Graham Kalton

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4.1 The Study Population and the Sample Design

This section describes the simulation study that was conducted to compare the efficiency of the small area estimates produced by the four HB models. The simulation study was based on the 2002 Natality public-use data file that covered all births occurring within the United States in that calendar year. The file contained data obtained from the certificates filed for births occurring in each state and territory (for details see U.S. National Center for Health Statistics 2009).

The finite population studied was restricted to the 4,024,378 records of live births that occurred in 2002 in the 50 states of U.S. and the District of Columbia (DC) and that had birth weights reported. The parameter of interest is the state level low birthweight rate $P_i$, $i=1,\mathrm{...},51$, where low birthweight is defined as less than 2,500 grams. The value of $P_i$ varied from 5 percent to 11 percent across the states.

Within each state, a stratified SRS design was used to draw samples from the birth records. Mother's race (White, Black, and Other) was used as the stratification variable. The national sample size was set to be about 1,500 birth records for each race group. A uniform sampling fraction was used across the states for each race group, subjecting to the condition that at least two birth records were sampled within each race group in each state. The resultant national sample size turned out to be $n=4,526$ birth records. The state sample sizes $n_i$ ranged from 7 (for small states such as Vermont) to 690 (for California), with a median sample size of 61. This sampling procedure was repeated $R=1,000$ times, creating 1,000 independent sample data sets. The sampling weights remained the same over different simulation runs.

4.2 Computation of the HB Estimates

For simplicity, the following assumptions were made for the HB models:

1. No auxiliary variables were used, so that $x_i\text{'}\beta =\mu$.
2. For Models 1 and 2, the sampling variances were taken to be given by ${\psi }_i=[p_w\text{(}1-p_w\text{)/}{n}_i\text{]}def{f}_{iw},$ where $p_w={\displaystyle \sum {\displaystyle \sum {\displaystyle \sum w_{ih}{y}_{ihk}}}}/{\displaystyle \sum {\displaystyle \sum n_i{w}_{ih}}}$ is the national estimate of the proportion of low birthweight live births. (A check on the use of $def{f}_{iw}$ as an approximation for $DEF{F}_i$ showed that the approximation was reasonable: the two quantities were close, with a product moment correlation of 0.96 and an average ratio of 1.08 between $def{f}_{iw}$ and $DEF{F}_i$.)
3. Flat prior for $\mu$, i.e., $f\text{(}\mu \text{)}\propto 1,$ and inverse gamma for ${\sigma }_v^2$, i.e., ${\sigma }_v^2~IG\text{(}0.001,0.001\text{)}$.

For each sample data set, the first step in the computations was to calculate the state direct sample estimates. The estimates for each sample data set were then used in turn as input to the WinBUGS software (Lunn, Thomas, Best and Spiegelhalter 2000), which was used to produce the HB estimates for all four models.

In a sizable number of the states with small $n_i$, the direct estimates were zero in some of the sample data sets. Since WinBUGS can handle direct estimates of zero only for Model 1, the zero direct estimates were perturbed to very small positive numbers for the other models.

For each WinBUGS run, three independent chains were used. For each chain, burn-ins of 10,000 samples were produced, with 10,000 samples after burn-in. The samples after burn-in were thinned by a factor of two to reduce auto-correlation of the MCMC samples. The resultant 15,000 MCMC samples from the three chains after burn-in were then used to compute the posterior mean and percentiles for each HB model based on each sample data set. The potential scale reduction factor $\widehat{R}$ was used as the primary measure for convergence (see Gelman and Rubin 1992). The WinBUGS code is given in Appendix B.

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