Model-assisted optimal allocation for planned domains using composite estimation 4. Numerical studyModel-assisted optimal allocation for planned domains using composite estimation 4. Numerical study

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We use data on the 26 cantons of Switzerland (Longford 2006); their population sizes range from 15,000 (Appenzell-Innerrhoden) to 1.23 million (Zürich). The population of Switzerland is 7.26 million. We assume that $n=\text{10,000},\rho =0.025$ and $\sigma /\mu =1$ (following Longford 2006). The last assumption only affects the magnitude of $F$ and the relative root mean squared errors (RRMSE) but not the relativities across methods. It is satisfied if, for example, a prevalence rate of 50% is estimated. All calculations were performed in the R statistical environment (R Development Core Team 2012). Values of $q=0,\mathrm{0.5,}\mathrm{1,}1.5$ and $2,$ and values of $G=0,10$ and $100$ were used, as in section 5.2 of Choudhry et al. (2012). The program used to produce all results is available in the appendix of Molefe and Clark (2014).

Six different allocations are evaluated in Tables 4.1-4.3. The value of $F$ is shown for each design, relative to the value for equal allocation. Strata sample sizes were constrained in all allocations to lie between 1 and the population sizes, while still summing to $n.$ The first design is equal allocation, then proportional allocation. The third design is the optimal design, which minimizes $F$ in (3.1) by NLP subject to all stratum sample sizes being at least 1. The fourth design minimizes $F$ subject to all stratum RRMSEs being 8% or less, which, from formula (3.1), is equivalent to a minimum stratum sample size of 113. For the third and fourth designs, NLP was carried out using the R package $Rsolnp$ (Ghalanos and Theussl 2011). The fifth design is power allocation, where the exponent $p$ is calculated to minimize $F.$ The sixth design is power allocation with all stratum sample sizes constrained to be 113 or more, and with $p$ calculated to minimize $F$ reflecting these constraints. In both the fifth and sixth cases, $p$ was calculated using the optimize function in R.

Table 4.1 shows the efficiency of the various methods when $G=0,$ where efficiency refers to the achieved values of $F$ from formula (3.1), which is a weighted combination of MSEs of area composite estimators and an overall grand mean estimator. When $q=0,$ equal allocation is then optimal for $F,$ and all of the allocation methods except proportional allocation return equal allocation. For larger values of $q,$ Optimal for Composite is the most efficient, as expected. Imposing the area maximum RRMSE constraint of 8% increases $F$ by 4% when $q=2,$ and has negligible effect (1.4% or less) for smaller $q.$ The optimal power allocation has virtually identical efficiency to the optimal-for-composite allocation, both with and without the area RRMSE constraint. The unconstrained optimal-for-composite and power allocations are more efficient than proportional allocation when $q$ is small, and about equally efficient for $q\ge \mathrm{1.5.}$ When the area RRMSE constraint is imposed, these designs suffer a small penalty, but are still more efficient than proportional except when $q=2.$

Table 4.2 shows relative efficiencies for $G=10.$ As for when $G=0,$ the optimal-for-composite and optimal power designs perform very similarly, with a similar effect of imposing the area RRMSE constraint. The major difference compared to $G=0$ is that proportional allocation is more efficient when $G$ is larger. The optimal designs, even with the constraint imposed, remain more efficient than proportional allocation except for $q\ge \mathrm{1.5.}$

Table 4.3 shows efficiencies for large $G\left(G=100\right).$ Here, proportional allocation is close to the best design for all $q.$ It is about equivalent to the unconstrained optimal designs for all $q\ge 0.5,$ and more efficient than the constrained optimal designs for all $q\ge 1.$ The relative performance of the four optimal designs is about the same as for $G=0$ and $G=10.$

Figure 4.1 shows the distribution of the area RRMSEs across the 26 cantons for $q\in \text{{0}\text{.5},\mathrm{1,}\mathrm{1.5,}\text{2}}$ when $G=0$ for the four optimal designs. The results for $q=0$ are not shown because the canton sample sizes are then all equal for the optimal designs. The optimal for composite allocation (top left) shows a fairly tight range of area RRMSEs when $q=0.5,$ becoming more dispersed as $q$ increases. The maximum RRMSEs are 6.6%, 9.4%, 13.8% and 15.6% for $q=0.5,\mathrm{1,}1.5$ and $2,$ respectively. Thus, for $q\ge 1,$ some of the RRMSEs are undesirably large. The optimal for composite allocation with constraints forces all area RRMSEs to be 8% or less, shown by the top right panel. The bottom two panels show the corresponding optimal power allocations. The unconstrained power allocation is broadly similar to the unconstrained optimal for composite allocation, but less dispersed, with lower maximum area RRMSEs. The two constrained designs are very similar.

Table 4.4 shows the values of the optimal exponents calculated for the optimal power designs for each $q$ and $G.$ When $G$ is 0 or 10, the optimal exponent $p$ of the power allocation is very close to $q/2,$ where $q$ is the exponent in the definition of $F$ in (3.1). For $G=100,$ the optimal exponent is quite close to 1, reflecting that for large $G,F$ essentially reflects the variance of the grand mean, so that proportional allocation is nearly optimal. Table 4.5 shows the optimal power exponents when the area RRMSE constraints are applied. Applying these constraints has little effect on the optimal $p.$

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