2. Notation

Benmei Liu, Partha Lahiri and Graham Kalton

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Let $N_{ih}$ denote the population size in stratum $h$ in area $i$ of a finite population $(i=1,\mathrm{...},m;h=1,\mathrm{...},H_i).$ Let $y_{ihk}$ be the binary response for the characteristic of interest for unit $k$ in stratum $h$ in area $i$ $\text{(}k=1,\mathrm{...},N_{ih}\text{)}$. The parameters to be estimated are the small area proportions $P_i={\displaystyle {\sum }_h{\displaystyle {\sum }_k{y}_{ihk}/N_{ih}}}$.

With the stratified simple random sample design under study, $n_{ih}$ units are selected from the $N_{ih}$ units in stratum $(ih)$. The standard direct survey estimator for $P_i$ is:

$$p_{iw}=\frac{{\displaystyle {\sum }_h^{H_i}{\displaystyle {\sum }_k^{n_{ih}}{w}_{ih}{y}_{ihk}}}}{{\displaystyle {\sum }_h^{H_i}{\displaystyle {\sum }_k^{n_{ih}}{w}_{ih}}}},i=1,\mathrm{...},m;(2.1)$$

where $w_{ih}$ denotes the sampling weight given by $w_{ih}=N_{ih}/n_{ih}.$

The variance of $p_{iw}$ can be expressed as

$$VA{R}_{st}\text{(}{p}_{iw}\text{)}=\frac{P_i\text{(}1-P_i\text{)}}{n_i}DEF{F}_i\text{, (}2.2)$$

where $DEF{F}_i$ is the design effect reflecting the effect of the complex sample design (Kish 1965). For a stratified simple random sample with negligible sampling fractions in all strata, the design effect is given approximately by:

$$DEF{F}_i=\frac{{\displaystyle {\sum }_h{W}_{ih}^2{P}_{ih}\left(1-P_{ih}\right)/n_{ih}}}{P\left(1-P_i\right)/n_i}\text{, (2}\text{.3)}$$

where $W_{ih}=N_{ih}/N_i$, $N_i={\displaystyle {\sum }_h{N}_{ih}}$, $n_i={\displaystyle {\sum }_h{n}_{ih}}$, and $P_{ih}$ is the population proportion in stratum $h$ in area $i$.

The design effect $DEF{F}_i$ is a function of the $P_{ih}$, which are unknown. If $P_{ih}(1-P_{ih})\approx P_i(1-P_i)$, $DEF{F}_i$ can be approximated by $def{f}_{iw}=n_i{\displaystyle {\sum }_h{W}_{ih}^2/n_{ih}}$. The value of $def{f}_{iw}$ can be readily computed since it does not depend on any unknown parameters.

Small area estimation procedures can be used to address the problem that $p_{iw}$ is very imprecise when the sample size $n_i$ is small. Section 3 describes the HB area level models investigated in this study.

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