3. Models Studied

Benmei Liu, Partha Lahiri and Graham Kalton

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A general area-level small area model has two components. One$—$the sampling model$—$is a model for the sampling error of the direct survey estimates. The other$—$the linking model$—$relates the population value for an area to area-specific auxiliary variables $x_i={\text{(}{x}_{i1},\mathrm{...},x_{ip}\text{)}}^{\text{'}}$ .

Section 3.1 describes two area models that are often used for estimating small area proportions and Section 3.2 outlines some problems associated with these models. Section 3.3 describes two alternative models that may serve to address these problems.

3.1 Two Commonly Used Models

We study two commonly used models for comparison with the new models described in Section 3.4. The first is the Fay-Herriot model (Fay and Herriot 1979), which assumes known sampling variances and normal distributions for both the sampling and the linking models. The second is the normal-logistic model, which differs from the Fay-Herriot model only by the replacement of a logit-normal distribution for the normal distribution in the linking model.

Model 1: (Fay-Herriot normal-normal model)

Sampling model:

$$p_{iw}|P_i\stackrel{ind}{~}N\text{(}{P}_i,{\psi }_i\text{) }(3.1)$$

Linking model:

$$P_i|\beta ,{\sigma }_v^2\stackrel{ind}{~}N\text{(}{x}_i^{\text{'}}\beta ,{\sigma }_v^2\text{) (3}\text{.2)}$$ 

Model 2: (normal-logistic model)

Sampling model:

$$p_{iw}|P_i\stackrel{ind}{~}N\text{(}{P}_i,{\psi }_i\text{) }(3.3)$$

Linking model:  

$$g\text{(}{P}_i\text{)}|\beta ,{\sigma }_v^2\stackrel{ind}{~}N\text{(}{x}_i^{\text{'}}\beta ,{\sigma }_v^2\text{) }(3.4)$$

In both models the sampling variance ${\psi }_i$ is assumed to be known. Model 1 is referred as a matched model because the sampling and linking models can be combined to produce a relatively simple linear mixed model. However, a nonlinear linking model is often preferred for modeling proportions, leading to unmatched sampling and linking models, as in Model 2 (see, for example, You and Rao 2002). The link function $g\text{(}\cdot \text{)}$ can be empirically determined by checking the model fit. The log and logit link functions have been used. The $\text{logit(}{P}_i)$ linking model is chosen here in order to guarantee that the estimate of $P_i$ always falls within the allowable range of (0,1).

3.2 Issues with Models 1 and 2

There are two main issues associated with Models 1 and 2. The first is that both models assume known sampling variances ${\psi }_i$, whereas in practice they have to be estimated. A simple approach is to use the direct variance estimate but that estimate is very imprecise when $P_i$ is either very small or very large and when the sample size $n_i$ is small. An alternative, more complex, approach is to develop an approximate estimate of $P_i$, say $p_{isyn}$, from a simple model such as a logistic model for $p_{iw}$ in terms of the auxiliary variables, and then use that estimate in the following synthetic variance estimator:

$${\mathrm{var}}_{stsyn}=\frac{p_{isyn}\text{(}1-p_{isyn}\text{)}}{n_i}def{f}_{iw}\text{. }(3.5)$$

When there are no auxiliary variables available, the overall sample proportion may be used for $p_{isyn}$ in the computation of the synthetic variance estimator.

The second issue concerns the normality assumption in the sampling model, which is based on a large sample approximation. As noted in Section 1, when the sample size $n_i$ is small and $P_i$ is near 0 or 1, as is often the case with small area estimation, that assumption is problematic.

3.3 Two Alternative Models

Under Models 1 and 2, the unknown sampling variances ${\psi }_i$ are estimated in some way, and then the resultant estimates are treated as if they were the known true values. A possible alternative approach is to treat the ${\psi }_i$ as unknown parameters in the HB model, as has been done in a number of studies. For example, Arora and Lahiri (1997) applied an HB model to model the design-based variances for the sample estimates. Singh, Folsom and Vaish (2005) proposed the use of a generalized design effect model to smooth the sampling covariance matrix in small area modeling with survey data. Recently, You (2008) proposed the use of equal design effects over time to model the sampling variances in estimating small area unemployment rates using a cross-sectional and time series log-linear model. The approach of treating the sampling variances ${\psi }_i$ as unknown is adopted in Model 3, as a variant of Model 2. One approach for addressing the non-normality of the sampling distributions of the survey-weighted small area proportions is to replace the normal distribution assumption by an alternative distribution. That approach is applied in Model 4 with the assumption of a beta sampling distribution, a distribution that has the desirable property of having a (0,1) range. In other regards Model 4 is the same as Model 3, including treating the ${\psi }_i,i=1,\mathrm{...},m$ as unknown parameters. Model 4 was previously considered by Jiang and Lahiri (2006b) in an illustrative example to estimate finite population domain means using an EBP approach.

Model 3: (normal-logistic model with unknown sampling variance)

Sampling model:

$$p_{iw}|P_i\stackrel{ind}{~}N\text{(}{P}_i,{\psi }_i\text{) (3}\text{.6)}$$

Linking model:

$$logit\text{(}{P}_i\text{)}|\beta ,{\sigma }_v^2\stackrel{ind}{~}N\text{(}{x}^{\text{'}}\beta ,{\sigma }_v^2\text{) }(3.7)$$

Model 4: (beta-logistic model with unknown sampling variance)

Sampling model:

$$p_{iw}|P_i\stackrel{ind}{~}beta\text{(}{a}_i,b_i\text{) }(3.8)$$   

Linking model:

$$logit\text{(}{P}_i\text{)}|\beta ,{\sigma }_v^2\stackrel{ind}{~}N\text{(}{x}_i^{\text{'}}\beta ,{\sigma }_v^2\text{) }(3.9)$$

For both Model 3 and Model 4, the approximate variance function ${\psi }_i=\text{[}{P}_i\text{(}1-P_i\text{)/}{n}_i\text{]}def{f}_{iw}$ is used. The parameters $a_i$ and $b_i$ in Model 4 are given by:

$$a_i=P_i\left(\frac{n_i}{def{f}_{iw}}\text{-}1\right)\text{, and }{b}_i=\text{(}1-P_i\text{)}\left(\frac{n_i}{def{f}_{iw}}-1\right).$$

HB small area estimates can be computed from all four models using the Metropolis-Hastings algorithm within the Gibbs sampler. Details of the algorithm, which draws random samples based on the full conditional distributions of the unknown parameters starting with one or multiple sets of initial values, are given by Robert and Casella (1999) and Chen, Shao, and Ibraham (2000). You and Rao (2002) also describe in detail how the Metropolis-Hastings algorithm works within the Gibbs sampler for models similar to Models 1 and 2. The algorithm works for Models 3 and 4 in the same way as for Model 2. The full conditional distributions under each model are provided in Appendix A.

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