In sae4health: Small Area Estimation for Key Health and Demographic Indicators from Household Surveys

Area-level (Fay-Herriot) Model

Fay-Herriot models provides smoothed estimates at the areal level using direct estimate $\hat p^{W}_{i}$ as input. The direct estimates are modeled as a noisy observation of the true prevalence, with the variance of noise determined by the design-based variance. We consider the spatial Fay-Herriot model for the logit transformed direct estimates, which is defined as follows:

$$\text{logit}(\hat p^{W}{i})|\lambda{i} \sim\textrm{Normal}(\lambda_{i}, V_{i}^{HT}),$$ $$\lambda_{i}= \alpha + e_i+S_i.$$ Here $\text{expit}(\lambda_{i})$ is the latent true prevalence, and $e_i$ and $S_i$ are unstructured and structured spatial random effects. Inference is carried out using Bayesian methods and so the model specific is completed by priors on $\alpha$, $e$ and $S$, and their hyperpriors. More details of the Bayesian model setup can be found in Wakefield, Okonek, and Pedersen (2020). Area level Fay-Herriot model are viewed as the most reliable model choice, since they acknowledge the design through the sue of a weighted estimate and its associated variance. See chapters 4 to 6 of Rao and Molina (2015).

As of now the package allows only an overall intercept $\alpha$, but future versions of the package will allow area level covariates to be included. The default prior for the intercept is $N(0, 1000)$. The structured and non-structured random effects are implemented used Besag-York-Mollié (BYM) via BYM2 parameterization, with default PC priors such that the marginal standard deviation has a prior such that $Prob(\sigma > 1) = 0.01$ and the proportion of variation explained by the spatial effect, $\phi$ has a prior such that $Prob(\phi \gt 0.5) = \frac{2}{3}$ (Riebler et al. 2016; Simpson et al. 2017).

The app implements spatial Fay-Herriot models at different administrative levels using surveyPrev::fhModel() and internally SUMMER::smoothSurvey().

A Fay-Herriot model at fine spatial level (over Admin-2) can be fitted by treating areas without direct estimates as missing data, though it is usually not recommended due to data sparsity. In addition, numerical issue arise when design-based variance of direct estimate is close to zero and logit precision become too large. Our implementation fixes this issue by identifying regions with too small design variance(< 1e-30), and deleting the clusters in these regions, before fitting the model. However, this step creates additional bias in the results if the number of clusters deleted is large.

sae4health documentation built on June 8, 2025, 10:43 a.m.