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Unit-level Model

General idea

This SAE method was introduced by Battese et al. 1 and is the closest to a conventional model-based approach, since a sampling model is used at the cluster-level, specifically an overdispersed binomial model. As with the area-level model, the area-specific prevalences across areas are linked via a hierarchical model. Again, the rationale is that the similarity of prevalences across the study region may be leveraged to fine-tune the estimates in each area. Since data from all areas is used for estimation in each area, this is another example of the production of “indirect” estimates.

Advantages

  • Sparse Data Handling: If the data are sparse, this is the only method that can be used, since the cluster-level overdispersed binomial model can deal with situations in which the responses are either all 0s or all 1s (this sparse data case leads to problems for methods based on weighted estimates since the estimates themselves are 0 or 1 and the methods collapse).
  • Efficiency: If the sampling model is a reasonable approximation to the true underlying data generating mechanism, the approach is an efficient use of the data.
  • Uncertainty Reduction: By modeling all the data in unison, uncertainty in each area is reduced (on average).
  • Reasonable Assumption: The model that links the areas together is an additional assumption, when compared to the direct estimates, but this is typically not a strong assumption.
  • Mean-squared Error Reduction: The linking model will shrink high and low weighted prevalences toward the national estimate, which is desirable when the data are sparse, but this does lead to (frequentist) bias. But the rationale for the approach is that the gain in precision will offset the shrinkage bias. Specifically, there is theory that suggests that on average the mean-squared error (\(\text{bias}^2\)+variance) will be reduced for estimates from hierarchical models, when compared to direct estimates.

Disadvantages

  • Sampling Model: The assumption of a sampling model for the cluster-level responses means this approach is fundamentally different from the direct and area-based methods which use weighted estimates.
  • Extra Design Adjustment Needed: Since the weights are not used, one must adjust for the design within the model specification by modeling the association between prevalence and urban/rural and including a cluster effect in the sampling model.
  • Over-Shrinkage: When the data are sparse, the estimates may be shrunk too much since the data are not sufficiently informative to discriminate from the “flat” map case.

Mathematical details

Unit-level models assume smoothing models for counts of events in each cluster (Wakefield, Okonek, and Pedersen, 20202; Li et al. 20203). In terms of traditional Small Area Estimation literature, cluster-level models are a type of unit-level model. Currently, the app implements an unstratified model without taking into account the urban/rural stratification in the sampling design.

Let \(Y_c\) be the number of events in cluster \(c\), and \(n_c\) be the number of individuals at risk, where \(c= 1,\dots,C\). The unstratified model assumes the hierarchical structure:

\[Y_c \mid p_c,d\sim \textrm{BetaBinomial}(n_c,p_c,d),\]
\[p_c=\textrm{expit}(\alpha+e_{i[s_c]}+S_{i[s_c]}),\]

where \(\alpha\) is the intercept, and \(i[s_c]\) indexes the area within which the cluster \(s_c\) resides. Similar to the area-level model, \(e_i\) and \(S_i\) are unstructured and structured spatial random effects with the same prior as before. The Beta-binomial distribution arise from a hierarchical model where the probability follows a \(\text{Beta}(a, b)\) prior. The overdispersion parameter, \(d=\frac{1}{\alpha+\beta+1}\), is between 0 and 1 and represent the the intracluster correlation between Bernoulli draws within cluster.

The default prior for \(d\) is \(\text{logit}(d) \sim \text{Normal}(0,0.4)\).

The app implements the unit-level model via surveyPrev::clusterModel() function, with BYM2 model for the spatial random effects. The app currently only supports unstratified models. Please refer to surveyPrev package for stratified unit-level model.

  1. George E Battese, Rachel M Harter, and Wayne A Fuller. An error-components model for prediction of county crop areas using survey and satellite data. Journal of the American Statistical Association, 83:28–36, 1988. ↩

  2. Jon Wakefield, Taylor Okonek, and Jon Pedersen. Small area estimation for disease prevalence mapping. International Statistical Review, 88:398–418, 2020. ↩

  3. Zehang R Li, Bryan D Martin, Tracy Q Dong, Geir-Arne Fuglstad, Jessica Godwin, John Paige, Andrea Riebler, Samuel Clark, and Jon Wakefield. Space-Time Smoothing of Demographic and Health Indicators using the R Package SUMMER. 2020. ↩