Small area model for binary and count data

Let be the value of the outcome of interest, a discrete or a categorical variable, for unit j in area i, and let denote a vector of unit level covariates (including an intercept). Working within a frequentist paradigm, one can follow Jiang and Lahiri (2001) who propose an empirical best predictor (EBP) for a binary response, or Jiang (2003) who extends these results to generalized linear mixed models (GLMMs). Nevertheless, use of EBP can be computationally challenging (Molina and Rao, 2010). Despite their attractive properties as far as modelling non-normal outcomes is concerned, fitting GLMMs requires numerical approximations. In particular, the likelihood function defined by a GLMM can involve high-dimensional integrals which can- not be evaluated analytically (see Mc Culloch 1994, 1997; Song et al. 2005). In such cases numerical approximations can be used, as for example in the R function glmer in the package lme4. Alternatively, estimation of the model parameters can be obtained by using an iterative procedure that combines Maximum Penalized Quasi-Likelihood (MPQL) and REML estimation (Saei & Chambers 2003). Furthermore, estimates of GLMM parameters can be very sensitive to outliers or departures from underlying distributional assumptions. Large deviations from the expected response as well as outlying points in the space of the explanatory variables are known to have a large influence on classical maximum likelihood inference based on generalized linear models (GLMs).

For discrete outcomes, model-based small area estimation conventionally employs a GLMM for of the form

\* MERGEFORMAT (..)

where g is a link function. When is binary-valued a popular choice for g is the logistic link function and the individual values in area i are taken to be independent Bernoulli outcomes with

and . When is a count outcome the logarithmic link function is commonly used and the individual values in area i are assumed to be independent Poisson random variables with

and . The q-dimensional vector is generally assumed to be independently distributed between areas according to a normal distribution with mean 0 and covariance matrix This matrix depends on parameters which are referred to as the variance components and in is the vector of fixed effects. If the target of inference is the small area i mean (proportion), and the Poisson or Bernoulli GLMM is assumed, the approximation to the minimum mean squared error predictor of is . Since depends on and , a further stage of approximation is required, where unknown parameters are replaced by suitable estimates. This leads to the Conditional Expectation Predictor (CEP) for the area i mean (proportion) under logarithmic or logistic,

\* MERGEFORMAT (..)

where or , , is the vector of the estimated fixed effects and denotes the vector of the predicted area-specific random effects. We refer to in this case as a ‘random intercepts’ CEP. For more details on this predictor, including estimation of its MSE, see Saei and Chambers (2003), Jiang and Lahiri (2006a) and Gonzalez-Manteiga et al. (2007). Note, however, that is not the proper Empirical Best Predictor by Jiang (2003). The proper EBP does not have closed form and needs to be computed by numerical approximations. For this reason, the CEP version is used in practice as is the case with the small area estimates of Labour Force activity currently produced by ONS in the UK.
4.3 Geostatistical methodsEquation Chapter 4 Section 3

Geostatistics is concerned with the problem of producing a map of a quantity of interest over a particular geographical region based on, usually noisy, measurement taken at a set of locations in the region. The aim of such a map is to describe and analyze the geographical pattern of the phenomenon of interest. Geostatistical methodologies are born and apply in areas such as environmental studies and epidemiology, where the spatial information is traditionally recorded and available. In the last years the diffusion of spatially detailed statistical data is considerably increased and these kind of procedures - possibly with appropriate modifications - can be used as well in other fields of application, for example to study demographic and socio- economic characteristics of a population living in a certain region.

Basically, to obtain a surface estimate one can exploit the exact knowledge of the spatial coordinates (latitude and longitude) of the studied phenomenon by using bivariate smoothing techniques, such as kernel estimate or kriging. Bivariate smoothing deals with the flexible smoothing of point clouds to obtain surface estimates that can be used to produce maps. The geographical application, however, is not the only use of bivariate smoothing as the method can be applied to handle the non-linear relation between any two continuous predictors and a response variable. (Cressie, 1993; Ruppert et al., 2003) Also kriging, a widely used method for interpolating or smoothing spatial data, has a close connection with penalized spline smoothing: the goals of kriging sound very much like nonparametric regression and the understanding of spatial estimates can be enriched through their interpretation as smoothing estimates (Nychka, 2000).

However, usually the spatial information alone does not properly explain the pattern of the response variable and we need to introduce some covariates in a more complex model.

Geoadditive models, introduced by Kammann and Wand (2003), answer this problem as they analyze the spatial distribution of the study variable while accounting for possible non-linear covariate effects. These models analyse the spatial distribution of the study variable while accounting for possible covariate effects through a linear mixed model representation. The first half of the model formulation involves a low rank mixed model representation of additive models; then, incorporation of the geographical component is achieved by expressing kriging as a linear mixed model and merging it with the additive model to obtain a single mixed model, the geoadditive model (Kammann and Wand, 2003; on kriging see also part 1 of this report).

The model is specified as

\* MERGEFORMAT (..)

where in the first part of the model , and represents measurements on two predictors s and t and a response variable y for unit i, f and g are smooth, but otherwise unspecified, functions of s and t respectively; the second part of the model is the simple universal kriging model with representing the geographical location, is a stationary zero-mean stochastic process. Since both the first and the second part of model can be specified as a linear mixed model, also the whole model can be formulated as a single linear mixed model that can be fitted using standard mixed model software. Thus, we can say that in a geoadditive model the linear mixed model structure allows to include the area-specific effect as an additional random components. In particular, a geoadditive SAE model has two random effect components: the area-specific effects and the spatial effects (Bocci, 2009). See Kammann and Wand (2003) for more details on geoadditive model specifications. Having a mixed model specification, geoadditive models can be used to obtain small area estimators under a non-parametric approach (Opsomer at al., 2008; see also part 4.2 of this report).

In this respect, Bocci et al. (2012) a two-part geoadditive small area estimation model to estimate the per farm average grapevine production, specified as a semicontinuous skewed variable, at Agrarian Region level using data from the fifth Italian Agricultural Census. More in detail, the response variable, assumed to have a significant spatial pattern, has a semicontinuous structure, which means that the variable has a fraction of values equal to zero and a continuous skewed distribution among the remaining values; thus, the variable can be recorded as

\* MERGEFORMAT (..)

and

For this variable, Bocci et al. (2012) specify two uncorrelated geoadditive small area models, one for the logit probability of and one for the conditional mean of the logarithm of the response .

Another extension to the work of Kammand and Wand (2003) is the geoadditive model proposed by Cafarelli and Castrignanò (2011), used to analyse the spatial distribution of grain weight, a commonly used indicator of wheat production, taking into account its nonlinear relations with other crop features.
4.3.2 Area-to-point kriging

Under a more general approach of changing of support for spatial data, Kyriakidis (2004) proposed an area-to-point interpolation technique that is a special case of kriging. This technique is a geostatistical framework that can explicitly and consistently account for the support differences between the available areal data and the sought point predictions, yields coherent (i.e. mass-preserving or pycnophylactic) predictions. Under this approach, a variable at point with coordinate vector u is considered a realization of a random variable . The collection of spattially correlated random variables , where A denotes the study region, is termed a random function. consists of two parts: a deterministic component , indicating the geographical trend or drift, and a stochastic residual component , autocorrelated in space:

. \* MERGEFORMAT (..)

Given a lag vector h, the expected difference between and is 0. The variance of the difference is

where is the so-called semivariogram of residuals. Considering the variance , the covariance between and is given by

.

Kyriakidis (2004) considers kriging of two types of data pertaining to the same attribute: the target residual , of point support and partially sampled, and the source residual , providing ancillary information and defined on a areal support. This kriging is called area-to-point residual kriging. Under this approach, any source residual is functionally linked to the point residual within it as

\* MERGEFORMAT (..)

where is the number of points within areal unit , is the weight associated with , assumed to be known. On different specifications and applications of area-to-point kriging see also Yoo and Kyriakidis (2006), Liu et al. (2008), Yoo and Kyriakidis (2009), Goovaerts (2010).

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