In this section we provide a update on the availability of SAE software. Although from an applied point of view many NSIs have a preference for software such as SAS, most of the recent developments in SAE are implemented in the open-source software R (R Core Team,
1000 2000 3000 4000 5000 1 2 3 4 5 6 7 8 9 10 11 12 13 Districts A v er
age household equiv
alised income
Direct EBP Box−Cox EBP Log EBP Log−Shift
Figure 3.8: Estimates for average household equivalised income at district level.
2017) via R packages.
A comprehensive review of relevant software is included in the CRAN task view on Official Statistics and Survey Methodology(Templ, 2015) with specific categories on Complex Survey Designs, Small Area Estimation and Microsimulations. In particular, the section on Complex Survey Designsincludes packages, like survey (Lumley, 2012) and sampling (Till´e and Matei, 2012) that can be used for point and variance estimation of direct estimators of means, totals, ratios, and quantiles under complex survey designs. Package laeken by Alfons and Templ (2013) provides functions for the estimation of different poverty and inequality indicators such as the at-risk of poverty-rate, Gini coefficient and quintile share ratio and the corresponding estimates of the variance. The sae package by Molina and Marhuenda (2015) can be used for computing synthetic and composite estimators and for implementing SAE with unit-level and area (Fay-Herriot) models that allow for complex correlations structures. A code in R for computing EBP estimates we discussed in Section 3.3.2 that includes an option for using the transformations discussed in the present paper, visualization and export of the results to Excel is proposed in the package emdi by Kreutzmann et al. (2018). Collections of R functions for implementing a wide range of SAE methods are available in the documentations of National and European funded research projects. Here we refer to the BIAS project (BIAS, 2005) which includes code for the unit-level EBLUP and spatial EBLUP with correlated random effects (Pratesi and Salvati, 2009). The SAMPLE project (SAMPLE, 2007) also provides a very wide range of code for implementing parametric, semi-parametric and outlier-robust small area esti- mation and allows for models with spatial and temporal correlations. We refer to Molina et al. (2010) for additional details. Small area estimation from a Bayesian perspective is provided in the packages hbsae (Boonstra, 2012) and BayesSAE (Shi and with contributions from Peng Zhang, 2013). It is also important to mention two packages namely, simPop (Meindl et al., 2016) and saeSim (Warnholz and Schmid, 2016) that support the prospective user in the setup of design- or model-based simulations that enable method evaluation at the evaluation stage.
Bank provides open-source software for poverty estimation called PovMap (The World Bank, 2013). PovMap implements the small area estimation procedure developed in Elbers et al. (2003) and is stand-alone software solution. The European funded project EURAREA (2001) delivered SAS codes for the computation of direct and indirect small area methods. For ad- ditional procedures in SAS we refer to Mukhopadhyay and McDowell (2011). Finally, all methods discussed in the paper are implemented by computationally efficient algorithms using R. The codes are available from the authors upon request.
3.6
Conclusions and Future Research Directions
In this paper we propose a general framework for the production of SA statistics and illustrate the SAE process in practice. As part of this framework we have touched upon three inter- related topics, namely specification of the problem, analysis of the data/ adaptation of the model, and method evaluation. While much can be said for each of these three areas, it is the interplay between them that provides the key to the successful application of SAE methods. There are no clear-cut ways of trading between them in a formal manner and mastering a balance between these three stages is in many ways the wisdom of applied statistics, which holds true also for SAE. We have illustrated some practical ways of keeping this balance. It is shown that specifying a sensible geography and defining targets of estimation that are supported by the data available are the first important steps for successful SAE. Careful model building using the principle of parsimony, model diagnostics and model adaptations are crucial steps for improving estimation without the need for additional data sources. Finally, obtaining uncertainty measures of good quality and designing method evaluation studies are of paramount importance for reassuring the users especially if interest is in using the estimates for official purposes, for example in the design of policy interventions. SAE is of course a large research area and hence it is not possible to capture all of its aspects in a single paper. Production of SA statistics with discrete outcomes and use of area level models are not covered although the proposed framework can be applied in most cases.
Nevertheless, there are questions that remain unresolved and which we would like to raise at this stage. Within the context of sample surveys there exists currently an apparent contrast between the prevalent preference for design-based approaches to statistics at the higher levels of aggregation and model-based approaches at the lower levels. This seems to imply that at some intermediate level of aggregation the choice between the two approaches may be somewhat blurred. Where are these intermediate levels of aggregation? Is it possible to develop a coherent framework for the different levels in the aggregation hierarchy? Should benchmarking towards aggregate-level estimates of acceptable quality actively drive the development of SAE methods or should benchmarking, as often it is, remain a side issue that one only pays attention to at the last stage of estimation?
Both area-specific and ensemble properties of a set of small area estimates are undoubtedly of interest. This is a distinctive feature of SA statistics in comparison to the national estimate that is a single number. Small area estimation is a simultaneous rather than a point estimation problem. Multi-purpose (multiple-goal) SAE aims to provide a compromise in a theoretical
manner. However, the usefulness of such an approach can only be explored together with users if the solution is to have an impact in practice. Can users ever be ready or willing to accept multiple sets of estimates, each optimal for a particular purpose? How can one avoid or limit the misuses of a particular set of estimates in practice? For now we leave these questions open, hoping that they will inform future discussions.
Acknowledgements
First of all, the authors are indebted to the Editor, Associate Editor and referees for comments that significantly improved the paper. Tzavidis, Zhang, Luna, Schmid and Rojas-Perilla grate- fully acknowledge support by grant ES/N011619/1 - Innovations in Small Area Estimation Methodologies from the UK Economic and Social Research Council. The authors are grateful to CONEVAL for providing the data used in empirical work. The views set out in this paper are those of the authors and do not reflect the official opinion of CONEVAL. The numerical results are not official estimates and are only produced for illustrating the methods.
Data-driven Transformations in Small
Area Estimation
4.1
Introduction
Model-based methods for small area estimation (SAE) are now widely used in practice for producing reliable estimates of linear and non-linear indicators for areas/domains with small sample sizes. Examples of indicators that are estimated by using model-based methods include poverty (income deprivation) and inequality measures such as the head count ratio, the poverty gap and the income quintile share ratio. Two popular small area methods in this case are the em- pirical best predictor (EBP), proposed by Molina and Rao (2010) and the World Bank method, proposed by Elbers et al. (2003). Both approaches are based on the use of unit-level linear mixed regression models. Although estimation of complex indicators can be also implemented with area-level linear mixed regression models (Fabrizi and Trivisano, 2016; Schmid et al., 2017), in this paper we focus on unit-level linear mixed regression models. In the original pa- per, Molina and Rao (2010) assumed that the error terms of the linear mixed regression model follow a Gaussian distribution. In case the model error terms significantly deviate from normal- ity, the EBP estimator can be biased. What are the options available to the data analyst when the normality assumptions are not met? One option is to formulate the EBP under alternative and more flexible parametric assumptions. Graf et al. (2014) study an EBP method under the gen- eralized beta distribution of the second kind (GB2), whereas Diallo and Rao (2014) propose the use of skewed-normal distributions in applications with income data. One complication with using the EBP under alternative parametric distributions is that new tools for estimation must be developed and training for the data analyst is needed. In addition, misspecification of the model assumptions is still possible. Another option when the Gaussian assumptions are not satisfied is to use a methodology that minimizes the use of parametric assumptions. For instance, Elbers and van der Weide (2014) proposed an EBP method based on normal mixture models. With this method the distribution of the error terms is described by normal mixtures. Weidenhammer et al. (2014) recently proposed a method that aims at estimating the quantiles of the empirical distribution function of the data. The estimation of the quantiles is facilitated by a nested error regression model using the asymmetric Laplace distribution for the unit-level error terms as a working assumption. The estimation of the random effects can be made com-
option, and the one we study in this paper, is to find an appropriate transformation such that the model assumptions (in this paper the Gaussian assumptions of the EBP method) hold. The aim is to find transformations that (a) are data-driven and optimal according to some criterion and (b) can be implemented by using standard software. To the best of our knowledge, the use and choice of transformations in SAE has not been extensively studied or it has been studied in fairly ad-hoc manner. Elbers et al. (2003) and Molina and Rao (2010) suggested the use of logarithmic-type transformations for income data. However, are such transformations the most appropriate choice? Can alternative transformations offer improved estimation? In order to answer these research questions, the paper investigates data-driven transformations for small area estimation.
The choice of transformations when modelling income-type outcomes - as is the case with poverty mapping applications - presents different challenges. Transformations should be suit- able for dealing with unimodal, leptokurtic and positively skewed data that may include zero and negative values. Besides the logarithmic transformation and its modifications (e.g. the log-shift transformation) a popular family of data-driven transformations that includes the log- arithmic one as a special case is the Box-Cox family (Box and Cox, 1964). Since the Box-Cox transformation is not defined for negative values, when negative values are present, the data must be shifted to the positive range. Another difficulty with the use of the Box-Cox trans- formation is the truncation on the transformation parameter described later in Section 4.4. A solution to this problem can be offered by using of the dual power transformation. Although ex- tensive literature on the use of transformations exists, see for example, John and Draper (1980), Bickel and Doksum (1981) and Yeo and Johnson (2000) among others. In this paper we focus on three types of transformations, namely log-shift, Box-Cox and dual power transformations. In addition to selecting the type of transformation, estimating the transformation parameter adds another layer of complexity. To the best of our knowledge the use of transformations in recent applications of SAE has employed visual residual diagnostics for finding a suitable transformation parameter. In this paper we propose a structured, data-driven approach for estimating the transformation parameter. In particular, we introduce maximum likelihood and residual maximum likelihood methods for estimating the transformation parameter under the linear mixed regression model following Gurka et al. (2006). Alternative estimation approaches based on the minimization of distances (Cram´er, 1928; Chakravarti and Laha, 1967) and on the minimization of the skewness (Carroll and Ruppert, 1987) are also discussed.
We study how the performance of the EBP method is affected by departures from normal- ity and how data-driven transformations can assist with improving the validity of the model assumptions and estimation. Emphasis is given to the estimation of poverty and inequality in- dicators due to their important socio-economic relevance and policy impact. We further study whether the impact of departures from Gaussian assumptions is different depending on the tar- get of estimation. For instance, departures from normality may have lesser impact on estimates of median income compared to estimates indicators that are more sensitive in the data distri- bution. The estimation for the latter indicator heavily depends on the entire distribution of the data. A parametric bootstrap for mean squared error (MSE) estimation under transformation is
the Gaussian assumptions after transformations is also proposed.
The rest of the paper is structured as follows. The EBP approach is introduced in Section 4.2. Section 4.3 presents the survey data we use in this paper and makes the case, via the use of residual diagnostics, for using transformations. In Section 4.4 selected transformations are introduced and extended for their use with model-based SAE methods under the linear mixed regression model. This section includes the theoretical details about the choice of an appro- priate scale and estimation of the transformation parameter. MSE estimation is discussed in Section 4.5. In Section 4.6 the proposed methods are applied to data from Guerrero in Mexico for estimating a range of deprivation and inequality indicators and corresponding estimates of uncertainty. In Section 4.7 the proposed methods are further evaluated by realistic - for income data - model-based simulations. Section 4.8 summarizes the main findings and outlines further research.