Small area estimation techniques are employed when sample data are insufficient for acceptably precise direct estimation in domains of interest. These techniques typically rely on regression models that use both covariates and random effects to explain variation between domains. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. We describe a new approach to small area estimation that is based on modelling quantile-like parameters of the conditional distribution of the target variable given the covariates. This avoids the problems associated with specification of random effects, allowing inter-domain differences to be characterized by the variation of area-specific M-quantile coefficients. The proposed approach is easily made robust against outlying data values and can be adapted for estimation of a wide range of area specific parameters, including that of the quantiles of the distribution of the target variable in the different small areas. Results from two simulation studies comparing the performance of the M-quantile modelling approach with more traditional mixed model approaches are also provided. SUMMARYSmall area estimation techniques are employed when sample data are insufficient for acceptably precise direct estimation in domains of interest. These techniques typically rely on regression models that use both covariates and random effects to explain variation between domains. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. We describe a new approach to small area estimation that is based on modelling quantile-like parameters of the conditional distribution of the target variable given the covariates. This avoids the problems associated with specification of random effects, allowing inter-domain differences to be characterized by the variation of area-specific M-quantile coefficients. The proposed approach is easily made robust against outlying data values and can be adapted for estimation of a wide range of area specific parameters, including that of the quantiles of the distribution of the target variable in the different small areas. Results from 2 two simulation studies comparing the performance of the M-quantile modelling approach with more traditional mixed model approaches are also provided.

Outliers are a well-known problem in survey estimation, and a variety of approaches have been suggested for dealing with them in this context. However, when the focus is on small area estimation using the survey data, much less is known -even though outliers within a small area sample are clearly much more influential than they are in the larger overall sample. To the best of our knowledge, Chambers and Tzavidis (2006) was the first published paper in small area estimation that explicitly addressed the issue of outlier robustness, using an approach based on fitting outlier robust M-quantile models to the survey data. Recently, Sinha and Rao (2009) have also addressed this issue from the perspective of linear mixed models. Both these approaches, however, use plug-in robust prediction. That is, they replace parameter estimates in optimal, but outlier sensitive, predictors by outlier robust versions. Unfortunately, this approach may involve an unacceptable prediction bias (but a low prediction variance) in situations where the outliers are drawn from a distribution that has a different mean to the rest of the survey data (Chambers, 1986), which then leads to the suggestion that outlier robust prediction should include an additional term that makes a correction for this bias.In this paper, we explore the extension of this idea to the small area estimation situation and we propose two different analytical mean squared error (MSE) estimators for outlier robust predictors of small area means. We use simulation based on realistic outlier contaminated data to evaluate how the extended small area estimation approach compares with the plug-in robust methods described earlier. The empirical results show that the biascorrected predictive estimators are less biased than the projective estimators especially when there are outliers in the area effects. Moreover, in the simulation experiments we contrast the proposed MSE estimators with those generally utilized for the plug-in robust predictors. The proposed bias-robust and linearization-based MSE estimators appear to perform well when used with the robust predictors of small area means that are considered in this paper.

SummarySmall area estimation techniques have typically relied on plug-in estimation based on models containing random area effects. More recently, regression M-quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug-in M-quantile estimator for the small area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small area estimation, based on representing a small area estimator as a functional of a predictor of this small area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small area means and quantiles and is not restricted to M-quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model-based and design-based simulations, with the latter using economic data collected in an Australian farm survey.

SummaryMultilevel modelling is a popular approach for longitudinal data analysis. Statistical models conventionally target a parameter at the centre of a distribution. However, when the distribution of the data is asymmetric, modelling other location parameters, e.g. percentiles, may be more informative. We present a new approach, M‐quantile random‐effects regression, for modelling multilevel data. The proposed method is used for modelling location parameters of the distribution of the strengths and difficulties questionnaire scores of children in England who participate in the Millennium Cohort Study. Quantile mixed models are also considered. The analyses offer insights to child psychologists about the differential effects of risk factors on children's outcomes.

Summary Small area models typically depend on the validity of model assumptions. For example, a commonly used version of the empirical best predictor relies on the Gaussian assumptions of the error terms of the linear mixed regression model: a feature rarely observed in applications with real data. The paper tackles the potential lack of validity of the model assumptions by using data‐driven scaled transformations as opposed to ad hoc chosen transformations. Different types of transformations are explored, the estimation of the transformation parameters is studied in detail under the linear mixed regression model and transformations are used in small area prediction of linear and non‐linear parameters. The use of scaled transformations is crucial as it enables fitting the linear mixed regression model with standard software and hence it simplifies the work of the data analyst. Mean‐squared error estimation that accounts for the uncertainty due to the estimation of the transformation parameters is explored by using the parametric and semiparametric (wild) bootstrap. The methods proposed are illustrated by using real survey and census data for estimating income deprivation parameters for municipalities in the Mexican state of Guerrero. Simulation studies and the results from the application show that using carefully selected, data‐driven transformations can improve small area estimation.

Summary Small area estimation is a research area in official and survey statistics of great practical relevance for national statistical institutes and related organizations. Despite rapid developments in methodology and software, researchers and users would benefit from having practical guidelines for the process of small area estimation. We propose a general framework for the production of small area statistics that is governed by the principle of parsimony and is based on three broadly defined stages, namely specification, analysis and adaptation, and evaluation. Emphasis is given to the interaction between a user of small area statistics and the statistician in specifying the target geography and parameters in the light of the available data. Model‐free and model‐dependent methods are described with a focus on model selection and testing, model diagnostics and adaptations such as use of data transformations. Uncertainty measures and the use of model and design‐based simulations for method evaluation are also at the centre of the paper. We illustrate the application of the proposed framework by using real data for the estimation of non‐linear deprivation indicators. Linear statistics, e.g. averages, are included as special cases of the general framework.

Ecological and transactional theories link child outcomes to neighbourhood disadvantage, family poverty and adverse life events. Traditionally, these three types of risk factors have been examined independently of one another or combined into one cumulative risk index. The first approach results in poor prediction of child outcomes, and the second is not well rooted in ecological theory as it does not consider that distal risk factors (such as poverty) may indirectly impact children through proximal risk factors (such as adverse life events). In this study, we modelled simultaneously the longitudinal effects of these three risk factors on children's internalising and externalising problems, exploring the role of parenting in moderating these effects. Our sample followed 16,916 children (at ages 3, 5 and 7 years; N = 16,916; 49% girls) from the UK Millennium Cohort Study. Parenting was characterised by quality of parent-child relationship, parental involvement in learning and parental discipline. Neighbourhood disadvantage, family poverty and adverse events were all simultaneously related to the trajectories of both outcomes. As expected, parenting moderated risk effects. Positive parent-child relationship, rather than greater involvement or authoritative discipline, most consistently 'buffered' risk effects. These findings suggest that a good parent-child relationship may promote young children's emotional and behavioural resilience to different types of environmental risk.

Few studies on resilience in young children model risk appropriately and test theory-led hypotheses about its moderation. This study addressed both issues. Our hypothesis was that for preschool children's emotional/behavioral adjustment in the face of contextual risk protective factors should be located in the cognitive domain. Data were from the first two sweeps of the UK's Millennium Cohort Study. The final study sample was 4,748 three-year-old children clustered in 1,549 Lower layer Super Output Areas in nine strata. Contextual risk was measured at both area (with the Index of Multiple Deprivation) and family (with proximal and distal adverse life events experienced) level. Moderator variables were parenting, verbal and non-verbal ability, developmental milestones, and temperament. Multivariate multilevel models-that allowed for correlated residuals at both individual and area level-and univariate multilevel models estimated risk effects on specific and broad psychopathology. At baseline, proximal family risk, distal family risk and area risk were all associated with broad psychopathology, although the most parsimonious was the proximal family risk model. The area risk/broad psychopathology association remained significant even after family risk was controlled but not after family level socioeconomic disadvantage was controlled. The cumulative family risk was more parsimonious than the specific family risks model. Non-verbal ability moderated the effect of proximal family risk on conduct and emotional problems, and developmental milestones moderated the effect of proximal family risk on conduct problems. The findings highlight the importance of modeling contextual risk appropriately and of locating in the cognitive domain factors that buffer its effect on young children's adjustment.