Summary A mixed generalized Akaike information criterion xGAIC is introduced and validated. It is derived from a quasi‐log‐likelihood that focuses on the random effect and the variability between the areas, and from a generalized degree‐of‐freedom measure, as a model complexity penalty, which is calculated by the bootstrap. To study the performance of xGAIC, we consider three popular mixed models in small area inference: a Fay–Herriot model, a monotone model and a penalized spline model. A simulation study shows the good performance of xGAIC. Besides, we show its relevance in practice, with two real applications: the estimation of employed people by economic activity and the prevalence of smokers in Galician counties. In the second case, where it is unclear which explanatory variables should be included in the model, the problem of selection between these explanatory variables is solved simultaneously with the problem of the specification of the functional form between the linear, monotone or spline options.

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INTRODUCTION Complete and accurate data on smoking prevalence at a local level would enable health authorities to plan context-dependent smoking interventions. However, national health surveys do not generally provide direct estimates of smoking prevalence by sex and age groups at the subnational level. This study uses a small-area model-based methodology to obtain precise estimations of smoking prevalence by sex, age group and region, from a population-based survey. METHODS The areas targeted for analysis consisted of 180 groups based on a combination of sex, age group (15–34, 35–54, 55–64, 65–74, and ≥75 years), and Autonomous Region. Data on tobacco use came from the 2017 Spanish National Health Survey (2017 SNHS). In each of the 180 groups, we estimated the prevalence of smokers (S), ex-smokers (ExS) and never smokers (NS), as well as their coefficients of variation (CV), using a weighted ratio estimator (direct estimator) and a multinomial logistic model with random area effects. RESULTS When smoking prevalence was estimated using the small-area model, the precision of direct estimates improved; the CV of S and ExS decreased on average by 26%, and those of NS by 25%. The range of S prevalence was 11–46% in men and 4–37% in women, excluding the group aged ≥75 years. CONCLUSIONS This study proposes a methodology for obtaining reliable estimates of smoking prevalence in groups or areas not covered in the survey design. The model applied is a good alternative for enhancing the precision of estimates at a detailed level, at a much lower cost than that involved in conducting large-scale surveys. This method could be easily integrated into routine data processing of population health surveys. Having such estimates directly after completing a health survey would help characterize the tobacco epidemic and/or any other risk factor more precisely.

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BackgroundEurope has had a large variability in COVID-19 incidence between and within countries, particularly after June 2020. We aim to assess the variability between European countries and regions located in a given country.MethodsWe used ECDC information including countries having 7 regions or more. The metric used to assess the regional variability within a country was the intercuartilic range in a weekly basis for 32 weeks between June 29th 2020 and February 1st 2021. We also calculated each country's overall variability across the 32 weeks using the distances from the regional curves of the 14-day incidence rates to the corresponding national curve, using the L2 metric for functional data. We afterwards standardised this metric to a scale from 0 to 100 points. We repeated the calculations excluding island regions.ResultsThe variability between and within countries was large. Slovenia, Spain and Portugal have the greatest variability. Spain and Slovenia held also the top three places for the greatest number of weeks (Spain for 19 weeks and Slovenia for 10) with the highest variability. For variability among the incidence curves across the 32-week period, Slovenia, Portugal and Spain ranked first in functional variability, when all the regions were analysed but also when the island regions were excluded.ConclusionsThese differences might be due to how countries tackled the epidemiological situation. The persistent variability in COVID-19 incidence between regions of a given country suggests that governmental action may have an important role in applying epidemiological control measures.