T. Goicoa

Spline smoothing in small area trend estimation and forecasting

Semiparametric models combining both non-parametric trends and small area random effects are now currently being investigated in small area estimation (SAE). These models can prevent bias when the functional form of the relationship between the response and the covariates is unknown. Furthermore, penalized spline regression can be a good tool to incorporate non-parametric regression models into the SAE techniques, as it can be represented as a mixed effects model. A penalized spline model is considered to analyze trends in small areas and to forecast future values of the response. The prediction mean squared error (MSE) for the fitted and the predicted values, together with estimators for those quantities, are derived. The procedure is illustrated with real data consisting of average prices per squared meter of used dwellings in nine neighborhoods of the city of Vitoria, Spain, during the period 1993-2007. Dwelling prices for the next five years are also forecast. A simulation study is conducted to assess the performance of both the small area trend estimator and the prediction MSE estimators. The results confirm a good behavior of the proposed estimators in terms of bias and variability.

On fitting spatio-temporal disease mapping models using approximate Bayesian inference

Spatio-temporal disease mapping comprises a wide range of models used to describe the distribution of a disease in space and its evolution in time. These models have been commonly formulated within a hierarchical Bayesian framework with two main approaches: an empirical Bayes (EB) and a fully Bayes (FB) approach. The EB approach provides point estimates of the parameters relying on the well-known penalized quasi-likelihood (PQL) technique. The FB approach provides the posterior distribution of the target parameters. These marginal distributions are not usually available in closed form and common estimation procedures are based on Markov chain Monte Carlo (MCMC) methods. However, the spatio-temporal models used in disease mapping are often very complex and MCMC methods may lead to large Monte Carlo errors and a huge computation time if the dimension of the data at hand is large. To circumvent these potential inconveniences, a new technique called integrated nested Laplace approximations (INLA), based on nested Laplace approximations, has been proposed for Bayesian inference in latent Gaussian models. In this paper, we show how to fit different spatio-temporal models for disease mapping with INLA using the Leroux CAR prior for the spatial component, and we compare it with PQL via a simulation study. The spatio-temporal distribution of male brain cancer mortality in Spain during the period 1986–2010 is also analysed.

Empirical Bayes and Fully Bayes procedures to detect high-risk areas in disease mapping

Disease mapping studies have experienced an enormous development in the last twenty years. Both an Empirical Bayes (EB) and a Fully Bayes (FB) approach have been used for smoothing purposes. However, an excess of smoothing might hinder the detection of true high-risk areas. Identifying these extreme regions minimizing the misclassification of background or normal areas, and then, avoiding false alarms is crucial in epidemiology. Bayesian decision rules, based on the posterior distribution of the relative risks, have been investigated for this task, but no similar studies have been conducted under the EB approach. Within this framework, second order correct estimators of the MSE of the log-relative risk predictor can be used to build appropriate confidence intervals for the relative risks. Their ability to detect high-risk areas is investigated through a simulation study using the geographical structure of the well-known Scottish lip cancer data. Bayesian credibility intervals and decision rules, based on the posterior distribution of the relative risks, are also investigated to check if any of the approaches outperforms the others when classifying high-risk regions. The conclusion is that Bayesian decision rules, exploiting the posterior distribution of the relative risks, are more powerful to detect high-risk areas than EB confidence intervals, but no general rules can be defined as a global criterion to be routinely applied in every real setting.

Age-Specific Spatio-Temporal Patterns of Female Breast Cancer Mortality in Spain (1975–2005)

PURPOSE In recent decades, a decline in breast cancer mortality has been observed across Europe, and also in Spain. Our objective is to assess the spatio-temporal pattern during the period 1975-2005 by specific age groups (<45, 45-64, ≥65) in the Spanish provinces. METHODS For each age group, a spatio-temporal P-spline model with a B-spline basis is used to smooth the mortality risks. Smoothing is carried out in three dimensions: longitude, latitude, and time, allowing for a different time evolution of both spatial components. The age-specific decline is calculated as the maximum of the estimated curve in each province. A confidence band for each curve is also provided. RESULTS For the first age group (<45), the decline in the different provinces is observed between 1986 and 1991. For women aged between 45 to 64 years, the change occurs between 1990 and 1993. For the third age group (≥65), change points range from 1992 to 2000, unlike Malaga and Cadiz where the change has not been observed in the studied period. Northern and some Mediterranean provinces are the areas with higher mortality risks for all the age groups. CONCLUSIONS A different behavior for breast cancer mortality risks is observed for different provinces among the age specific groups. The decline of mortality is delayed for the oldest age group. Province differences in the implementation of screening programs could explain some of the observed differences.

A P-spline ANOVA type model in space-time disease mapping

One of the main objectives in disease mapping is the identification of temporal trends and the production of a series of smoothed maps from which spatial patterns of mortality risks can be monitored over time. When studying rare diseases, conditional autoregressive models have been commonly used for smoothing risks. In this work, a P-spline ANOVA type model is used instead. The model is anisotropic and explicitly considers different smooth terms for space, time, and space-time interaction avoiding, in addition, model identifiability problems. The mean squared error of the log-risk predictor is derived accounting for the variability associated to the estimation of the smoothing parameters. The procedure is illustrated analyzing Spanish prostate cancer mortality data in the period 1975–2008.

Spatio-temporal trends in gastric cancer mortality in Spain: 1975–2008

AIM OF THE STUDY There has been a downward trend in gastric cancer mortality worldwide. In Spain, a marked spatial aggregation of areas with excess mortality due to this cause has long been reported. This paper sought to analyse the evolution of gastric cancer mortality risk in Spanish provinces and explore the possible attenuation of the geographical pattern. METHODS We studied a series of gastric cancer mortality data by province, year of death, sex and age group using a conditional autoregressive (CAR) model that incorporated space, time and spatio-temporal interactions. RESULTS Gastric cancer mortality risk decreased in all Spanish provinces in both males and females. Overall, decreasing trends were more pronounced during the first years of the study period, largely due to a sharper fall in gastric cancer mortality risk among the older population. Recent decades have witnessed a slowing in the rate of decrease, especially among the younger age groups. In most areas, risk declined at a similar rate, thus serving to maintain interprovincial differences and the persistence of the geographical pattern, though with some differences. The north and northwest provinces were the areas with higher mortality risks in both sexes and age groups over the entire study period. CONCLUDING STATEMENT Despite the decline in gastric cancer mortality risk observed for the 50 Spanish provinces studied, geographical differences still persist in Spain, and the cluster of excess mortality in the north-west of the country remains in evidence.

Modelling aboveground tree biomass while achieving the additivity property

Measuring forest tree biomass is becoming a very important issue due to the general environmental awareness motivated by global warming and climate change. However, weighing a tree is a very complicated, expensive, and destructive process. The tree is divided into several parts, and the total weight is obtained by adding the weight of the different components. The biomass information of a forest is obtained using statistical models, but one of the main difficulties is that the additivity property is not generally satisfied, i.e., when adding the predicted weights for the different tree components, the result does not match up with the total weight predicted for the tree. In this work, alternative methods for obtaining biomass predictions satisfying the additivity property are analyzed. In particular, segmented regression models with a common break point and penalized splines with the same smoothing parameter achieve the additivity property without any further adjustments. Some classical models will be also used for comparison purposes. Results are illustrated with real data from a beech forest (European project FORSEE-020) in the province of Navarre, Spain.

In spatio-temporal disease mapping models, identifiability constraints affect PQL and INLA results

Disease mapping studies the distribution of relative risks or rates in space and time, and typically relies on generalized linear mixed models (GLMMs) including fixed effects and spatial, temporal, and spatio-temporal random effects. These GLMMs are typically not identifiable and constraints are required to achieve sensible results. However, automatic specification of constraints can sometimes lead to misleading results. In particular, the penalized quasi-likelihood fitting technique automatically centers the random effects even when this is not necessary. In the Bayesian approach, the recently-introduced integrated nested Laplace approximations computing technique can also produce wrong results if constraints are not well-specified. In this paper the spatial, temporal, and spatio-temporal interaction random effects are reparameterized using the spectral decompositions of their precision matrices to establish the appropriate identifiability constraints. Breast cancer mortality data from Spain is used to illustrate the ideas.

Using Small Area Models to Estimate the Total Area Occupied by Olive Trees

This article aims to estimate the total area occupied by olive trees in a region called Comarca IV, located in a central region of Navarra, Spain. Traditionally, small area linear mixed models have been used for similar purposes using regular quadrats (also called segments) as sampling units, and assuming that the majority of segments are fully included in the study domain. When this does not happen, the sampling units are of different size, and there exists an extra variability that can be very different within areas. In this case it is advisable to include model weights in the model. In this article, we propose a weighted unit level linear mixed model where both the variance components and the coefficients of the model are estimated using these weights. We also discuss, the model performance and compare it with alternatives.

SEARCHING FOR HOUSING SUBMARKETS USING MIXTURES OF LINEAR MODELS

This paper presents a mixture of linear models (or hedonic regressions) for defining housing submarkets. Two different mixture models are considered: the first model allows all the regression coefficients to vary among the clusters (random coefficients); and the second model allows only the intercept term to change (random intercept). The model with a random intercept can be seen as a linear mixed model where the random effects distribution is estimated via non-parametric maximum likelihood (NPML). The models are illustrated using a real data set of 293 properties in Pamplona, Spain. These mixture models provide a classification of the dwellings into homogeneous groups that determine the structure of the submarkets.