Ana F. Militino

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.

Penalized quasi-likelihood with spatially correlated data

Abstract This article discusses and evaluates penalized quasi-likelihood (PQL) estimation techniques for the situation where random effects are correlated, as is typical in mapping studies. This is an approximate fitting technique which uses a Laplace approximation to the integrated mixed model likelihood. It is much easier to implement than usual maximum likelihood estimation. Our results show that the PQL estimates are reasonably unbiased for analysis of mixed Poisson models when there is correlation in the random effects, except when the means are sufficiently small to yield sparse data. However, although the normal approximation to the distribution of the parameter estimates works fairly well for the parameters in the mean it does not perform as well for the variance components. In addition, when the mean mortality counts are small, the estimated standard errors of the variance components tend to become more biased than those for the mean. We illustrate our approaches by applying PQL for mapping mortality in British Columbia, Canada, over the five-year period 1985–1989.

Analyzing Censored Spatial Data

Spatial data that are incomplete because of observations arising below or above a detection limit occur in many settings, for example, in mining, hydrology, and pollution monitoring. These observations are referred to as censored observations. For example, in a life test, censoring may occur at random times because of accident or breakdown of equipment. Also, censoring may occur when failures are discovered only at periodic inspections. Because the informational content of censored observations is less than that of uncensored ones, censored data create difficulties in an analysis, particularly when such data are spatially dependent. Traditional methodology applicable for uncensored data needs to be adapted to deal with censorship. In this paper we propose an adaptation of the traditional methodology using the so-called Expectation-Maximization (EM) algorithm. This approach permits estimation of the drift coefficients of a spatial linear model when censoring is present. As a by-product, predictions of unobservable values of the response variable are possible. Some aspects of the spatial structure of the data related to the implicit correlation also are discussed. We illustrate the results with an example on uranium concentrations at various depths.

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.

Modelling risks in disease mapping

In this article, we propose a strategy of analysis of mortality data with the aim of providing a guideline for epidemiologists and public health researchers to choose a reasonable model for estimating mortality (or incidence) risks. Maps displaying the crude mortality rates or ratios are usually misleading because of the instability of the estimators in low populated areas. As an alternative, many smoothing methods have been presented in the literature based on Poisson inference. They account for the extra-Poisson variation (overdispersion), frequently present in the homogeneous Poisson model, by incorporating random effects. Here, we recommend to test for the potential sources of extra-Poisson variation because, depending on them, the models which fit better the data may be different. Overdispersion can be mainly due to spatial autocorrelation, unstructured heterogeneity or to a combination of these two, and also, when studying very rare diseases, it can be due to an excess of zeros in the data. In this article, different situations the analyst may encounter are detailed and appropriate procedures for each case are presented. The alternative models are illustrated using mortality data provided by the Statistical Institute of Navarra, Spain.

Assessing the covariance function in geostatistics

In geostatistics, one of the crucial problems is the choice of the covariance function. In this paper we show how to improve the cross-validation criterion, traditionally used for evaluating the fit of a covariance function, in the case of unequally spaced data.

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.