María Durbán

Smoothing and forecasting mortality rates

The prediction of future mortality rates is a problem of fundamental importance for the insurance and pensions industry. We show how the method of P-splines can be extended to the smoothing and forecasting of two-dimensional mortality tables. We use a penalized generalized linear model with Poisson errors and show how to construct regression and penalty matrices appropriate for two-dimensional modelling. An important feature of our method is that forecasting is a natural consequence of the smoothing process. We illustrate our methods with two data sets provided by the Continuous Mortality Investigation Bureau, a central body for the collection and processing of UK insurance and pensions data.

Fast and compact smoothing on large multidimensional grids

A framework of penalized generalized linear models and tensor products of B-splines with roughness penalties allows effective smoothing of data in multidimensional arrays. A straightforward application of the penalized Fisher scoring algorithm quickly runs into storage and computational difficulties. A novel algorithm takes advantage of the special structure of both the data as an array and the model matrix as a tensor product; the algorithm is fast, uses only a moderate amount of memory and works for any number of dimensions. Examples are given of how the method is used to smooth life tables and image data.

Flexible smoothing with P-splines: a unified approach

We consider the application of P-splines (Eilers and Marx, 1996) to three classes of models with smooth components: semiparametric models, models with serially correlated errors, and models with heteroscedastic errors. We show that P-splines provide a common approach to these problems. We set out a simple nonparametric strategy for the choice of the P-spline parameters (the number of knots, the degree of the P-spline, and the order of the penalty) and use mixed model (REML) methods for smoothing parameter selection. We give an example of a model in each of the three classes and analyse appropriate data sets.

Decreasing prevalence of disability in activities of daily living, functional limitations and poor self-rated health: a 6-year follow-up study in Spain

Background and aims: Forecasting functional status in elderly populations is uncertain. Our aim is to provide evidence of population trends of Activities of Daily Living (ADL) disability, functional limitations and self-rated health. Methods: Data come from a longitudinal study of aging in Leganés (Spain), collected in 1993, 1995, 1997 and 1999, on a representative sample of 1560 community dwelling people over 65. Response rate at baseline was 82%. ADL disability was defined as needing help in at least one of the following: walking across a small room, taking a shower, toileting, getting out of bed, getting up from a chair, using the toilet, dressing and eating. Functional limitations were based on questions of difficulty with upper and lower limbs. Self-rated health was assessed with a single question. ADL disability, functional limitations and self-rated health were regressed on age, survey year, sex and education. Results: There are significant declines in ADL disability, functional limitations and poor self-rated health at every age and up to very advanced ages. Over 90, the ADL disability trend may be reversed, with the emergence of a very old and disabled population. Women and people with little education have a higher prevalence of disability, functional limitations and poor health, when compared with men and those with higher education. Conclusions: Results suggest the postponement of severe disability onset in this Spanish population, leading to longer healthy life expectancy, and support the emergence of a very disabled population over 90 years of age.

P-spline ANOVA-type interaction models for spatio-temporal smoothing

In recent years, spatial and spatio-temporal modelling have become an important area of research in many fields (epidemiology, environmental studies, disease mapping, etc.). However, most of the models developed are constrained by the large amounts of data available. We propose the use of penalized splines ( -splines) in a mixed model framework for smoothing spatio-temporal data. Our approach allows the consideration of interaction terms which can be decomposed as a sum of smooth functions similarly as an analysis of variance decomposition. The properties of the bases used for regression allow the use of algorithms that can handle large amount of data. We show that on imposing the same constraints as in a factorial design it is possible to avoid identifiability problems. We illustrate the methodology for Europe ozone levels in the period 1999–2005.

Intimate partner violence: last year prevalence and association with socio-economic factors among women in Madrid, Spain

BACKGROUND Intimate partner violence (IPV) is a public health problem with significant consequences on womens health. This study estimates the prevalence of intimate partner violence by type among Madrids female population and assesses the association with socio-economic variables. METHODS We conducted a cross-sectional study in 2004, 2136 women aged 18-70 years, living in the Madrid region with a partner or who had been in contact with an ex-partner in the previous year, were interviewed by telephone. The questionnaire used to measure past-year intimate partner violence, consisted of a Spanish translation of the psychological and sexual violence module of the French National Survey on Violence against Women, and the physical violence module of the Conflict Tactics Scale-1. To assess the association with socio-economic factors, logistic regression models were fitted. RESULTS About 10.1% [confidence interval (CI) 8.9-11.5] of the women had suffered some type of IPV in the previous year. 8.6% (CI 7.4-9.8) experienced psychological violence, 2.4% (CI 1.8-3.1) physical violence and 1.1% (CI 0.68-1.6) sexual violence; the prevalence of psychological-only violence (non-physical/non-sexual) was 6.9% (CI 5.8-8.0). Factors associated with psychological-only violence were divorced or separated status and Group III (clerical workers; supervisors of manual workers) or V (unskilled manual workers) occupation. Unemployment and divorced or separated status were associated with physical violence. CONCLUSIONS Spanish women in our study, experienced past year partner violence at a similar level as in other industrialized countries. Unemployment and low occupational status are associated with physical and psychological-only violence, respectively.

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.

Fast smoothing parameter separation in multidimensional generalized P-splines: the SAP algorithm

A new computational algorithm for estimating the smoothing parameters of a multidimensional penalized spline generalized linear model with anisotropic penalty is presented. This new proposal is based on the mixed model representation of a multidimensional P-spline, in which the smoothing parameter for each covariate is expressed in terms of variance components. On the basis of penalized quasi-likelihood methods, closed-form expressions for the estimates of the variance components are obtained. This formulation leads to an efficient implementation that considerably reduces the computational burden. The proposed algorithm can be seen as a generalization of the algorithm by Schall (1991)—for variance components estimation—to deal with non-standard structures of the covariance matrix of the random effects. The practical performance of the proposed algorithm is evaluated by means of simulations, and comparisons with alternative methods are made on the basis of the mean square error criterion and the computing time. Finally, we illustrate our proposal with the analysis of two real datasets: a two dimensional example of historical records of monthly precipitation data in USA and a three dimensional one of mortality data from respiratory disease according to the age at death, the year of death and the month of death.

Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases

Low-rank smoothing techniques have gained much popularity in non-standard regression modeling. In particular, penalized splines and tensor product smooths are used as flexible tools to study non-parametric relationships among several covariates. The use of standard statistical software facilitates their use for several types of problems and applications. However, when interaction terms are considered in the modeling, and multiple smoothing parameters need to be estimated standard software does not work well when datasets are large or higher-order interactions are included or need to be tested. In this paper, a general approach for constructing and estimating bivariate smooth models for additive and interaction terms using penalized splines is proposed. The formulation is based on the mixed model representation of the smooth-ANOVA model by Lee and Durban (in?press), and several nested models in terms of random effects components are proposed. Each component has a clear interpretation in terms of function shape and model identifiability constraints. The term P S -ANOVA is coined for this type of models. The estimation method is relatively straightforward based on the algorithm by Schall (1991) for generalized linear mixed models. Further, a simplification of the smooth interaction term is used by constructing lower-rank basis (nested basis). Finally, some simulation studies and real data examples are presented to evaluate the new model and the estimation method.

The Practical Use of Semiparametric Models in Field Trials

This article examines the practical use of semiparametric models in the analysis of field trials—that is, models with parameterized treatment effects and additive terms derived by a data-driven approach using a locally weighted running line smoother (loess). We discuss graphical methods to identify spatial structure in the data and model selection procedures to choose the degree of smoothing. Once the spatial part of the model has been chosen, hypotheses about the treatment effects may be tested. Semiparametric models are used to analyze two barley field trials exhibiting spatial trends. The first has a single experimental treatment and a row-column design. The second has a split-plot design, and we use a semiparametric model which accounts for the randomization at the different strata of this design. We compare the semiparametric analyses with classical analyses of variance and with alternative spatial models. We find that semiparametric models give a good insight into spatial variation in the field and can improve the precision of parameter estimates.