Local mixture models of exponential families

Exponential families are the workhorses of parametric modelling theory. One reason for their popularity is their associated inference theory, which is very clean, both from a theoretical and a computational point of view. One way in which this set of tools can be enriched in a natural and interpretable way is through mixing. This paper develops and applies the idea of local mixture modelling to exponential families. It shows that the highly interpretable and flexible models which result have enough structure to retain the attractive inferential properties of exponential families. In particular, results on identification, parameter orthogonality and log-concavity of the likelihood are proved.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6170 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Orthogonal simple component analysis: A new, exploratory approach

Combining principles with pragmatism, a new approach and accompanying algorithm are presented to a longstanding problem in applied statistics: the interpretation of principal components. Following Rousson and Gasser [53 (2004) 539--555] @p250pt@ the ultimate goal is not to propose a method that leads automatically to a unique solution, but rather to develop tools for assisting the user in his or her choice of an interpretable solution. Accordingly, our approach is essentially exploratory. Calling a vector 'simple' if it has small integer elements, it poses the open question: @p250pt@ What sets of simply interpretable orthogonal axes---if any---are angle-close to the principal components of interest? its answer being presented in summary form as an automated visual display of the solutions found, ordered in terms of overall measures of simplicity, accuracy and star quality, from which the user may choose. Here, 'star quality' refers to striking overall patterns in the sets of axes found, deserving to be especially drawn to the user's attention precisely because they have emerged from the data, rather than being imposed on it by (implicitly) adopting a model. Indeed, other things being equal, explicit models can be checked by seeing if their fits occur in our exploratory analysis, as we illustrate. Requiring orthogonality, attractive visualization and dimension reduction features of principal component analysis are retained.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS374 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatial spillover analysis of a cluster-randomized trial against dengue vectors in Trujillo, Venezuela.

BACKGROUND: The ability of cluster-randomized trials to capture mass or indirect effects is one reason for their increasing use to test interventions against vector-borne diseases such as malaria and dengue. For the same reason, however, the independence of clusters may be compromised if the distances between clusters is too small to ensure independence. In other words they may be subject to spillover effects. METHODS: We distinguish two types of spatial spillover effect: between-cluster dependence in outcomes, or spillover dependence; and modification of the intervention effect according to distance to the intervention arm, or spillover indirect effect. We estimate these effects in trial of insecticide-treated materials against the dengue mosquito vector, Aedes aegypti, in Venezuela, the endpoint being the Breteau index. We use a novel random effects Poisson spatial regression model. Spillover dependence is incorporated via an orthogonalized intrinsic conditional autoregression (ICAR) model. Spillover indirect effects are incorporated via the number of locations within a certain radius, set at 200m, that are in the intervention arm. RESULTS: From the model with ICAR spatial dependence, and the degree of surroundedness, the intervention effect is estimated as 0.74-favouring the intervention-with a 95% credible interval of 0.34 to 1.69. The point estimates are stronger with increasing surroundedness within intervention locations. CONCLUSION: In this trial there is some evidence of a spillover indirect effect of the intervention, with the Breteau index tending to be lower in locations which are more surrounded by locations in the intervention arm

Using geometry to select one dimensional exponential families that are monotone likelihood ratio in the sample space, are weakly unimodal and can be parametrized by a measure of central tendency

One dimensional exponential families on finite sample spaces are studied using the geometry of the simplex Δn°-1 and that of a transformation Vn-1 of its interior. This transformation is the natural parameter space associated with the family of multinomial distributions. The space Vn-1 is partitioned into cones that are used to find one dimensional families with desirable properties for modeling and inference. These properties include the availability of uniformly most powerful tests and estimators that exhibit optimal properties in terms of variability and unbiasedness

Antipsychotic drugs and risks of myocardial infarction: a self-controlled case series study.

AIM: Antipsychotics increase the risk of stroke. Their effect on myocardial infarction remains uncertain because people prescribed and not prescribed antipsychotic drugs differ in their underlying vascular risk making between-person comparisons difficult to interpret. The aim of our study was to investigate this association using the self-controlled case series design that eliminates between-person confounding effects. METHODS AND RESULTS: All the patients with a first recorded myocardial infarction and prescription for an antipsychotic identified in the Clinical Practice Research Datalink linked to the Myocardial Ischaemia National Audit Project were selected for the self-controlled case series. The incidence ratio of myocardial infarction during risk periods following the initiation of antipsychotic use relative to unexposed periods was estimated within individuals. A classical case-control study was undertaken for comparative purposes comparing antipsychotic exposure among cases and matched controls. We identified 1546 exposed cases for the self-controlled case series and found evidence of an association during the first 30 days after the first prescription of an antipsychotic, for first-generation agents [incidence rate ratio (IRR) 2.82, 95% confidence interval (CI) 2.0-3.99] and second-generation agents (IRR: 2.5, 95% CI: 1.18-5.32). Similar results were found for the case-control study for new users of first- (OR: 3.19, 95% CI: 1.9-5.37) and second-generation agents (OR: 2.55, 95% CI: 0.93-7.01) within 30 days of their myocardial infarction. CONCLUSION: We found an increased risk of myocardial infarction in the period following the initiation of antipsychotics that was not attributable to differences between people prescribed and not prescribed antipsychotics

The relative frailty variance and shared frailty models

The relative frailty variance among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We discuss the properties of the relative frailty variance, show that it characterizes frailty distributions, and that, suitably rescaled, it may be used to compare patterns of dependence across models and data sets. In shared frailty models, the relative frailty variance is closely related to the cross-ratio function, which is estimable from bivariate survival data. We investigate the possible shapes of the relative frailty variance function for the purpose of model selection, and review available frailty distribution families in this context. We introduce several new families with contrasting properties, including simple but flexible time-varying frailty models. The benefits of the approach we propose are illustrated with two applications to bivariate current status data obtained from serological surveys

Estimation of basic reproduction numbers: individual heterogeneity and robustness to perturbation of the contact function

The basic reproduction number of an infection in a given population, R0, is inflated by individual heterogeneity in contact rates. Recently, new methods for estimating R0 using social contact data and serological survey data have been proposed. These methods, like most of their predecessors, ignore individual heterogeneity, and are sensitive to perturbation of the contact function. Using a frailty framework, we derive expressions for R0 in the presence of age-varying heterogeneity. In this case, R0 is the spectral radius of a population version of the next generation operator, which involves the variance function of the age-dependent frailty. This variance can be estimated within a shared frailty framework from paired data on two infections transmitted by the same route. We propose two estimators of R0 for infections in endemic equilibrium. We investigate their performance by simulation, and find that one is generally less efficient but more robust than the other to perturbation of the effective contact function. These methods are applied to data on varicella zoster virus infection from two European countries

On the Geometric Interplay Between Goodness-of-Fit and Estimation: Illustrative Examples

We show how information geometry throws new light on the interplay between goodness-of-fit and estimation, a fundamental issue in statistical inference. A geometric analysis of simple, yet representative, models involving the same population parameter compellingly establishes the main theme of the paper: namely, that goodness-of-fit is necessary but not sufficient for model selection. Visual examples vividly communicate this. Specifically, for a given estimation problem, we define a class of least-informative models, linking these to both nonparametric and maximum entropy methods. Any other model is then seen to involve an informative rotation, often embodying extra-data considerations. We also look at the way that translation of models generates a form of bias-variance trade-off. Overall, our approach is a global extension of pioneering local work by Copas and Eguchi which, we note, was also geometrically inspired