Estimation with Interval Censored Data in Longitudinal Studies

In biostatistical applications interest often focuses on the estimation of the distribution of a time-until-event variable T. If one observes whether or not T exceeds an observed monitoring time at a random number of monitoring times, then the data structure is called interval censored data. We extend this data structure by allowing the presence of a possibly time-dependent covariate process that is observed until end of follow up. If one only assumes that the censoring mechanism satisfies coarsening at random, then, by the curve of dimensionality, typically no regular estimators will exist. To fight the curse of dimensionality we follow the approach of Robins and Rotnitzky (1992) by modeling parameters of the censoring mechanism. We model the right-censoring mechanism by modeling the hazard of the follow up time, conditional on T and the covariate process. For the monitoring mechanism we avoid modeling the joint distribution of the monitoring times by only modeling a univariate hazard of the pooled monitoring times, conditional on the follow up time, T, and the covariates process, which can be estimated by treating the pooled sample of monitoring times as i.i.d. In particular, it is assumed that the monitoring times and the right-censoring times only depend on T through the observed covariate process. We introduce inverse probability of censoring weighted (IPCW) estimator of the distribution of T and of smooth functionals thereof which are guaranteed to be consistent and asymptotically normal if we have available correctly specified semiparametric models for the two hazards of the censoring process. Furthermore, given such correctly specified models for these hazards of the censoring process, we propose a one-step estimator which will improve on the IPCW estimator if we correctly specify a lower-dimensional working model for the conditional distribution of T, given the covariate process, that remains consistent and asymptotically normal if this latter working model is misspecified. It is shown that the one-step estimator is efficient if each subject is at most monitored once and the working model contains the truth. In general, it is shown that the one-step estimator optimally uses the surrogate information if the working model contains the truth. It is not optimal in using the interval information provided by the current status indicators at the monitoring times, but simulations in Peterson, van der Laan (1997) show that the efficiency loss is small.

Locally Efficient Estimation with Current Status Data and High Dimensional Covariates

In biostatistical applications, interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed monitoring time, then the data is described by the well known singly-censored current status model, also known as interval censored data, case I. We extend this current status model by allowing the presence of a time-dependent process, which is partly observed and allowing C to depend on T through the observed part of this time-dependent process. Because of the high dimension of the covariate process, no globally efficient estimators exist with a good practical performance at moderate sample sizes. We follow the approach of Robins and Rotnitzky (1992) by modeling the censoring variable, given the time-variable and the covariate-process, i.e., the missingness process, under the restriction that it satisfied coarsening at random. We propose a generalization of the simple current status estimator of the distribution of T and of smooth functionals of the distribution of T, which is based on an estimate of the missingness. In this estimator the covariates enter only through the estimate of the missingness process. Due to the coarsening at random assumption, the estimator has the interesting property that if we estimate the missingness process more nonparametrically, then we improve its efficiency. We show that by local estimation of an optimal model or optimal function of the covariates for the missingness process, the generalized current status estimator for smooth functionals become locally efficient; meaning it is efficient if the right model or covariate is consistently estimated and it is consistent and asymptotically normal in general. Estimation of the optimal model requires estimation of the conditional distribution of T, given the covariates. Any (prior) knowledge of this conditional distribution can be used at this stage without any risk of losing root-n consistency. We also propose locally efficient one step estimators. Finally, we show some simulation results.

Fitting of Mixtures with Unspecified Number of Components Using Cross Validation Distance Estimate

Estimation of the number of mixture components (k) is an unsolved problem. Available methods for estimation of k include bootstrapping the likelihood ratio test statistics and optimizing a variety of validity functionals such as AIC, BIC/MDL, and ICOMP. We investigate the minimization of distance between fitted mixture model and the true density as a method for estimating k. The distances considered are Kullback-Leibler (KL) and “L sub 2”. We estimate these distances using cross validation. A reliable estimate of k is obtained by voting of B estimates of k corresponding to B cross validation estimates of distance. This estimation methods with KL distance is very similar to Monte Carlo cross validated likelihood methods discussed by Smyth (2000). With focus on univariate normal mixtures, we present simulation studies that compare the cross validated distance method with AIC, BIC/MDL, and ICOMP. We also apply the cross validation estimate of distance approach along with AIC, BIC/MDL and ICOMP approach, to data from an osteoporosis drug trial in order to find groups that differentially respond to treatment.

A BAYESIAN SHRINKAGE MODEL FOR INCOMPLETE LONGITUDINAL BINARY DATA WITH APPLICATION TO THE BREAST CANCER PREVENTION TRIAL

We consider inference in randomized studies, in which repeatedly measured outcomes may be informatively missing due to drop out. In this setting, it is well known that full data estimands are not identified unless unverified assumptions are imposed. We assume a non-future dependence model for the drop-out mechanism and posit an exponential tilt model that links non-identifiable and identifiable distributions. This model is indexed by non-identified parameters, which are assumed to have an informative prior distribution, elicited from subject-matter experts. Under this model, full data estimands are shown to be expressed as functionals of the distribution of the observed data. To avoid the curse of dimensionality, we model the distribution of the observed data using a Bayesian shrinkage model. In a simulation study, we compare our approach to a fully parametric and a fully saturated model for the distribution of the observed data. Our methodology is motivated and applied to data from the Breast Cancer Prevention Trial.

OPTIMAL SHAPE IN ELECTROMAGNETIC SCATTERING BY SMALL ASPHERICAL PARTICLES

We consider the question of optimal shapes, e.g., those causing minimal extinction among all shapes of equal volume. Guided by the isoperimetric property of a sphere, relevant in the geometrical optics limit of scattering by large particles, we examine an analogous question in the low frequency (electrostatics) approximation, seeking to disentangle electric and geometric contributions. To that end, we survey the literature on shape functionals and focus on ellipsoids, giving a simple proof of spherical optimality for the coated ellipsoidal particle. Monotonic increase with asphericity in the low frequency regime for orientation-averaged induced dipole moments and scattering cross-sections is also shown. Additional physical insight is obtained from the Rayleigh-Gans (transparent) limit and eccentricity expansions. We propose connecting low and high frequency regime in a single minimum principle valid for all size parameters, provided that reasonable size distributions of randomly oriented aspherical particles wash out the resonances for intermediate size parameters. This proposal is further supported by the sum rule for integrated extinction.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Scaling relations between trabecular bone volume fraction and microstructure at different skeletal sites

In this study, we investigated the scaling relations between trabecular bone volume fraction (BV/TV) and parameters of the trabecular microstructure at different skeletal sites. Cylindrical bone samples with a diameter of 8mm were harvested from different skeletal sites of 154 human donors in vitro: 87 from the distal radius, 59/69 from the thoracic/lumbar spine, 51 from the femoral neck, and 83 from the greater trochanter. μCT images were obtained with an isotropic spatial resolution of 26μm. BV/TV and trabecular microstructure parameters (TbN, TbTh, TbSp, scaling indices (< > and σ of α and αz), and Minkowski Functionals (Surface, Curvature, Euler)) were computed for each sample. The regression coefficient β was determined for each skeletal site as the slope of a linear fit in the double-logarithmic representations of the correlations of BV/TV versus the respective microstructure parameter. Statistically significant correlation coefficients ranging from r=0.36 to r=0.97 were observed for BV/TV versus microstructure parameters, except for Curvature and Euler. The regression coefficients β were 0.19 to 0.23 (TbN), 0.21 to 0.30 (TbTh), −0.28 to −0.24 (TbSp), 0.58 to 0.71 (Surface) and 0.12 to 0.16 (<α>), 0.07 to 0.11 (<αz>), −0.44 to −0.30 (σ(α)), and −0.39 to −0.14 (σ(αz)) at the different skeletal sites. The 95% confidence intervals of β overlapped for almost all microstructure parameters at the different skeletal sites. The scaling relations were independent of vertebral fracture status and similar for subjects aged 60–69, 70–79, and >79years. In conclusion, the bone volume fraction–microstructure scaling relations showed a rather universal character.

Fast Update of Conditional Simulation Ensembles

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at mini- mal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on n ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 1. Compared to direct approaches, the proposed formulae proveto substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the q new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

Regularity conditions in the realisability problem in applications to point processes and random closed sets

We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive ex-tension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extens ion can be chosen to possess some invariance properties. The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

On degeneracy and invariances of random fields paths with applications in Gaussian process modelling

We study pathwise invariances and degeneracies of random fields with motivating applications in Gaussian process modelling. The key idea is that a number of structural properties one may wish to impose a priori on functions boil down to degeneracy properties under well-chosen linear operators. We first show in a second order set-up that almost sure degeneracy of random field paths under some class of linear operators defined in terms of signed measures can be controlled through the two first moments. A special focus is then put on the Gaussian case, where these results are revisited and extended to further linear operators thanks to state-of-the-art representations. Several degeneracy properties are tackled, including random fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process modelling. In a first numerical experiment, it is shown that dedicated kernels can be used to infer an axis of symmetry. Our second numerical experiment deals with conditional simulations of a solution to the heat equation, and it is found that adapted kernels notably enable improved predictions of non-linear functionals of the field such as its maximum.

Some Identification Issues in Nonparametric Linear Models with Endogenous Regressors

In applied work economists often seek to relate a given response variable y to some causal parameter mu* associated with it. This parameter usually represents a summarization based on some explanatory variables of the distribution of y, such as a regression function, and treating it as a conditional expectation is central to its identification and estimation. However, the interpretation of mu* as a conditional expectation breaks down if some or all of the explanatory variables are endogenous. This is not a problem when mu* is modelled as a parametric function of explanatory variables because it is well known how instrumental variables techniques can be used to identify and estimate mu*. In contrast, handling endogenous regressors in nonparametric models, where mu* is regarded as fully unknown, presents di±cult theoretical and practical challenges. In this paper we consider an endogenous nonparametric model based on a conditional moment restriction. We investigate identification related properties of this model when the unknown function mu* belongs to a linear space. We also investigate underidentification of mu* along with the identification of its linear functionals. Several examples are provided in order to develop intuition about identification and estimation for endogenous nonparametric regression and related models.

La resistencia especial en el fútbol

La finalidad de esta tesis es establecer un análisis de la metodología de entrenamiento de la resistencia especial en el futbolista. Su objetivo no está vinculado a realizar propuestas prácticas de entrenamiento, sino más bien, se tratará de abordar una posible justificación fisiológico - metabólica, a partir de la relevancia bioenergética de la creatina, en función de la creciente especialización que debe ir adquiriendo el proceso del entrenamiento deportivo a largo plazo, enfocado al logro de altos rendimientos deportivos. A partir del análisis de conceptos terminológicos de referencia, se asienta la idea general de este trabajo, es decir, la estructuración y desarrollo de la resistencia en los deportes de conjunto, como el fútbol. Los pilares de una adecuada planificación son el conocimiento y la aplicación de distintas leyes y principios del entrenamiento deportivo y su relación con los distintos medios y métodos de entrenamiento, como así también, los efectos de adaptación que provocan. Por lo tanto, a partir del análisis de los requerimientos morfológicos - funcionales de las competiciones de elite en fútbol, se pueden elaborar modelos que servirán de base y como objetivo final al cual debe ser orientado el proceso de entrenamiento. Es decir, que un entrenamiento multianual con miras a la formación de futbolistas de elite, debe respetar la especialización creciente de las cargas de entrenamiento, estableciendo una sucesión metodológica adecuada en función de los objetivos de cada etapa. En función de lo expuesto, se realiza un análisis que va desde la resistencia como capacidad física y su metodología de entrenamiento, recorriendo distintos conceptos y manifestaciones, pasando por el análisis de distintas zonas de intensidad o áreas funcionales, y desembocando en la metodología de entrenamiento intermitente de la resistencia o resistencia especial -en los deportes de conjunto-. Y es a partir de todo el análisis precedente que estamos en condiciones de abordar el entrenamiento específico en el fútbol, y más detalladamente la resistencia específica o intermitente que requiere este deporte. El entrenamiento intermitente puede ser considerado como una metodología cuyo énfasis es puesto en modificaciones que se producen a nivel muscular, por sobre factores centrales de rendimiento, presentándose como una variante óptima para el entrenamiento de la resistencia muscular local y específica del futbolista. Básicamente, el entrenamiento intermitente actuaría sobre dos puntos centrales: la mejora del sistema shuttle de la CrP, y sobre la rapidez de entrega de oxígeno al inicio del ejercicio. Aquí aparece la importancia de la suplementación con Cr: que al aumentar las concentraciones del sustrato, y junto con el entrenamiento, que mejora las reacciones enzimáticas implicadas, potenciaría las mejoras buscadas con este tipo de metodología. Queda por determinar cual es el preciso mecanismo de acción por el cual la recuperación de los fosfatos altamente energéticos se produce: si por biogénesis mitocondrial en las fibras reclutadas - generalmente FT -; o mediante el sistema de proteínas transportadoras de Cr - destacando la importancia de las ST - o por algún otro mecanismo no conocido. Su descubrimiento permitiría direccionar más precisamente el entrenamiento deportivo.

La resistencia especial en el fútbol

La finalidad de esta tesis es establecer un análisis de la metodología de entrenamiento de la resistencia especial en el futbolista. Su objetivo no está vinculado a realizar propuestas prácticas de entrenamiento, sino más bien, se tratará de abordar una posible justificación fisiológico - metabólica, a partir de la relevancia bioenergética de la creatina, en función de la creciente especialización que debe ir adquiriendo el proceso del entrenamiento deportivo a largo plazo, enfocado al logro de altos rendimientos deportivos. A partir del análisis de conceptos terminológicos de referencia, se asienta la idea general de este trabajo, es decir, la estructuración y desarrollo de la resistencia en los deportes de conjunto, como el fútbol. Los pilares de una adecuada planificación son el conocimiento y la aplicación de distintas leyes y principios del entrenamiento deportivo y su relación con los distintos medios y métodos de entrenamiento, como así también, los efectos de adaptación que provocan. Por lo tanto, a partir del análisis de los requerimientos morfológicos - funcionales de las competiciones de elite en fútbol, se pueden elaborar modelos que servirán de base y como objetivo final al cual debe ser orientado el proceso de entrenamiento. Es decir, que un entrenamiento multianual con miras a la formación de futbolistas de elite, debe respetar la especialización creciente de las cargas de entrenamiento, estableciendo una sucesión metodológica adecuada en función de los objetivos de cada etapa. En función de lo expuesto, se realiza un análisis que va desde la resistencia como capacidad física y su metodología de entrenamiento, recorriendo distintos conceptos y manifestaciones, pasando por el análisis de distintas zonas de intensidad o áreas funcionales, y desembocando en la metodología de entrenamiento intermitente de la resistencia o resistencia especial -en los deportes de conjunto-. Y es a partir de todo el análisis precedente que estamos en condiciones de abordar el entrenamiento específico en el fútbol, y más detalladamente la resistencia específica o intermitente que requiere este deporte. El entrenamiento intermitente puede ser considerado como una metodología cuyo énfasis es puesto en modificaciones que se producen a nivel muscular, por sobre factores centrales de rendimiento, presentándose como una variante óptima para el entrenamiento de la resistencia muscular local y específica del futbolista. Básicamente, el entrenamiento intermitente actuaría sobre dos puntos centrales: la mejora del sistema shuttle de la CrP, y sobre la rapidez de entrega de oxígeno al inicio del ejercicio. Aquí aparece la importancia de la suplementación con Cr: que al aumentar las concentraciones del sustrato, y junto con el entrenamiento, que mejora las reacciones enzimáticas implicadas, potenciaría las mejoras buscadas con este tipo de metodología. Queda por determinar cual es el preciso mecanismo de acción por el cual la recuperación de los fosfatos altamente energéticos se produce: si por biogénesis mitocondrial en las fibras reclutadas - generalmente FT -; o mediante el sistema de proteínas transportadoras de Cr - destacando la importancia de las ST - o por algún otro mecanismo no conocido. Su descubrimiento permitiría direccionar más precisamente el entrenamiento deportivo.

La resistencia especial en el fútbol

La finalidad de esta tesis es establecer un análisis de la metodología de entrenamiento de la resistencia especial en el futbolista. Su objetivo no está vinculado a realizar propuestas prácticas de entrenamiento, sino más bien, se tratará de abordar una posible justificación fisiológico - metabólica, a partir de la relevancia bioenergética de la creatina, en función de la creciente especialización que debe ir adquiriendo el proceso del entrenamiento deportivo a largo plazo, enfocado al logro de altos rendimientos deportivos. A partir del análisis de conceptos terminológicos de referencia, se asienta la idea general de este trabajo, es decir, la estructuración y desarrollo de la resistencia en los deportes de conjunto, como el fútbol. Los pilares de una adecuada planificación son el conocimiento y la aplicación de distintas leyes y principios del entrenamiento deportivo y su relación con los distintos medios y métodos de entrenamiento, como así también, los efectos de adaptación que provocan. Por lo tanto, a partir del análisis de los requerimientos morfológicos - funcionales de las competiciones de elite en fútbol, se pueden elaborar modelos que servirán de base y como objetivo final al cual debe ser orientado el proceso de entrenamiento. Es decir, que un entrenamiento multianual con miras a la formación de futbolistas de elite, debe respetar la especialización creciente de las cargas de entrenamiento, estableciendo una sucesión metodológica adecuada en función de los objetivos de cada etapa. En función de lo expuesto, se realiza un análisis que va desde la resistencia como capacidad física y su metodología de entrenamiento, recorriendo distintos conceptos y manifestaciones, pasando por el análisis de distintas zonas de intensidad o áreas funcionales, y desembocando en la metodología de entrenamiento intermitente de la resistencia o resistencia especial -en los deportes de conjunto-. Y es a partir de todo el análisis precedente que estamos en condiciones de abordar el entrenamiento específico en el fútbol, y más detalladamente la resistencia específica o intermitente que requiere este deporte. El entrenamiento intermitente puede ser considerado como una metodología cuyo énfasis es puesto en modificaciones que se producen a nivel muscular, por sobre factores centrales de rendimiento, presentándose como una variante óptima para el entrenamiento de la resistencia muscular local y específica del futbolista. Básicamente, el entrenamiento intermitente actuaría sobre dos puntos centrales: la mejora del sistema shuttle de la CrP, y sobre la rapidez de entrega de oxígeno al inicio del ejercicio. Aquí aparece la importancia de la suplementación con Cr: que al aumentar las concentraciones del sustrato, y junto con el entrenamiento, que mejora las reacciones enzimáticas implicadas, potenciaría las mejoras buscadas con este tipo de metodología. Queda por determinar cual es el preciso mecanismo de acción por el cual la recuperación de los fosfatos altamente energéticos se produce: si por biogénesis mitocondrial en las fibras reclutadas - generalmente FT -; o mediante el sistema de proteínas transportadoras de Cr - destacando la importancia de las ST - o por algún otro mecanismo no conocido. Su descubrimiento permitiría direccionar más precisamente el entrenamiento deportivo.

Optimal fits of diffusion constants from single-time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.