Spatiotemporal prediction for log-Gaussian Cox processes

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.     

Multivariate geometric anisotropic Cox processes

This paper introduces a new modeling and inference framework for multivariate and anisotropic point processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields, and from them a family of anisotropic point processes. We give conditions that make the proposed models valid. We also propose a Palm likelihood-based inference method for this type of point process, circumventing issues of likelihood tractability. Finally we illustrate the utility of the proposed modeling framework by analyzing spatial ecological observations of plants and trees in the Barro Colorado Island data.     

Regularized estimation for highly multivariate log Gaussian Cox processes

Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper, we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.     

Partial least squares Cox regression for genome-wide data

Most methods for survival prediction from high-dimensional genomic data combine the Cox proportional hazards model with some technique of dimension reduction, such as partial least squares regression (PLS). Applying PLS to the Cox model is not entirely straightforward, and multiple approaches have been proposed. The method of Park et al. (Bioinformatics 18(Suppl. 1):S120–S127, 2002) uses a reformulation of the Cox likelihood to a Poisson type likelihood, thereby enabling estimation by iteratively reweighted partial least squares for generalized linear models. We propose a modification of the method of park et al. (2002) such that estimates of the baseline hazard and the gene effects are obtained in separate steps. The resulting method has several advantages over the method of park et al. (2002) and other existing Cox PLS approaches, as it allows for estimation of survival probabilities for new patients, enables a less memory-demanding estimation procedure, and allows for incorporation of lower-dimensional non-genomic variables like disease grade and tumor thickness. We also propose to combine our Cox PLS method with an initial gene selection step in which genes are ordered by their Cox score and only the highest-ranking k% of the genes are retained, obtaining a so-called supervised partial least squares regression method. In simulations, both the unsupervised and the supervised version outperform other Cox PLS methods.     

Cox regression for current status data with mismeasured covariates

Covariate measurement error problems have been extensively studied in the context of right‐censored data but less so for current status data. Motivated by the zebrafish basal cell carcinoma (BCC) study, where the occurrence time of BCC was only known to lie before or after a sacrifice time and where the covariate (Sonic hedgehog expression) was measured with error, the authors describe a semiparametric maximum likelihood method for analyzing current status data with mismeasured covariates under the proportional hazards model. They show that the estimator of the regression coefficient is asymptotically normal and efficient and that the profile likelihood ratio test is asymptotically Chi‐squared. They also provide an easily implemented algorithm for computing the estimators. They evaluate their method through simulation studies, and illustrate it with a real data example. The Canadian Journal of Statistics 39: 73–88; 2011 © 2011 Statistical Society of Canada     

Piecewise Cox models with right-censored data

We study a general class of piecewise Cox models. We discuss the computation of the semi-parametric maximum likelihood estimates (SMLE) of the parameters, with right-censored data, and a simplified algorithm for the maximum partial likelihood estimates (MPLE). Our simulation study suggests that the relative efficiency of the PMLE of the parameter to the SMLE ranges from 96% to 99.9%, but the relative efficiency of the existing estimators of the baseline survival function to the SMLE ranges from 3% to 24%. Thus, the SMLE is much better than the existing estimators.     

Bayesian inference for Gaussian excursion set generated Cox processes with set-marking

This work considers spatial Cox point processes where the random intensity is defined by a random closed set such that different point intensities appear in the two phases formed by the random set and its complement. The point pattern is observed as a set of point coordinates in a bounded region W?? ^d together with the information on the phase of the location of each point. This phase information, called set-marking, is not a representative sample from the random set, and hence it cannot be directly used for deducing properties of the random set. Excursion sets of continuous-parameter Gaussian random fields are applied as a flexible model for the random set. Fully Bayesian method and Markov chain Monte Carlo (MCMC) simulation is adopted for inferring the parameters of the model and estimating the random set. The performance of the new approach is studied by means of simulation experiments. Further, two forestry data sets on point patterns of saplings are analysed. The saplings grow in a clear-cut forest area where, before planting and natural seeding, the soil has been mounded forming a blotched soil structure. The tree densities tend to be different in the tilled patches and in the area outside the patches. The coordinates of each sapling have been measured and it is known whether this location is in a patch or outside. This example has been a motivation for the study.     

Cox regression for mixed case interval-censored data with covariate errors

Covariate measurement error problems have been extensively studied in the context of right-censored data but less so for interval-censored data. Motivated by the AIDS Clinical Trial Group 175 study, where the occurrence time of AIDS was examined only at intermittent clinic visits and the baseline covariate CD4 count was measured with error, we describe a semiparametric maximum likelihood method for analyzing mixed case interval-censored data with mismeasured covariates under the proportional hazards model. We show that the estimator of the regression coefficient is asymptotically normal and efficient and provide a very stable and efficient algorithm for computing the estimators. We evaluate the method through simulation studies and illustrate it with AIDS data.     

Bayesian computation for Log-Gaussian Cox processes: a comparative analysis of methods

The Log-Gaussian Cox process is a commonly used model for the analysis of spatial point pattern data. Fitting this model is difficult because of its doubly stochastic property, that is, it is a hierarchical combination of a Poisson process at the first level and a Gaussian process at the second level. Various methods have been proposed to estimate such a process, including traditional likelihood-based approaches as well as Bayesian methods. We focus here on Bayesian methods and several approaches that have been considered for model fitting within this framework, including Hamiltonian Monte Carlo, the Integrated nested Laplace approximation, and Variational Bayes. We consider these approaches and make comparisons with respect to statistical and computational efficiency. These comparisons are made through several simulation studies as well as through two applications, the first examining ecological data and the second involving neuroimaging data.     

Using informative priors for handling missing data problem in Cox regression

The aim of this study is to determine the effect of informative priors for variables with missing value and to compare Bayesian Cox regression and Cox regression analysis. For this purpose, firstly simulated data sets with different sample size within different missing rate were generated and each of data sets were analysed by Cox regression and Bayesian Cox regression with informative prior. Secondly lung cancer data set as real data set was used for analysis. Consequently, using informative priors for variables with missing value solved the missing data problem.     

Information criteria for inhomogeneous spatial point processes

The theoretical foundation for a number of model selection criteria is established in the context of inhomogeneous point processes and under various asymptotic settings: infill, increasing domain and combinations of these. For inhomogeneous Poisson processes we consider Akaike's information criterion and the Bayesian information criterion, and in particular we identify the point process analogue of ‘sample size’ needed for the Bayesian information criterion. Considering general inhomogeneous point processes we derive new composite likelihood and composite Bayesian information criteria for selecting a regression model for the intensity function. The proposed model selection criteria are evaluated using simulations of Poisson processes and cluster point processes.     

Two-step estimation for inhomogeneous spatial point processes

Summary.  The paper is concerned with parameter estimation for inhomogeneous spatial point processes with a regression model for the intensity function and tractable second-order properties ( K -function). Regression parameters are estimated by using a Poisson likelihood score estimating function and in the second step minimum contrast estimation is applied for the residual clustering parameters. Asymptotic normality of parameter estimates is established under certain mixing conditions and we exemplify how the results may be applied in ecological studies of rainforests.     

Analysis of dependently truncated data in Cox framework

Truncation is a known feature of bone marrow transplant (BMT) registry data, for which the survival time of a leukemia patient is left truncated by the waiting time to transplant. It was recently noted that a longer waiting time was linked to poorer survival. A straightforward solution is a Cox model on the survival time with the waiting time as both truncation variable and covariate. The Cox model should also include other recognized risk factors as covariates. In this article, we focus on estimating the distribution function of waiting time and the probability of selection under the aforementioned Cox model.     

Two-scale spatial models for binary data

A spatial lattice model for binary data is constructed from two spatial scales linked through conditional probabilities. A coarse grid of lattice locations is specified, and all remaining locations (which we call the background) capture fine-scale spatial dependence. Binary data on the coarse grid are modelled with an autologistic distribution, conditional on the binary process on the background. The background behaviour is captured through a hidden Gaussian process after a logit transformation on its Bernoulli success probabilities. The likelihood is then the product of the (conditional) autologistic probability distribution and the hidden Gaussian–Bernoulli process. The parameters of the new model come from both spatial scales. A series of simulations illustrates the spatial-dependence properties of the model and likelihood-based methods are used to estimate its parameters. Presence–absence data of corn borers in the roots of corn plants are used to illustrate how the model is fitted.     

Nonparametric spectrum estimation for spatial data

Smoothed nonparametric kernel spectral density estimates are considered for stationary data observed on a d-dimensional lattice. The implications for edge effect bias of the choice of kernel and bandwidth are considered. Under some circumstances the bias can be dominated by the edge effect. We show that this problem can be mitigated by tapering. Some extensions and related issues are discussed.     

Mantel test for spatial functional data

Statistics for spatial functional data is an emerging field in statistics which combines methods of spatial statistics and functional data analysis to model spatially correlated functional data. Checking for spatial autocorrelation is an important step in the statistical analysis of spatial data. Several statistics to achieve this goal have been proposed. The test based on the Mantel statistic is widely known and used in this context. This paper proposes an application of this test to the case of spatial functional data. Although we focus particularly on geostatistical functional data, that is functional data observed in a region with spatial continuity, the test proposed can also be applied with functional data which can be measured on a discrete set of areas of a region (areal functional data) by defining properly the distance between the areas. Based on two simulation studies, we show that the proposed test has a good performance. We illustrate the methodology by applying it to an agronomic data set.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

Local influence for incomplete data models

This paper proposes a method to assess the local influence in a minor perturbation of a statistical model with incomplete data. The idea is to utilize Cook's approach to the conditional expectation of the complete-data log-likelihood function in the EM algorithm. It is shown that the method proposed produces analytic results that are very similar to those obtained from a classical local influence approach based on the observed data likelihood function and has the potential to assess a variety of complicated models that cannot be handled by existing methods. An application to the generalized linear mixed model is investigated. Some illustrative artificial and real examples are presented.