Showing papers in "Biometrika in 2016"

Going off grid: computationally efficient inference for log-Gaussian Cox processes

Abstract: This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making novel use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, while an approximation based on a counting process on a partition of the domain only achieves first-order convergence. The given results improve on the general theory of convergence of the stochastic partial differential equation models, introduced by Lindgren et al. (2011). The new method is demonstrated on a standard point pattern data set and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of Chakraborty et al. (2011). The second extension constructs a log-Gaussian Cox process on the world's oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

Partially functional linear regression in high dimensions

Abstract: SUMMARY In modern experiments, functional and nonfunctional data are often encountered simultaneously when observations are sampled from random processes and high-dimensional scalar covariates. It is difficult to apply existing methods for model selection and estimation. We propose a new class of partially functional linear models to characterize the regression between a scalar response and covariates of both functional and scalar types. The new approach provides a unified and flexible framework that simultaneously takes into account multiple functional and ultrahigh-dimensional scalar predictors, enables us to identify important features, and offers improved interpretability of the estimators. The underlying processes of the functional predictors are considered to be infinite-dimensional, and one of our contributions is to characterize the effects of regularization on the resulting estimators. We establish the consistency and oracle properties of the proposed method under mild conditions, demonstrate its performance with simulation studies, and illustrate its application using air pollution data.

Fast sampling with Gaussian scale mixture priors in high-dimensional regression

Abstract: We propose an efficient way to sample from a class of structured multivariate Gaussian distributions. The proposed algorithm only requires matrix multiplications and linear system solutions. Its computational complexity grows linearly with the dimension, unlike existing algorithms that rely on Cholesky factorizations with cubic complexity. The algorithm is broadly applicable in settings where Gaussian scale mixture priors are used on high-dimensional parameters. Its effectiveness is illustrated through a high-dimensional regression problem with a horseshoe prior on the regression coefficients. Other potential applications are outlined.

Combining eigenvalues and variation of eigenvectors for order determination

Abstract: In applying statistical methods such as principal component analysis, canonical correlation analysis, and sufficient dimension reduction, we need to determine how many eigenvectors of a random matrix are important for estimation. This problem is known as order determination, and amounts to estimating the rank of a matrix. Previous order-determination procedures rely either on the decreasing pattern, or elbow, of the eigenvalues, or on the increasing pattern of the variability in the directions of the eigenvectors. In this paper we propose a new order-determination procedure by exploiting both patterns: when the eigenvalues of a random matrix are close together, their eigenvectors tend to vary greatly; when the eigenvalues are far apart, their variability tends to be small. The combination of both helps to pinpoint the rank of a matrix more precisely than the previous methods. We establish the consistency of the new order-determination procedure, and compare it with other such procedures by simulation and in an applied setting.

Semiparametric inverse propensity weighting for nonignorable missing data

Abstract: To estimate unknown population parameters based on data having nonignorable missing values with a semiparametric exponential tilting propensity, Kim & Yu (2011) assumed that the tilting parameter is known or can be estimated from external data, in order to avoid the identifiability issue. To remove this serious limitation on the methodology, we use an instrument, i.e., a covariate related to the study variable but unrelated to the missing data propensity, to construct some estimating equations. Because these estimating equations are semiparametric, we profile the nonparametric component using a kernel-type estimator and then estimate the tilting parameter based on the profiled estimating equations and the generalized method of moments. Once the tilting parameter is estimated, so is the propensity, and then other population parameters can be estimated using the inverse propensity weighting approach. Consistency and asymptotic normality of the proposed estimators are established. The finite-sample performance of the estimators is studied through simulation, and a real-data example is also presented.

Empirical Bayes deconvolution estimates

Abstract: An unknown prior density $g(\theta )$ has yielded realizations $\Theta _1,\ldots ,\Theta _N$. They are unobservable, but each $\Theta _i$ produces an observable value $X_i$ according to a known probability mechanism, such as $X_i\sim {\rm Po}(\Theta _i)$. We wish to estimate $g(\theta )$ from the observed sample $X_1,\ldots ,X_N$. Traditional asymptotic calculations are discouraging, indicating very slow nonparametric rates of convergence. In this article we show that parametric exponential family modelling of $g(\theta )$ can give useful estimates in moderate-sized samples. We illustrate the approach with a variety of real and artificial examples. Covariate information can be incorporated into the deconvolution process, leading to a more detailed theory of generalized linear mixed models.

Maximum likelihood estimation for semiparametric transformation models with interval-censored data

Abstract: Interval censoring arises frequently in clinical, epidemiological, financial and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We devise an EM-type algorithm that converges stably, even in the presence of time-dependent covariates, and show that the estimators for the regression parameters are consistent, asymptotically normal, and asymptotically efficient with an easily estimated covariance matrix. Finally, we demonstrate the performance of our procedures through simulation studies and application to an HIV/AIDS study conducted in Thailand.

Exact simulation of max-stable processes.

Abstract: Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

On inverse probability-weighted estimators in the presence of interference

Abstract: We consider inference about the causal effect of a treatment or exposure in the presence of interference, i.e., when one individual's treatment affects the outcome of another individual. In the observational setting where the treatment assignment mechanism is not known, inverse probability-weighted estimators have been proposed when individuals can be partitioned into groups such that there is no interference between individuals in different groups. Unfortunately this assumption, which is sometimes referred to as partial interference, may not hold, and moreover existing weighted estimators may have large variances. In this paper we consider weighted estimators that could be employed when interference is present. We first propose a generalized inverse probability-weighted estimator and two Hajek-type stabilized weighted estimators that allow any form of interference. We derive their asymptotic distributions and propose consistent variance estimators assuming partial interference. Empirical results show that one of the Hajek estimators can have substantially smaller finite-sample variance than the other estimators. The different estimators are illustrated using data on the effects of rotavirus vaccination in Nicaragua.

An adaptive two-sample test for high-dimensional means.

Abstract: Several two-sample tests for high-dimensional data have been proposed recently, but they are powerful only against certain limited alternative hypotheses. In practice, since the true alternative hypothesis is unknown, it is unclear how to choose a powerful test. We propose an adaptive test that maintains high power across a wide range of situations, and study its asymptotic properties. Its finite sample performance is compared with existing tests. We apply it and other tests to detect possible associations between bipolar disease and a large number of single nucleotide polymorphisms on each chromosome based on a genome-wide association study dataset. Numerical studies demonstrate the superior performance and high power of the proposed test across a wide spectrum of applications.

On varieties of doubly robust estimators under missingness not at random with a shadow variable.

Abstract: Suppose we are interested in the mean of an outcome variable missing not at random. Suppose however that one has available a fully observed shadow variable, which is associated with the outcome but independent of the missingness process conditional on covariates and the possibly unobserved outcome. Such a variable may be a proxy or a mismeasured version of the outcome and is available for all individuals. We have previously established necessary and sufficient conditions for identification of the full data law in such a setting, and have described semiparametric estimators including a doubly robust estimator of the outcome mean. Here, we propose two alternative estimators, which may be viewed as extensions of analogous methods under missingness at random, but enjoy different properties. We assess the correctness of the required working models via straightforward goodness-of-fit tests.

On the win-ratio statistic in clinical trials with multiple types of event

Abstract: Pocock et al. (2012), following Finkelstein & Schoenfeld (1999), has popularized the win ratio for analysis of controlled clinical trials with multiple types of outcome event. The approach uses pairwise comparisons between patients in the treatment and control groups using a primary outcome, say the time to death, with ties broken using a secondary outcome, say the time to hospitalization. In general the observed pairwise preferences and the weight they attach to the component rankings will depend on the distribution of potential follow-up time. We present expressions for the win and loss probabilities for general bivariate survival models when follow-up of all patients is limited to a specified time horizon. In the special case of a bivariate Lehmann model we show that the win ratio does not depend on this horizon. We show how the win ratio may be estimated nonparametrically or from a parametric model. Extensions to events of three or more types are described. Application of the method of marginal estimation due to Wei et al. (1989) to this problem is described.

Bootstrap-based testing of equality of mean functions or equality of covariance operators for functional data

Abstract: We investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data. Theoretical size and power results are derived for certain test statistics, whose limiting distributions depend on unknown infinite-dimensional parameters. Simulations demonstrate good size and power of the bootstrap-based functional tests.

High-dimensional and banded vector autoregressions

Abstract: We consider a class of vector autoregressive models with banded coefficient matrices. The setting represents a type of sparse structure for high-dimensional time series, though the implied autocovariance matrices are not banded. The structure is also practically meaningful when the order of component time series is arranged appropriately. The convergence rates for the estimated banded autoregressive coefficient matrices are established. We also propose a Bayesian information criterion for determining the width of the bands in the coefficient matrices, which is proved to be consistent. By exploring some approximate banded structure for the autocovariance functions of banded vector autoregressive processes, consistent estimators for the auto-covariance matrices are constructed.

Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding.

Abstract: It is often of interest to decompose the total effect of an exposure into a component that acts on the outcome through some mediator and a component that acts independently through other pathways. Said another way, we are interested in the direct and indirect effects of the exposure on the outcome. Even if the exposure is randomly assigned, it is often infeasible to randomize the mediator, leaving the mediator-outcome confounding not fully controlled. We develop a sensitivity analysis technique that can bound the direct and indirect effects without parametric assumptions about the unmeasured mediator-outcome confounding.

A Bayesian view of doubly robust causal inference

Abstract: In causal inference the effect of confounding may be controlled using regression adjustment in an outcome model, propensity score adjustment, inverse probability of treatment weighting or a combination of these. Approaches based on modelling the treatment assignment mechanism, along with their doubly robust extensions, have been difficult to motivate using formal likelihood-based or Bayesian arguments, as the treatment assignment model plays no part in inferences concerning the expected outcomes. On the other hand, forcing dependency between the outcome and treatment assignment models by allowing the former to be misspecified results in loss of the balancing property of the propensity scores and the loss of any double robustness. In this paper, we explain in the framework of misspecified models why doubly robust inferences cannot arise from purely likelihood-based arguments. As an alternative to Bayesian propensity score analysis, we propose a Bayesian posterior predictive method for constructing doubly robust estimation procedures by incorporating the inverse treatment assignment probabilities as importance sampling weights in Monte Carlo integration.

Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression

Abstract: The envelope model allows efficient estimation in multivariate linear regression. In this paper, we propose the sparse envelope model, which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients. The envelope estimator is consistent but not sparse, and in many situations it is important to identify the response variables for which the regression coefficients are zero. The sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model. Response variable selection arises naturally in many applications, but has not been studied as thoroughly as predictor variable selection. In this paper, we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts. In response variable selection, even if a response has zero coefficients, it should still be retained to improve the estimation efficiency of the nonzero coefficients. This is different from the practice in predictor variable selection. We establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator.

Approximating fragmented functional data by segments of Markov chains

Abstract: We consider curve extension and linear prediction for functional data observed only on a part of their domain, in the form of fragments. We suggest an approach based on a combination of Markov chains and nonparametric smoothing techniques, which enables us to extend the observed fragments and construct approximated prediction intervals around them, construct mean and covariance function estimators, and derive a linear predictor. The procedure is illustrated on real and simulated data.

Spatial regression models over two-dimensional manifolds

Abstract: We propose a regression model for data spatially distributed over general two-dimensional Riemannian manifolds. This is a generalized additive model with a roughness penalty term involving a differential operator computed over the non-planar domain. By virtue of a semiparametric framework, the model allows inclusion of space-varying covariate information. Estimation can be performed by conformally parameterizing the non-planar domain and then generalizing existing models for penalized spatial regression over planar domains. The conformal coordinates and the estimation problem are both computed with a finite element approach.

High-dimensional classification via nonparametric empirical Bayes and maximum likelihood inference

Abstract: We propose new nonparametric empirical Bayes methods for high-dimensional classification. Our classifiers are designed to approximate the Bayes classifier in a hypothesized hierarchical model, where the prior distributions for the model parameters are estimated nonparametrically from the training data. As is common with nonparametric empirical Bayes, the proposed classifiers are effective in high-dimensional settings even when the underlying model parameters are in fact nonrandom. We use nonparametric maximum likelihood estimates of the prior distributions, following the elegant approach studied by Kiefer & Wolfowitz in the 1950s. However, our implementation is based on a recent convex optimization framework for approximating these estimates that is well-suited for large-scale problems. We derive new theoretical results on the accuracy of the approximate estimator, which help control the misclassification rate of one of our classifiers. We show that our methods outperform several existing methods in simulations and perform well when gene expression microarray data is used to classify cancer patients.

Multivariate spatial covariance models: a conditional approach

Abstract: Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables at any sets of locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model must be nonnegative-definite. In this article, we develop a conditional approach for spatial-model construction whose validity conditions are easy to check. We start with bivariate spatial covariance models and go on to demonstrate the approach’s connection to multivariate models defined by networks of spatial variables. In some circumstances, such as modelling respiratory illness conditional on air pollution, the direction of conditional dependence is clear. When it is not, the two directional models can be compared. More generally, the graph structure of the network reduces the number of possible models to compare. Model selection then amounts to finding possible causative links in the network. We demonstrate our conditional approach on surface temperature and pressure data, where the role of the two variables is seen to be asymmetric.

Default Bayesian analysis with global-local shrinkage priors

Abstract: We provide a framework for assessing the default nature of a prior distribution using the property of regular variation, which we study for global-local shrinkage priors. In particular, we show that the horseshoe priors, originally designed to handle sparsity, are regularly varying and thus are appropriate for default Bayesian analysis. To illustrate our methodology, we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions. In each case, we show that global-local horseshoe priors perform as required. Global-local shrinkage priors can separate a low-dimensional signal from high-dimensional noise even for nonlinear functions.

Nonparametric identification and maximum likelihood estimation for hidden Markov models

Abstract: Nonparametric identification and maximum likelihood estimation for finite-state hidden Markov models are investigated. We obtain identification of the parameters as well as the order of the Markov chain if the transition probability matrices have full-rank and are ergodic, and if the state-dependent distributions are all distinct, but not necessarily linearly independent. Based on this identification result, we develop a nonparametric maximum likelihood estimation theory. First, we show that the asymptotic contrast, the Kullback-Leibler divergence of the hidden Markov model, also identifies the true parameter vector nonparametrically. Second, for classes of state-dependent densities which are arbitrary mixtures of a parametric family, we establish the consistency of the nonparametric maximum likelihood estimator. Here, identification of the mixing distributions need not be assumed. Numerical properties of the estimates and of nonparametric goodness of fit tests are investigated in a simulation study.

Particle Metropolis-adjusted Langevin algorithms

Abstract: This paper proposes a new sampling scheme based on Langevin dynamics that is applicable within pseudo-marginal and particle Markov chain Monte Carlo algorithms. We investigate this algorithm's theoretical properties under standard asymptotics, which correspond to an increasing dimension of the parameters, $n$. Our results show that the behaviour of the algorithm depends crucially on how accurately one can estimate the gradient of the log target density. If the error in the estimate of the gradient is not sufficiently controlled as dimension increases, then asymptotically there will be no advantage over the simpler random-walk algorithm. However, if the error is sufficiently well-behaved, then the optimal scaling of this algorithm will be $O(n^{-1/6})$ compared to $O(n^{-1/2})$ for the random walk. Our theory also gives guidelines on how to tune the number of Monte Carlo samples in the likelihood estimate and the proposal step-size.

A note on multiple imputation for method of moments estimation

Abstract: Multiple imputation is a popular imputation method for general purpose estimation. Rubin(1987) provided an easily applicable formula for the variance estimation of multiple imputation. However, the validity of the multiple imputation inference requires the congeniality condition of Meng(1994), which is not necessarily satisfied for method of moments estimation. This paper presents the asymptotic bias of Rubin's variance estimator when the method of moments estimator is used as a complete-sample estimator in the multiple imputation procedure. A new variance estimator based on over-imputation is proposed to provide asymptotically valid inference for method of moments estimation.

Intrinsic efficiency and multiple robustness in longitudinal studies with drop-out

Abstract: Intrinsic efficiency and multiple robustness are desirable properties in missing data analysis. We establish both for estimating the mean of a response at the end of a longitudinal study with drop-out. The idea is to calibrate the estimated missingness probability at each visit using data from past visits. We consider one working model for the missingness probability and multiple working models for the data distribution. Intrinsic efficiency guarantees that, when the missingness probability is correctly modelled, the multiple data distribution models, combined with data prior to the end of the study, are optimally accommodated to maximize efficiency. The efficiency generally increases with the number of data distribution models, except where one such model is correctly specified as well, in which case all the proposed estimators attain the semiparametric efficiency bound. Multiple robustness ensures estimation consistency if the missingness probability model is misspecified but one data distribution model is correct. Our proposed estimators are all convex combinations of the observed responses, and thus always fall within the parameter space.

Fréchet integration and adaptive metric selection for interpretable covariances of multivariate functional data

Abstract: SUMMARY For multivariate functional data recorded for a sample of subjects on a common domain, one 10 is often interested in the covariance between pairs of the component functions, extending the notion of a covariance matrix for multivariate data to the functional case. A straightforward approach is to integrate the pointwise covariance matrices over the functional time domain. We generalize this approach by defining the Frintegral, which depends on the metric chosen for the space of covariance matrices, and demonstrate that the ordinary integration is a special 15 case when the Frobenius metric is used. As the space of covariance matrices is nonlinear, we propose a class of power metrics as alternatives to the Frobenius metric. For any such power metric, the calculation of Frintegrals is equivalent to transforming the covariance matrices with the chosen power, applying the classical Riemann integral to the transformed matrices, and finally applying the inverse transformation to return to the original scale. We also propose data- 20 adaptive metric selection with respect to a user-specified target criterion, for example fastest decline of the eigenvalues, establish consistency of the proposed procedures and demonstrate their effectiveness in a simulation. The proposed functional covariance approach through Fr´ echet integration is illustrated in a comparison of connectivity between brain voxels for normal subjects and Alzheimer's patients based on fMRI data. 25

Data augmentation for models based on rejection sampling.

Abstract: We present a data augmentation scheme to perform Markov chain Monte Carlo inference for models where data generation involves a rejection sampling algorithm. Our idea is a simple scheme to instantiate the rejected proposals preceding each data point. The resulting joint probability over observed and rejected variables can be much simpler than the marginal distribution over the observed variables, which often involves intractable integrals. We consider three problems: modelling flow-cytometry measurements subject to truncation; the Bayesian analysis of the matrix Langevin distribution on the Stiefel manifold; and Bayesian inference for a nonparametric Gaussian process density model. The latter two are instances of doubly-intractable Markov chain Monte Carlo problems, where evaluating the likelihood is intractable. Our experiments demonstrate superior performance over state-of-the-art sampling algorithms for such problems.

Semiparametric approach to regression with a covariate subject to a detection limit

Abstract: We consider generalized linear regression with a covariate left-censored at a lower detection limit. Complete-case analysis, where observations with values below the limit are eliminated, yields valid estimates for regression coefficients but loses efficiency, ad hoc substitution methods are biased, and parametric maximum likelihood estimation relies on parametric models for the unobservable tail probability distribution and may suffer from model misspecification. To obtain robust and more efficient results, we propose a semiparametric likelihood-based approach using an accelerated failure time model for the covariate subject to the detection limit. A two-stage estimation procedure is developed, where the conditional distribution of this covariate given other variables is estimated prior to maximizing the likelihood function. The proposed method outperforms complete-case analysis and substitution methods in simulation studies. Technical conditions for desirable asymptotic properties are provided.