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  • Articles  (544)
  • Oxford University Press  (544)
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  • 2015-2019  (544)
  • Biometrika  (170)
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  • 1
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    Tree-based methods for individualized treatment regimes (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Individualized treatment rules recommend treatments on the basis of individual patient characteristics. A high-quality treatment rule can produce better patient outcomes, lower costs and less treatment burden. If a treatment rule learned from data is to be used to inform clinical practice or provide scientific insight, it is crucial that it be interpretable; clinicians may be unwilling to implement models they do not understand, and black-box models may not be useful for guiding future research. The canonical example of an interpretable prediction model is a decision tree. We propose a method for estimating an optimal individualized treatment rule within the class of rules that are representable as decision trees. The class of rules we consider is interpretable but expressive. A novel feature of this problem is that the learning task is unsupervised, as the optimal treatment for each patient is unknown and must be estimated. The proposed method applies to both categorical and continuous treatments and produces favourable marginal mean outcomes in simulation experiments. We illustrate it using data from a study of major depressive disorder.
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    Diagnostic studies in sufficient dimension reduction (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information. Numerous dimension reduction methods have been developed based on this paradigm. However, little effort has been devoted to diagnostic studies within the context of dimension reduction. In this paper we introduce methods to check goodness-of-fit for a given dimension reduction subspace. The key idea is to extend the so-called distance correlation to measure the conditional dependence relationship between the covariates and the response given a reduction subspace. Our methods require minimal assumptions, which are usually much less restrictive than the conditions needed to justify the original methods. Asymptotic properties of the test statistic are studied. Numerical examples demonstrate the effectiveness of the proposed approach.
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    Bayesian sensitivity analysis with the Fisher-Rao metric (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: We propose a geometric framework to assess sensitivity of Bayesian procedures to modelling assumptions based on the nonparametric Fisher–Rao metric. While the framework is general, the focus of this article is on assessing local and global robustness in Bayesian procedures with respect to perturbations of the likelihood and prior, and on the identification of influential observations. The approach is based on a square-root representation of densities, which enables analytical computation of geodesic paths and distances, facilitating the definition of naturally calibrated local and global discrepancy measures. An important feature of our approach is the definition of a geometric $\epsilon$ -contamination class of sampling distributions and priors via intrinsic analysis on the space of probability density functions. We demonstrate the applicability of our framework to generalized mixed-effects models and to directional and shape data.
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    Efficient estimation of nonparametric genetic risk function with censored data (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: With the discovery of an increasing number of causal genes for complex human disorders, it is crucial to assess the genetic risk of disease onset for individuals who are carriers of these causal mutations and to compare the distribution of the age-at-onset for such individuals with the distribution for noncarriers. In many genetic epidemiological studies that aim to estimate causal gene effect on disease, the age-at-onset of disease is subject to censoring. In addition, the mutation carrier or noncarrier status of some individuals may be unknown, due to the high cost of in-person ascertainment by collecting DNA samples or because of the death of older individuals. Instead, the probability of such individuals’ mutation status can be obtained from various other sources. When mutation status is missing, the available data take the form of censored mixture data. Recently, various methods have been proposed for risk estimation using such data, but none is efficient for estimating a nonparametric distribution. We propose a fully efficient sieve maximum likelihood estimation method, in which we estimate the logarithm of the hazard ratio between genetic mutation groups using B-splines, while applying nonparametric maximum likelihood estimation to the reference baseline hazard function. Our estimator can be calculated via an expectation-maximization algorithm which is much faster than existing methods. We show that our estimator is consistent and semiparametrically efficient and establish its asymptotic distribution. Simulation studies demonstrate the superior performance of the proposed method, which is used to estimate the distribution of the age-at-onset of Parkinson's disease for carriers of mutations in the leucine-rich repeat kinase 2, LRRK2, gene.
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    Efficient computation of smoothing splines via adaptive basis sampling (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Smoothing splines provide flexible nonparametric regression estimators. However, the high computational cost of smoothing splines for large datasets has hindered their wide application. In this article, we develop a new method, named adaptive basis sampling, for efficient computation of smoothing splines in super-large samples. Except for the univariate case where the Reinsch algorithm is applicable, a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally O ( n 3 ). We achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions, obtained by an adaptive sampling scheme that uses values of the response variable. Our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines. Using simulation studies and a large-scale deep earth core-mantle boundary imaging study, we show that the proposed method outperforms a sampling method that does not use the values of response variables.
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    Entropy testing for nonlinear serial dependence in time series (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: We propose tests for nonlinear serial dependence in time series under the null hypothesis of general linear dependence, in contrast to the more widely studied null hypothesis of independence. The approach is based on combining an entropy dependence metric, which possesses many desirable properties and is used as a test statistic, with a suitable extension of surrogate data methods, a class of Monte Carlo distribution-free tests for nonlinearity, and a smoothed sieve bootstrap scheme. We show how, in the same way as the autocorrelation function is used for linear models, our tests can in principle be employed to detect the lags at which a significant nonlinear relationship is present. We prove the asymptotic validity of the proposed procedures and the corresponding inferences. The small-sample performance of the tests in terms of power and size is assessed through a simulation study. Applications to real datasets of different kinds are also presented.
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    Robust estimation under heavy contamination using unnormalized models (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Contamination caused by outliers is inevitable in data analysis, and robust statistical methods are often needed. In this paper we develop a new approach for robust data analysis on the basis of scoring rules. A scoring rule is a discrepancy measure to assess the quality of probabilistic forecasts. We propose a simple method of estimating not only parameters in the statistical model but also the contamination ratio, i.e., the ratio of outliers. The outliers are detected based on the estimated contamination ratio. For this purpose, we use scoring rules with extended statistical models called unnormalized models. Regression problems are also considered. We study complex heterogeneous contamination wherein the contamination ratio in a response variable may depend on covariate variables, and propose a simple method to estimate a robust regression function and expected contamination ratio. Simulation studies demonstrate the effectiveness of our method.
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    A cautionary note on robust covariance plug-in methods (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: The sample covariance matrix, which is well known to be highly nonrobust, plays a central role in many classical multivariate statistical methods. A popular way of making such multivariate methods more robust is to replace the sample covariance matrix with some robust scatter matrix. The aim of this paper is to point out that multivariate methods often require that certain properties of the covariance matrix hold also for the robust scatter matrix in order for the corresponding robust plug-in method to be a valid approach, but that not all scatter matrices possess the desired properties. Plug-in methods for independent components analysis, observational regression and graphical modelling are considered in more detail. For each case, it is shown that replacing the sample covariance matrix with a symmetrized robust scatter matrix yields a valid robust multivariate procedure.
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    Big data and precision (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: So-called big data are likely to have complex structure, in particular implying that estimates of precision obtained by applying standard statistical procedures are likely to be misleading, even if the point estimates of parameters themselves may be reasonably satisfactory. While this possibility is best explored in the context of each special case, here we outline a fairly general representation of the accretion of error in large systems and explore the possible implications for the estimation of regression coefficients. The discussion raises issues broadly parallel to the distinction between short-range and long-range dependence in time series theory.
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    Hysteretic autoregressive time series models (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: This paper extends the classical two-regime threshold autoregressive model by introducing hysteresis to its regime-switching structure, which leads to a new model: the hysteretic autoregressive model. The proposed model enjoys the piecewise linear structure of a threshold model but has a more flexible regime switching mechanism. A sufficient condition is given for geometric ergodicity. Conditional least squares estimation is discussed, and the asymptotic distributions of its estimators and information criteria for model selection are derived. Simulation results and an example support the model.
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    A note on convergence of an iterative algorithm for semiparametric odds ratio models (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: This paper points out an error in Davidov and Iliopoulos's ( Biometrika 100 , 778–80) proof of convergence of an iterative algorithm for the proportional likelihood ratio model. It is shown that the iterative algorithm increases the likelihood in each iteration and converges under mild additional conditions when the odds ratio function is bounded.
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    Order selection in finite mixture models: complete or observed likelihood information criteria? (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Choosing the number of components in a finite mixture model is a challenging task. In this article, we study the behaviour of information criteria for selecting the mixture order, based on either the observed likelihood or the complete likelihood including component labels. We propose a new observed likelihood criterion called aic mix , which is shown to be order consistent. We further show that when there is a nontrivial level of classification uncertainty in the true model, complete likelihood criteria asymptotically underestimate the true number of components. A simulation study illustrates the potentially poor finite-sample performance of complete likelihood criteria, while aic mix and the Bayesian information criterion perform strongly regardless of the level of classification uncertainty.
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    Outlier detection for high-dimensional data (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Outlier detection is an integral component of statistical modelling and estimation. For high-dimensional data, classical methods based on the Mahalanobis distance are usually not applicable. We propose an outlier detection procedure that replaces the classical minimum covariance determinant estimator with a high-breakdown minimum diagonal product estimator. The cut-off value is obtained from the asymptotic distribution of the distance, which enables us to control the Type I error and deliver robust outlier detection. Simulation studies show that the proposed method behaves well for high-dimensional data.
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    Nonparametric Bayesian testing for monotonicity (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: This paper adopts a nonparametric Bayesian approach to testing whether a function is monotone. Two new families of tests are constructed. The first uses constrained smoothing splines with a hierarchical stochastic-process prior that explicitly controls the prior probability of monotonicity. The second uses regression splines together with two proposals for the prior over the regression coefficients. Via simulation, the finite-sample performance of the tests is shown to improve upon existing frequentist and Bayesian methods. The asymptotic properties of the Bayes factor for comparing monotone versus nonmonotone regression functions in a Gaussian model are also studied. Our results significantly extend those currently available, which chiefly focus on determining the dimension of a parametric linear model.
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    Benchmarked empirical Bayes methods in multiplicative area-level models with risk evaluation (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: The paper develops hierarchical empirical Bayes and benchmarked hierarchical empirical Bayes estimators of positive small area means under multiplicative models. The usual benchmarking requirement is that the small area estimates, when aggregated, should equal the direct estimates for the larger geographical areas. However, while estimating positive small area parameters, the conventional squared error or weighted squared error loss subject to the usual benchmark constraint may not produce positive estimators, so it is necessary to seek other loss functions. We consider a multiplicative model for the original data for estimating positive small area means, and suggest a variant of the Kullback–Leibler divergence as a loss function. The prediction errors of the suggested hierarchical empirical Bayes estimators are investigated asymptotically, and their second-order unbiased estimators are provided. Bootstrapped estimators of these prediction errors for both hierarchical empirical Bayes and benchmarked hierarchical empirical Bayes estimators are also given. The performance of the suggested procedures is investigated through simulation as well as with an example.
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    Semiparametric causal inference in matched cohort studies (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Odds ratios can be estimated in case-control studies using standard logistic regression, ignoring the outcome-dependent sampling. In this paper we discuss an analogous result for treatment effects on the treated in matched cohort studies. Specifically, in studies where a sample of treated subjects is observed along with a separate sample of possibly matched controls, we show that efficient and doubly robust estimators of effects on the treated are computationally equivalent to standard estimators, which ignore the matching and exposure-based sampling. This is not the case for general average effects. We also show that matched cohort studies are often more efficient than random sampling for estimating effects on the treated, and derive the optimal number of matches for a given set of matching variables. We illustrate our results via simulation and in a matched cohort study of the effect of hysterectomy on the risk of cardiovascular disease.
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    Sieve maximum likelihood regression analysis of dependent current status data (2015)
    Oxford University Press
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    Publication Date: 2015-08-21
    Description: Current status data occur in contexts including demographic studies and tumorigenicity experiments. In such cases, each subject is observed only once and the failure time of interest is either left- or right-censored (Kalbfleisch & Prentice, 2002). Many methods have been developed for the analysis of such data (Huang, 1996; Sun, 2006), most of which assume that the failure time and the observation time are independent completely or given covariates. In this paper, we present a sieve maximum likelihood approach for current status data when independence does not hold. A copula model and monotone I-splines are used and the asymptotic properties of the resulting estimators are established. In particular, the estimated regression parameters are shown to be semiparametrically efficient. An illustrative example is provided.
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    The correlation space of Gaussian latent tree models and model selection without fitting (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: We provide a complete description of possible distributions consistent with any Gaussian latent tree model. This description consists of polynomial equations and inequalities involving covariances between the observed variables. Testing inequality constraints can be done using the inverse Wishart distribution and this leads to simple preliminary assessment of tree-compatibility. To test equality constraints we employ general techniques of tetrad analyses. This approach is effective even for small sample sizes and can be easily adjusted to test either entire models or just particular macrostructures of a tree. Our methods are simple to implement and do not require fitting of the model. The versatility of the techniques is illustrated by performing exploratory and confirmatory tetrad analyses in linguistic and biological settings respectively.
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    Automatic structure recovery for additive models (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We propose an automatic structure recovery method for additive models, based on a backfitting algorithm coupled with local polynomial smoothing, in conjunction with a new kernel-based variable selection strategy. Our method produces estimates of the set of noise predictors, the sets of predictors that contribute polynomially at different degrees up to a specified degree M , and the set of predictors that contribute beyond polynomially of degree M . We prove consistency of the proposed method, and describe an extension to partially linear models. Finite-sample performance of the method is illustrated via Monte Carlo studies and a real-data example.
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    A useful variant of the Davis-Kahan theorem for statisticians (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: The Davis–Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and provide an improvement in many cases to the usual bound in the statistical literature. We also give an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.
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    A validated information criterion to determine the structural dimension in dimension reduction models (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: A crucial component of performing sufficient dimension reduction is to determine the structural dimension of the reduction model. We propose a novel information criterion-based method for this purpose, a special feature of which is that when examining the goodness-of-fit of the current model, one needs to perform model evaluation by using an enlarged candidate model. Although the procedure does not require estimation under the enlarged model of dimension k +1, the decision as to how well the current model of dimension k fits relies on the validation provided by the enlarged model; thus we call this procedure the validated information criterion, vic ( k ). Our method is different from existing information criterion-based model selection methods; it breaks free from dependence on the connection between dimension reduction models and their corresponding matrix eigenstructures, which relies heavily on a linearity condition that we no longer assume. We prove consistency of the proposed method, and its finite-sample performance is demonstrated numerically.
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    Effective dimension reduction for sparse functional data (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We propose a method of effective dimension reduction for functional data, emphasizing the sparse design where one observes only a few noisy and irregular measurements for some or all of the subjects. The proposed method borrows strength across the entire sample and provides a way to characterize the effective dimension reduction space, via functional cumulative slicing. Our theoretical study reveals a bias-variance trade-off associated with the regularizing truncation and decaying structures of the predictor process and the effective dimension reduction space. A simulation study and an application illustrate the superior finite-sample performance of the method.
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    Envelopes and reduced-rank regression (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We incorporate the nascent idea of envelopes (Cook et al., Statist. Sinica 20 , 927–1010) into reduced-rank regression by proposing a reduced-rank envelope model, which is a hybrid of reduced-rank and envelope regressions. The proposed model has total number of parameters no more than either of reduced-rank regression or envelope regression. The resulting estimator is at least as efficient as both existing estimators. The methodology of this paper can be adapted to other envelope models, such as partial envelopes (Su & Cook, Biometrika 98 , 133–46) and envelopes in predictor space (Cook et al., J. R. Statist. Soc. B 75 , 851–77).
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    On the degrees of freedom of reduced-rank estimators in multivariate regression (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We study the effective degrees of freedom of a general class of reduced-rank estimators for multivariate regression in the framework of Stein's unbiased risk estimation. A finite-sample exact unbiased estimator is derived that admits a closed-form expression in terms of the thresholded singular values of the least-squares solution and hence is readily computable. The results continue to hold in the high-dimensional setting where both the predictor and the response dimensions may be larger than the sample size. The derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator. The use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection, as demonstrated by simulation studies and a data example.
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    Effective degrees of freedom: a flawed metaphor (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: To most applied statisticians, a fitting procedure’s degrees of freedom is synonymous with its model complexity, or its capacity for overfitting to data. In particular, the degrees of freedom is often used to parameterize the bias-variance trade-off in model selection. We argue that, on the contrary, model complexity and degrees of freedom may correspond very poorly. We exhibit and theoretically explore various fitting procedures for which the degrees of freedom is not monotonic in the model complexity parameter and can exceed the total dimension of the ambient space even in very simple settings. We show that the degrees of freedom for any nonconvex projection method can be unbounded.
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    Semiparametric exponential families for heavy-tailed data (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We propose a semiparametric method for fitting the tail of a heavy-tailed population given a relatively small sample from that population and a larger sample from a related background population. We model the tail of the small sample as an exponential tilt of the better-observed large-sample tail, using a robust sufficient statistic motivated by extreme value theory. In particular, our method induces an estimator of the small-population mean, and we give theoretical and empirical evidence that this estimator outperforms methods that do not use the background sample. We demonstrate substantial efficiency gains over competing methods in simulation and on data from a large controlled experiment conducted by Facebook.
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    Optimum designs for two treatments with unequal variances in the presence of covariates (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Optimum designs are described for two treatments with different variances when covariates are included in the model. The designs, a generalization of Neyman allocation, are required in personalized medicine to model the effect of covariates on the choice of treatment. The use of the designs in clinical trials is indicated. D-optimality of the designs is established using results from Kiefer’s general equivalence theorem. The results are obtained with the use of surprisingly elementary algebra.
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    Testing differential networks with applications to the detection of gene-gene interactions (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Model organisms and human studies have yielded increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits. In the presence of gene-gene interactions, the dimensionality of the feature space becomes extremely high relative to the sample size. This poses a significant methodological challenge in the identification of gene-gene interactions. In this paper, by using a Gaussian graphical model framework, we translate the problem of identifying gene-gene interactions associated with a binary trait D into an inference problem on the difference of two high-dimensional precision matrices that summarize the conditional dependence network structures of the genes. We propose a procedure for testing the differential network globally, which is particularly powerful against sparse alternatives. In addition, a multiple testing procedure with false discovery rate control is developed to infer the specific structure of the differential network. Theoretical justification is provided to ensure the validity of the proposed tests, and optimality results are derived under sparsity assumptions. Through a simulation study we demonstrate that the proposed tests maintain the desired error rates under the null hypothesis and have good power under the alternative hypothesis. The methods are applied to a breast cancer gene expression study.
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    Hierarchical recognition of sparse patterns in large-scale simultaneous inference (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We study how to separate signals from noisy data accurately and determine the patterns of the selected signals. Controlling the inflation of false positive errors is important in large-scale simultaneous inference but has not been addressed in the pattern recognition literature. We develop a decision-theoretic framework and formulate the sparse pattern recognition problem as a simultaneous inference problem with multiple decision trees. Oracle and adaptive classifiers are proposed for maximizing the expected number of true positives subject to a constraint on the overall false positive rate. Existing results on multiple testing are extended by allowing more than two states of nature, hierarchical decision-making and new error rate concepts.
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    Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: When an unbiased estimator of the likelihood is used within a Metropolis–Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of the averages computed under this chain. Using many Monte Carlo samples will typically result in Metropolis–Hastings averages with lower asymptotic variances than the corresponding averages that use fewer samples; however, the computing time required to construct the likelihood estimator increases with the number of samples. Under the assumption that the distribution of the additive noise introduced by the loglikelihood estimator is Gaussian with variance inversely proportional to the number of samples and independent of the parameter value at which it is evaluated, we provide guidelines on the number of samples to select. We illustrate our results by considering a stochastic volatility model applied to stock index returns.
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    Information-theoretic optimality of observation-driven time series models for continuous responses (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We investigate information-theoretic optimality properties of the score function of the predictive likelihood as a device for updating a real-valued time-varying parameter in a univariate observation-driven model with continuous responses. We restrict our attention to models with updates of one lag order. The results provide theoretical justification for a class of score-driven models which includes the generalized autoregressive conditional heteroskedasticity model as a special case. Our main contribution is to show that only parameter updates based on the score will always reduce the local Kullback–Leibler divergence between the true conditional density and the model-implied conditional density. This result holds irrespective of the severity of model misspecification. We also show that use of the score leads to a considerably smaller global Kullback–Leibler divergence in empirically relevant settings. We illustrate the theory with an application to time-varying volatility models. We show that the reduction in Kullback–Leibler divergence across a range of different settings can be substantial compared to updates based on, for example, squared lagged observations.
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    On the dependence structure of bivariate recurrent event processes: inference and estimation (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Bivariate or multivariate recurrent event processes are often encountered in longitudinal studies in which more than one type of event is of interest. There has been much research on regression analysis for such data, but little has been done to measure the dependence between recurrent event processes. We propose a time-dependent measure, termed the rate ratio, to assess the local dependence between two types of recurrent event processes. We model the rate ratio as a parametric function of time, and leave unspecified all other aspects of the distribution. We develop a composite likelihood procedure for model fitting and parameter estimation. We show that the proposed estimator is consistent and asymptotically normal. Its finite sample performance is evaluated by simulation and illustrated by an application to a soft tissue sarcoma study.
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    Maximum projection designs for computer experiments (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Space-filling properties are important in designing computer experiments. The traditional maximin and minimax distance designs consider only space-filling in the full-dimensional space; this can result in poor projections onto lower-dimensional spaces, which is undesirable when only a few factors are active. Restricting maximin distance design to the class of Latin hypercubes can improve one-dimensional projections but cannot guarantee good space-filling properties in larger subspaces. We propose designs that maximize space-filling properties on projections to all subsets of factors. We call our designs maximum projection designs. Our design criterion can be computed at no more cost than a design criterion that ignores projection properties.
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    On random-effects meta-analysis (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Meta-analysis is widely used to compare and combine the results of multiple independent studies. To account for between-study heterogeneity, investigators often employ random-effects models, under which the effect sizes of interest are assumed to follow a normal distribution. It is common to estimate the mean effect size by a weighted linear combination of study-specific estimators, with the weight for each study being inversely proportional to the sum of the variance of the effect-size estimator and the estimated variance component of the random-effects distribution. Because the estimator of the variance component involved in the weights is random and correlated with study-specific effect-size estimators, the commonly adopted asymptotic normal approximation to the meta-analysis estimator is grossly inaccurate unless the number of studies is large. When individual participant data are available, one can also estimate the mean effect size by maximizing the joint likelihood. We establish the asymptotic properties of the meta-analysis estimator and the joint maximum likelihood estimator when the number of studies is either fixed or increases at a slower rate than the study sizes and we discover a surprising result: the former estimator is always at least as efficient as the latter. We also develop a novel resampling technique that improves the accuracy of statistical inference. We demonstrate the benefits of the proposed inference procedures using simulated and empirical data.
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    Jump information criterion for statistical inference in estimating discontinuous curves (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: Nonparametric regression analysis when the regression function is discontinuous has many applications. Existing methods for estimating a discontinuous regression curve usually assume that the number of jumps in the regression curve is known beforehand, which is unrealistic in some situations. Although there has been research on estimation of a discontinuous regression curve when the number of jumps is unknown, the problem remains mostly open because such research often requires assumptions on other related quantities, such as a known minimum jump size. In this paper we propose a jump information criterion which consists of a term measuring the fidelity of the estimated regression curve to the observed data and a penalty related to the number of jumps and the jump sizes. The number of jumps can then be determined by minimizing our criterion. Theoretical and numerical studies show that our method works well.
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    A Mobius transformation-induced distribution on the torus (2015)
    Oxford University Press
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    Publication Date: 2015-05-24
    Description: We propose a five-parameter bivariate wrapped Cauchy distribution as a unimodal model for toroidal data. It is highly tractable, displays numerous desirable properties, including marginal and conditional distributions that are all wrapped Cauchy, and arises as an appealing submodel of a six-parameter distribution obtained by applying Möbius transformation to a pre-existing bivariate circular model. Method of moments and maximum likelihood estimation of its parameters are fast, and tests for independence and goodness-of-fit are available. An analysis involving dihedral angles of the proteinogenic amino acid Tyrosine illustrates the distribution’s application. A Markov process for circular data is also explored.
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    Going off grid: computationally efficient inference for log-Gaussian Cox processes (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making 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, whereas an approximation based on a counting process on a partition of the domain achieves only first-order convergence. The results improve upon the general theory of convergence for stochastic partial differential equation models introduced by Lindgren et al. (2011) . The new method is demonstrated on a standard point pattern dataset, 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.
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    Spatial regression models over two-dimensional manifolds (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    Improving Holm's procedure using pairwise dependencies (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: Seneta & Chen (2005) tightened the familywise error rate control of Holm's procedure by sharpening its critical values using pairwise dependencies of the $p$ -values. In this paper we further sharpen these critical values in the case where the distribution functions of the pairwise maxima of null $p$ -values are convex, a property shown to hold in some applications of Holm's procedure. The newer critical values are uniformly larger, providing tighter familywise error rate control than the approach of Seneta & Chen (2005) , significantly so under high pairwise positive dependencies. The critical values can be further improved under exchangeable null $p$ -values.
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    Empirical Bayes deconvolution estimates (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    Partially functional linear regression in high dimensions (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    Semiparametric inverse propensity weighting for nonignorable missing data (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    An estimating equation approach to dimension reduction for longitudinal data (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: Sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data. In this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace. The proposed method accounts for the covariance structure within each subject and improves estimation efficiency when the covariance structure is correctly specified. Even if the covariance structure is misspecified, our estimator remains consistent. In addition, our method relaxes distributional assumptions on the covariates and is doubly robust. To determine the structural dimension of the central mean subspace, we propose a Bayesian-type information criterion. We show that the estimated structural dimension is consistent and that the estimated basis directions are root- $n$ consistent, asymptotically normal and locally efficient. Simulations and an analysis of the Framingham Heart Study data confirm the effectiveness of our approach.
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    Some matched comparisons of two distributions of survival time (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: A theoretical analysis is made of the properties of various methods for comparing two distributions of survival time. The results are intended primarily to guide the choice of method of analysis for such simple comparisons as of a treatment versus a control, but the main implications are fairly general, illustrating the performance of different models in a range of conditions. For most of the models there is a parameter specifying the comparison of interest and the Fisher information per observation can be calculated for that parameter, and provides a succinct basis for comparison. Two of the models are semiparametric and the others are based on exponential distributions with or without extra sources of variability.
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    Semiparametric approach to regression with a covariate subject to a detection limit (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    On distributions of ratios (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: Inversion formulae are derived that express the density and distribution function of a ratio of random variables in terms of the joint characteristic function of the numerator and denominator. The resulting expressions are amenable to numerical evaluation and lead to simple asymptotic expansions. The expansions reduce to known results when the denominator is almost surely positive. Their accuracy is demonstrated with numerical examples.
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    High-dimensional classification via nonparametric empirical Bayes and maximum likelihood inference (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: 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.
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    Nonparametric Bayes inference on conditional independence (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: In many application areas, a primary focus is on assessing evidence in the data refuting the assumption of independence of $Y$ and $X$ conditionally on $Z$ , with $Y$ response variables, $X$ predictors of interest, and $Z$ covariates. Ideally, one would have methods available that avoid parametric assumptions, allow $Y, X, Z$ to be random variables on arbitrary spaces with arbitrary dimension, and accommodate rapid consideration of different candidate predictors. As a formal decision-theoretic approach has clear disadvantages in this context, we instead rely on an encompassing nonparametric Bayes model for the joint distribution of $Y$ , $X$ and $Z$ , with conditional mutual information used as a summary of the strength of conditional dependence. We construct a functional of the encompassing model and empirical measure for estimation of conditional mutual information. The implementation relies on a single Markov chain Monte Carlo run under the encompassing model, with conditional mutual information for candidate models calculated as a byproduct. We provide an asymptotic theory supporting the approach, and apply the method to variable selection. The methods are illustrated through simulations and criminology applications.
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    Goodness-of-fit test in parametric mixed effects models based on estimation of the error distribution (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: We address the problem of testing for a parametric function of fixed effects in mixed models. We propose a test based on the distance between two empirical error distribution functions, which are constructed from residuals calculated under the opposing hypotheses. The proposed test statistic has power against all alternatives, and its asymptotic distribution is derived. A simulation study shows that the test outperforms others in the literature. The test is applied to longitudinal data from an AIDS clinical trial and a growth study.
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    A unified framework for spline estimators (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: This article develops a unified framework to study the asymptotic properties of all periodic spline-based estimators, that is, of regression, penalized and smoothing splines. The explicit form of the periodic Demmler–Reinsch basis in terms of exponential splines allows the derivation of an expression for the asymptotic equivalent kernel on the real line for all spline estimators simultaneously. The corresponding bandwidth, which drives the asymptotic behaviour of spline estimators, is shown to be a function of the number of knots and the smoothing parameter. Strategies for the selection of the optimal bandwidth and other model parameters are discussed.
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    Frechet integration and adaptive metric selection for interpretable covariances of multivariate functional data (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: For multivariate functional data recorded from a sample of subjects on a common domain, one 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 Fréchet integral, which depends on the metric chosen for the space of covariance matrices, and demonstrate that ordinary integration is a special case where 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 Fréchet integrals is equivalent to transforming the covariance matrices with the chosen power, applying the classical Riemann integral to the transformed matrices, and finally using the inverse transformation to return to the original scale. We also propose data-adaptive metric selection with respect to a user-specified target criterion, such as fastest decline of the eigenvalues, establish consistency of the proposed procedures, and demonstrate their effectiveness in a simulation. The proposed functional covariance approach through Fréchet integration is illustrated by a comparison of connectivity between brain voxels for normal subjects and Alzheimer's patients based on fMRI data.
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    Stochastic mechanistic interaction (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: We define mechanistic interaction between the effects of two variables on an outcome in terms of departure of these effects from a generalized noisy-OR model in a stratum of the population. We develop a fully probabilistic framework for the observational identification of this type of interaction via excess risk or superadditivity, one novel feature of which is its applicability when the interacting variables have been generated by arbitrarily dichotomizing continuous exposures. The method allows for stochastic mediators of the interacting effects. The required assumptions are provided in the form of conditional independencies between the problem variables, which may relate to a causal-graph representation of the problem. We also develop a theory of mechanistic interaction between effects associated with specific paths of the causal graph.
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    Towards a unification of second-order theory for likelihood and marginal composite likelihood (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: An adjustment for marginal composite likelihoods is derived to match the second-order theory of the likelihood when inference is for a vector-valued parameter in the absence of nuisance components. The adjustment overcomes the failure of Bartlett identities for marginal composite likelihoods and leads to a Bartlett-correctable marginal composite likelihood ratio statistic.
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    Higher dimensional Clayton-Oakes models for multivariate failure time data (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: The Clayton–Oakes bivariate failure time model is extended to dimensions $m 〉 2$ in a manner that allows unspecified marginal survivor functions for all dimensions less than $m$ . Special cases that allow unspecified marginal survivor functions of dimension $q$ or less with $q 〈 m$ , while making some provisions for dependencies of dimension greater than $q$ , are also described.
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    A note on multiple imputation for method of moments estimation (2016)
    Oxford University Press
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    Publication Date: 2016-03-01
    Description: Multiple imputation is widely used for estimation in situations where there are missing data. Rubin (1987) provided an easily applicable formula for multiple imputation variance estimation, but its validity requires the congeniality condition of Meng (1994) , which may not be satisfied for method of moments estimation. We give the asymptotic bias of Rubin's variance estimator when method of moments estimation is used in the complete-sample analysis for each imputed dataset. A new variance estimator based on over-imputation is proposed to provide asymptotically valid inference in this case.
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    Strong control of the familywise error rate in observational studies that discover effect modification by exploratory methods (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: An effect modifier is a pretreatment covariate that affects the magnitude of the treatment effect or its stability. When there is effect modification, an overall test that ignores an effect modifier may be more sensitive to unmeasured bias than a test that combines results from subgroups defined by the effect modifier. If there is effect modification, one would like to identify specific subgroups for which there is evidence of effect that is insensitive to small or moderate biases. In this paper, we propose an exploratory method for discovering effect modification, and combine it with a confirmatory method of simultaneous inference that strongly controls the familywise error rate in a sensitivity analysis, despite the fact that the groups being compared are defined empirically. A new form of matching, strength- $k$ matching, permits a search through more than $k$ covariates for effect modifiers, in such a way that no pairs are lost, provided that at most $k$ covariates are selected to group the pairs. In a strength- $k$ match, each set of $k$ covariates is exactly balanced, although a set of more than $k$ covariates may exhibit imbalance. We apply the proposed method to study the effects of the earthquake that struck Chile in 2010.
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    Direct estimation of the mean outcome on treatment when treatment assignment and discontinuation compete (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Several authors have investigated the challenges of statistical analyses and inference in the presence of early treatment termination, including a loss of efficiency in randomized controlled trials and a connection to dynamic regimes in observational studies. Popular estimation strategies for causal estimands in dynamic regimes lend themselves to studies where treatment is assigned at a finite number of points and the extension to continuous treatment assignment is nontrivial. We re-examine this from a different perspective and propose a new estimator for the mean outcome of a target treatment length policy that does not involve a treatment model. Because this strategy avoids modelling the treatment assignment mechanism, the estimator works for both discrete and continuous treatment length data and eschews bias and imprecision that arise as a result of coarsening continuous time data into intervals. We show how the competition of treatment length assignment and terminating event lead to a competing risks problem. We exemplify the direct estimator through numerical studies and the analysis of two real datasets. When all modelling assumptions for both the direct and inverse weighted estimators are correct, our simulation studies suggest that the direct estimator is more precise.
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    Bayesian inference for partially observed stochastic differential equations driven by fractional Brownian motion (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a nontrivial likelihood given the latent path. Due to the non-Markovian and high-dimensional nature of the latent path, estimating posterior expectations is computationally challenging. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use it to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The algorithm is based on a version of hybrid Monte Carlo simulation that delivers increased efficiency when used on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model, allowing for memory in the volatility increments through a fractional specification. The method is demonstrated on simulated data and on the S&P 500/VIX time series. In the latter case, the posterior distribution favours values of the Hurst parameter smaller than $1/2$ , pointing towards medium-range dependence.
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    Singular value shrinkage priors for Bayesian prediction (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We develop singular value shrinkage priors for the mean matrix parameters in the matrix-variate normal model with known covariance matrices. Our priors are superharmonic and put more weight on matrices with smaller singular values. They are a natural generalization of the Stein prior. Bayes estimators and Bayesian predictive densities based on our priors are minimax and dominate those based on the uniform prior in finite samples. In particular, our priors work well when the true value of the parameter has low rank.
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    Efficient inference and simulation for elliptical Pareto processes (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Recent advances in extreme value theory have established $\ell $ -Pareto processes as the natural limits for extreme events defined in terms of exceedances of a risk functional. In this paper we provide methods for the practical modelling of data based on a tractable yet flexible dependence model. We introduce the class of elliptical $\ell $ -Pareto processes, which arise as the limits of threshold exceedances of certain elliptical processes characterized by a correlation function and a shape parameter. An efficient inference method based on maximizing a full likelihood with partial censoring is developed. Novel procedures for exact conditional and unconditional simulation are proposed. These ideas are illustrated using precipitation extremes in Switzerland.
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    Nonparametric methods for group testing data, taking dilution into account (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Group testing methods are used widely to assess the presence of a contaminant, based on measurements of the concentration of a biomarker, for example to test the presence of a disease in pooled blood samples. The test would be perfect if it produced a positive result whenever the contaminant was present, and a negative result otherwise. However, in practice the test is always at least somewhat imperfect, for example because it is sensitive to the proportion of contaminated items in the group, rather than to the sheer existence of one or more contaminated items. We develop a nonparametric method for accommodating this dilution effect. Our approach allows us to estimate, under minimal assumptions, the probability $m(x)$ that an item is contaminated, conditional on the value $x$ of an explanatory variable, and to estimate the probability, $q$ , that an individual chosen at random is disease free, and the specificity Sp, and the sensitivity Se, of the test. These are all ill-posed problems, where poor convergence rates are usually encountered, but despite this, our estimators of $q$ , Sp and Se are root- $N$ consistent, where $N$ denotes the total number of individuals in all the groups, and our estimator of $m(x)$ converges at the rate it would enjoy if $q$ , Sp and Se were known.
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    A new specification of generalized linear models for categorical responses (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Many regression models for categorical responses have been introduced, motivated by different paradigms, but it is difficult to compare them because of their different specifications. In this paper we propose a unified specification of regression models for categorical responses, based on a decomposition of the link function into an inverse continuous cumulative distribution function and a ratio of probabilities. This allows us to define a new family of reference models for nominal responses, comparable to the families of adjacent, cumulative and sequential models for ordinal responses. A new equivalence between cumulative and sequential models is shown. Invariances under permutations of the categories are studied for each family of models. We introduce a reversibility property that distinguishes adjacent and cumulative models from sequential models. The new family of reference models is tested on three benchmark classification datasets.
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    Shared kernel Bayesian screening (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: This article concerns testing for equality of distribution between groups. We focus on screening variables with shared distributional features such as common support, modes and patterns of skewness. We propose a Bayesian testing method using kernel mixtures, which improves performance by borrowing information across the different variables and groups through shared kernels and a common probability of group differences. The inclusion of shared kernels in a finite mixture, with Dirichlet priors on the weights, leads to a simple framework for testing that scales well for high-dimensional data. We provide closed asymptotic forms for the posterior probability of equivalence in two groups and prove consistency under model misspecification. The method is applied to DNA methylation array data from a breast cancer study, and compares favourably to competitors when Type I error is estimated via permutation.
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    General weighted optimality of designed experiments (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: The standard approach to finding optimal experimental designs employs conventional measures of design efficacy, such as the $A$ , $E$ , and $D$ -criterion, that assume equal interest in all estimable functions of model parameters. This paper develops a general theory for weighted optimality, allowing precise design selection according to expressed relative interest in different functions in the estimation space. The approach employs a very general class of matrix-specified weighting schemes that produce easily interpretable weighted optimality criteria. In particular, for any set of estimable functions, and any selected corresponding weights, analogs of standard optimality criteria are found that guide design selection according to the weighted variances of estimators of those particular functions. The results are applied to solve the $A$ -optimal design problem for baseline factorial effects in unblocked experiments.
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    Locally optimal designs for errors-in-variables models (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We consider the construction of optimal designs for nonlinear regression models when there are measurement errors in the covariates. Corresponding approximate design theory is developed for maximum likelihood and least-squares estimation, with the latter leading to nonconcave optimization problems. Analytical characterizations of the locally D-optimal saturated designs are provided for the Michaelis–Menten, $E_{\rm max}$ and exponential regression models. Through concrete applications, we illustrate how measurement errors in the covariates affect the optimal choice of design and show that the locally D-optimal saturated designs are highly efficient for relatively small misspecifications of the parameter values.
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    Optimal multiple testing under a Gaussian prior on the effect sizes (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We develop a new method for large-scale frequentist multiple testing with Bayesian prior information. We find optimal $p$ -value weights that maximize the average power of the weighted Bonferroni method. Due to the nonconvexity of the optimization problem, previous methods that account for uncertain prior information are suitable for only a small number of tests. For a Gaussian prior on the effect sizes, we give an efficient algorithm that is guaranteed to find the optimal weights nearly exactly. Our method can discover new loci in genome-wide association studies and compares favourably to competitors. An open-source implementation is available.
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    Consistent testing for recurrent genomic aberrations (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We consider the detection and identification of recurrent departures from stationary behaviour in genomic or similarly arranged data containing measurements at an ordered set of variables. Our primary focus is on departures that occur only at a single variable, or within a small window of contiguous variables, but involve more than one sample. This encompasses the identification of aberrant markers in genome-wide measurements of DNA copy number and DNA methylation, as well as meta-analyses of genome-wide association studies. We propose and analyse a cyclic shift-based procedure for testing recurrent departures from stationarity. Our analysis establishes the consistency of cyclic shift $p$ -values for datasets with a fixed set of samples as the number of observed variables tends to infinity, under the assumption that each sample is an independent realization of a stationary Markov chain. Our results apply to any test statistic satisfying a simple invariance condition.
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    Diagnostic measures for the Cox regression model with missing covariates (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We investigate diagnostic measures for assessing the influence of observations and model misspecification on the Cox regression model when there are missing covariate data. Our diagnostics include case-deletion measures, conditional martingale residuals, and score residuals. The Q-distance is introduced to examine the effects of deleting individual observations on the estimates of finite- and infinite-dimensional parameters. Conditional martingale residuals are used to construct goodness-of-fit statistics for testing misspecification of the model assumptions. A resampling method is developed to approximate the $p$ -values of the goodness-of-fit statistics. We conduct simulation studies to evaluate our methods, and analyse a real dataset to illustrate their use.
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    Changepoint estimation: another look at multiple testing problems (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We consider large scale multiple testing for data that have locally clustered signals. With this structure, we apply techniques from changepoint analysis and propose a boundary detection algorithm so that the clustering information can be utilized. Consequently the precision of the multiple testing procedure is substantially improved. We study tests with independent as well as dependent $p$ -values. Monte Carlo simulations suggest that the methods perform well with realistic sample sizes and show improved detection ability compared with competing methods. Our procedure is applied to a genome-wide association dataset of blood lipids.
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    Optimal two-level choice designs for any number of choice sets (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: For two-level choice experiments, we obtain a simple form of the information matrix of a choice design for estimating the main effects, and provide $D$ - and MS -optimal paired choice designs with distinct choice sets under the main effects model for any number of choice sets. It is shown that the optimal designs under the main effects model are also optimal under the broader main effects model. We find that optimal choice designs with a choice set size two often outperform their counterparts with larger choice set sizes.
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    Designing dose-finding studies with an active control for exponential families (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Optimal design of dose-finding studies with an active control has only been considered in the literature for regression models with normally distributed errors and known variances, where the focus is on estimating the smallest dose that achieves the same treatment effect as the active control. This paper discusses such dose-finding studies from a broader perspective. We consider a general class of optimality criteria and models arising from an exponential family. Optimal designs are constructed for several situations and their efficiency is illustrated with examples.
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    Clarifying missing at random and related definitions, and implications when coupled with exchangeability (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We clarify the key concept of missingness at random in incomplete data analysis. We first distinguish between data being missing at random and the missingness mechanism being a missing-at-random one, which we call missing always at random and which is more restrictive. We further discuss how, in general, neither of these conditions is a statement about conditional independence. We then consider the implication of the more restrictive missing-always-at-random assumption when coupled with full unit-exchangeability for the matrix of the variables of interest and the missingness indicators: the conditional distribution of the missingness indicators for any variable that can have a missing value can depend only on variables that are always fully observed. We discuss implications of this for modelling missingness mechanisms.
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    Space-filling properties of good lattice point sets (2015)
    Oxford University Press
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    Description: We study space-filling properties of good lattice point sets and obtain some general theoretical results. We show that linear level permutation does not decrease the minimum distance for good lattice point sets, and we identify several classes of such sets with large minimum distance. Based on good lattice point sets, some maximin distance designs are also constructed.
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    Score tests for association under response-dependent sampling designs for expensive covariates (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: Response-dependent sampling is widely used in settings where certain variables are expensive to obtain. Estimation has been thoroughly investigated but recent applications have emphasized tests of association for expensive covariates and a response variable. We consider testing and provide easily implemented likelihood score tests for generalized linear models under a broad range of sampling plans. We show that when there are no additional covariates, the score statistics are identical for conditional and full likelihood approaches, and are of the same form as for ordinary random sampling. Applications in genetics are discussed briefly.
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    On the validity of the pairs bootstrap for lasso estimators (2015)
    Oxford University Press
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    Publication Date: 2015-11-28
    Description: We study the validity of the pairs bootstrap for lasso estimators in linear regression models with random covariates and heteroscedastic error terms. We show that the naive pairs bootstrap does not provide a valid method for approximating the distribution of the lasso estimator. To overcome this deficiency, we introduce a modified pairs bootstrap procedure and prove its consistency. Finally, we consider the adaptive lasso and show that the modified pairs bootstrap consistently estimates the distribution of the adaptive lasso estimator.
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    Data augmentation for models based on rejection sampling (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    On varieties of doubly robust estimators under missingness not at random with a shadow variable (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    Calibrated propensity score method for survey nonresponse in cluster sampling (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: Weighting adjustment is commonly used in survey sampling to correct for unit nonresponse. In cluster sampling, the missingness indicators are often correlated within clusters and the response mechanism is subject to cluster-specific nonignorable missingness. Based on a parametric working model for the response mechanism that incorporates cluster-specific nonignorable missingness, we propose a method of weighting adjustment. We provide a consistent estimator of the mean or totals in cases where the study variable follows a generalized linear mixed-effects model. The proposed method is robust in the sense that the consistency of the estimator does not require correct specification of the functional forms of the response and outcome models. A consistent variance estimator based on Taylor linearization is also proposed. Numerical results, including a simulation and a real-data application, are presented.
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    Regression analysis of networked data (2016)
    Oxford University Press
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    Description: This paper concerns regression methodology for assessing relationships between multi-dimensional response variables and covariates that are correlated within a network. To address analytical challenges associated with the integration of network topology into the regression analysis, we propose a hybrid quadratic inference method that uses both prior and data-driven correlations among network nodes. A Godambe information-based tuning strategy is developed to allocate weights between the prior and data-driven network structures, so the estimator is efficient. The proposed method is conceptually simple and computationally fast, and has appealing large-sample properties. It is evaluated by simulation, and its application is illustrated using neuroimaging data from an association study of the effects of iron deficiency on auditory recognition memory in infants.
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    A pairwise interaction model for multivariate functional and longitudinal data (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: Functional data vectors consisting of samples of multivariate data where each component is a random function are encountered increasingly often but have not yet been comprehensively investigated. We introduce a simple pairwise interaction model that leads to an interpretable and straightforward decomposition of multivariate functional data and of their variation into component-specific processes and pairwise interaction processes. The latter quantify the degree of pairwise interactions between the components of the functional data vectors, while the component-specific processes reflect the functional variation of a particular functional vector component that cannot be explained by the other components. Thus the proposed model provides an extension of the usual notion of a covariance or correlation matrix for multivariate vector data to functional data vectors and generates an interpretable functional interaction map. The decomposition provided by the model can also serve as a basis for subsequent analysis, such as study of the network structure of functional data vectors. The decomposition of the total variance into componentwise and interaction contributions can be quantified by an $R^2$ -like decomposition. We provide consistency results for the proposed methods and illustrate the model by applying it to sparsely sampled longitudinal data from the Baltimore Longitudinal Study of Aging, examining the relationships between body mass index and blood fats.
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    Quantile-based classifiers (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: Classification with small samples of high-dimensional data is important in many application areas. Quantile classifiers are distance-based classifiers that require a single parameter, regardless of the dimension, and classify observations according to a sum of weighted componentwise distances of the components of an observation to the within-class quantiles. An optimal percentage for the quantiles can be chosen by minimizing the misclassification error in the training sample. It is shown that this choice is consistent for the classification rule with the asymptotically optimal quantile and that under some assumptions, as the number of variables goes to infinity, the probability of correct classification converges to unity. The effect of skewness of the distributions of the predictor variables is discussed. The optimal quantile classifier gives low misclassification rates in a comprehensive simulation study and in a real-data application.
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    Skew-normal antedependence models for skewed longitudinal data (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: Antedependence models, also known as transition models, have proven to be useful for longitudinal data exhibiting serial correlation, especially when the variances and/or same-lag correlations are time-varying. Statistical inference procedures associated with normal antedependence models are well-developed and have many nice properties, but they are not appropriate for longitudinal data that exhibit considerable skewness. We propose two direct extensions of normal antedependence models to skew-normal antedependence models. The first is obtained by imposing antedependence on a multivariate skew-normal distribution, and the second is a sequential autoregressive model with skew-normal innovations. For both models, necessary and sufficient conditions for $p$ th-order antedependence are established, and likelihood-based estimation and testing procedures for models satisfying those conditions are developed. The procedures are applied to simulated data and to real data from a study of cattle growth.
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    Partial least squares for dependent data (2016)
    Oxford University Press
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    Description: We consider the partial least squares algorithm for dependent data and study the consequences of ignoring the dependence both theoretically and numerically. Ignoring nonstationary dependence structures can lead to inconsistent estimation, but a simple modification yields consistent estimation. A protein dynamics example illustrates the superior predictive power of the proposed method.
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    On exchangeable multinomial distributions (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: We derive an expression for the joint distribution of exchangeable multinomial random variables, which generalizes the multinomial distribution based on independent trials while retaining some of its important properties. Unlike de Finneti's representation theorem for a binary sequence, the exchangeable multinomial distribution derived here does not require that the finite set of random variables under consideration be a subset of an infinite sequence. Using expressions for higher moments and correlations, we show that the covariance matrix for exchangeable multinomial data has a different form from that usually assumed in the literature, and we analyse data from developmental toxicology studies. The proposed analyses have been implemented in R and are available on CRAN in the CorrBin package.
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    Nonparametric identification and maximum likelihood estimation for hidden Markov models (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    'Clarifying missing at random and related definitions, and implications when coupled with exchangeability (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
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    Exponential tilting in Bayesian asymptotics (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: We use exponential tilting to obtain versions of asymptotic formulae for Bayesian computation that do not involve conditional maxima of the likelihood function, yielding a more stable computational procedure and significantly reducing computational time. In particular we present an alternative version of the Laplace approximation for a marginal posterior density. Implementation of the asymptotic formulae and a modified signed root based importance sampler are illustrated with an example.
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    Maximum likelihood estimation for semiparametric transformation models with interval-censored data (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    A pairwise likelihood-based approach for changepoint detection in multivariate time series models (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: This paper develops a composite likelihood-based approach for multiple changepoint estimation in multivariate time series. We derive a criterion based on pairwise likelihood and minimum description length for estimating the number and locations of changepoints and for performing model selection in each segment. The number and locations of the changepoints can be consistently estimated under mild conditions and the computation can be conducted efficiently with a pruned dynamic programming algorithm. Simulation studies and real data examples demonstrate the statistical and computational efficiency of the proposed method.
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    Modelling complex survey data with population level information: an empirical likelihood approach (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: Survey data are often collected with unequal probabilities from a stratified population. In many modelling situations, the parameter of interest is a subset of a set of parameters, with the others treated as nuisance parameters. We show that in this situation the empirical likelihood ratio statistic follows a chi-squared distribution asymptotically, under stratified single and multi-stage unequal probability sampling, with negligible sampling fractions. Simulation studies show that the empirical likelihood confidence interval may achieve better coverages and has more balanced tail error rates than standard approaches involving variance estimation, linearization or resampling.
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    Retraction Notice (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
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    Exact simulation of max-stable processes (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: 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.
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    Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem (2016)
    Oxford University Press
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    Publication Date: 2016-05-25
    Description: We study nonparametric maximum likelihood estimation for the distribution of spherical radii using samples containing a mixture of one-dimensional, two-dimensional biased and three-dimensional unbiased observations. Since direct maximization of the likelihood function is intractable, we propose an expectation-maximization algorithm for implementing the estimator, which handles an indirect measurement problem and a sampling bias problem separately in the E- and M-steps, and circumvents the need to solve an Abel-type integral equation, which creates numerical instability in the one-sample problem. Extensions to ellipsoids are studied and connections to multiplicative censoring are discussed.
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    A differential-geometric approach to generalized linear models with grouped predictors (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: We propose an extension of the differential-geometric least angle regression method to perform sparse group inference in a generalized linear model. An efficient algorithm is proposed to compute the solution curve. The proposed group differential-geometric least angle regression method has important properties that distinguish it from the group lasso. First, its solution curve is based on the invariance properties of a generalized linear model. Second, it adds groups of variables based on a group equiangularity condition, which is shown to be related to score statistics. An adaptive version, which includes weights based on the Kullback–Leibler divergence, improves its variable selection features and is shown to have oracle properties when the number of predictors is fixed.
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    Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: 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.
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    Accurate directional inference for vector parameters (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: We consider statistical inference for a vector-valued parameter of interest in a regular asymptotic model with a finite-dimensional nuisance parameter. We use highly accurate likelihood theory to derive a directional test, in which the $p$ -value is obtained by one-dimensional numerical integration. This extends the results of Davison et al. (2014) for linear exponential families to nonlinear parameters of interest and to more general models. Examples and simulations provide comparisons with the likelihood ratio test and adjusted versions of the likelihood ratio test. The directional approach gives extremely accurate inference, even in high-dimensional settings where the likelihood ratio versions can fail catastrophically.
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    A theoretical study of Stein's covariance estimator (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: Stein proposed an estimator to address the poor performance of the sample covariance matrix for samples of small size. The estimator does not impose sparsity conditions and uses an isotonizing algorithm to preserve the order of the sample eigenvalues. Despite its superior numerical performance, its theoretical properties are not well understood. We demonstrate that Stein's covariance estimator gives modest risk reductions when it is not isotonized, and when it is isotonized the risk reductions are significant. Three broad regimes of the estimator's behaviour are identified.
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    Variable selection for case-cohort studies with failure time outcome (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: Case-cohort designs are widely used in large cohort studies to reduce the cost associated with covariate measurement. In many such studies the number of covariates is very large, so an efficient variable selection method is necessary. In this paper, we study the properties of a variable selection procedure using the smoothly clipped absolute deviation penalty in a case-cohort design with a diverging number of parameters. We establish the consistency and asymptotic normality of the maximum penalized pseudo-partial-likelihood estimator, and show that the proposed variable selection method is consistent and has an asymptotic oracle property. Simulation studies compare the finite-sample performance of the procedure with tuning parameter selection methods based on the Akaike information criterion and the Bayesian information criterion. We make recommendations for use of the proposed procedures in case-cohort studies, and apply them to the Busselton Health Study.
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    A goodness-of-fit test for structural nested mean models (2016)
    Oxford University Press
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    Publication Date: 2016-09-04
    Description: Coarse structural nested mean models are tools for estimating treatment effects from longitudinal observational data with time-dependent confounding. There is, however, no guidance on how to specify the treatment effect model, and model misspecification can lead to bias. We derive a goodness-of-fit test based on modified over-identification restrictions tests for evaluating a treatment effect model, and show that our test is doubly robust in the sense that, with a correct treatment effect model, the test has the correct Type I error if either the treatment initiation model or a nuisance regression outcome model is correctly specified. In a simulation study, we show that the test has correct Type I error and can detect model misspecification. We use the test to study how the timing of antiretroviral treatment initiation after HIV infection predicts the effect of one year of treatment in HIV-positive patients with acute and early infection.
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