Showing papers in "Biometrika in 2006"

Adaptive linear step-up procedures that control the false discovery rate

Abstract: We provide a new two-stage procedure in which the linear step-up procedure is used in stage one to estimate mo, providing a new level q' which is used in the linear step-up procedure in the second stage. We prove that a general form of the two-stage procedure controls the false discovery rate at the desired level q. This framework enables us to study analytically the properties of other procedures that exist in the literature. A simulation study is presented that shows that two-stage adaptive procedures improve in power over the original procedure, mainly because they provide tighter control of the false discovery rate. We further study the performance of the current suggestions, some variations of the procedures, and previous suggestions, in the case where the test statistics are positively dependent, a case for which the original procedure controls the false discovery rate. In the setting studied here the newly proposed two-stage procedure is the only one that controls the false discovery rate. The procedures are illustrated with two examples of biological importance.

Covariance matrix selection and estimation via penalised normal likelihood

Abstract: SUMMARY We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L1 and L2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while, because of its singularity, the L1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix.

An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants

Abstract: Maximum likelihood parameter estimation and sampling from Bayesian posterior distributions are problematic when the probability density for the parameter of interest involves an intractable normalising constant which is also a function of that parameter. In this paper, an auxiliary variable method is presented which requires only that independent samples can be drawn from the unnormalised density at any particular parameter value. The proposal distribution is constructed so that the normalising constant cancels from the Metropolis-Hastings ratio. The method is illustrated by producing posterior samples for parameters of the Ising model given a particular lattice realisation.

False discovery control with p-value weighting

Abstract: We present a method for multiple hypothesis testing that maintains control of the False Discovery Rate while incorporating prior information about the hypotheses. The prior information takes the form of p-value weights. If the assignment of weights is positively associated with the null hypotheses being false, the procedure improves power, except in cases where power is already near one. Even if the assignment of weights is poor, power is only reduced slightly, as long as the weights are not too large. We also provide a similar method to control False Discovery Exceedance.

Studies in the history of probability and statistics XLIX On the Matérn correlation family

Abstract: SUMMARY Handcock & Stein (1993) introduced the Matern family of spatial correlations into statistics as a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field We document the varied history of this family, which includes contributions by eminent physical scientists and statisticians

Efficient Bayesian inference for Gaussian copula regression models

Abstract: A Gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non-Gaussian. Our paper presents a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous marginals, and generalises Gaussian graphical models to the Gaussian copula framework. Posterior inference is carried out using a novel and efficient simulation method. The methods in the paper are applied to simulated and real data.

M-quantile models for small area estimation

Abstract: Small area estimation techniques are employed when sample data are insufficient for acceptably precise direct estimation in domains of interest. These techniques typically rely on regression models that use both covariates and random effects to explain variation between domains. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. We describe a new approach to small area estimation that is based on modelling quantile-like parameters of the conditional distribution of the target variable given the covariates. This avoids the problems associated with specification of random effects, allowing inter-domain differences to be characterized by the variation of area-specific M-quantile coefficients. The proposed approach is easily made robust against outlying data values and can be adapted for estimation of a wide range of area specific parameters, including that of the quantiles of the distribution of the target variable in the different small areas. Results from two simulation studies comparing the performance of the M-quantile modelling approach with more traditional mixed model approaches are also provided.

Efficient estimation of semiparametric transformation models for counting processes

Abstract: SUMMARY A class of semiparametric transformation models is proposed to characterise the effects of possibly time-varying covariates on the intensity functions of counting processes. The class includes the proportional intensity model and linear transformation models as special cases. Nonparametric maximum likelihood estimators are developed for the regression parameters and cumulative intensity functions of these models based on censored data. The estimators are shown to be consistent and asymptotically normal. The limiting variances for the estimators of the regression parameters achieve the semi parametric efficient bounds and can be consistently estimated. The limiting variances for the estimators of smooth functionals of the cumulative intensity function can also be consistently estimated. Simulation studies reveal that the proposed inference procedures perform well in practical settings. Two medical studies are provided.

Variable selection in clustering via Dirichlet process mixture models

Abstract: SUMMARY The increased collection of high-dimensional data in various fields has raised a strong interest in clustering algorithms and variable selection procedures. In this paper, we pro pose a model-based method that addresses the two problems simultaneously. We introduce a latent binary vector to identify discriminating variables and use Dirichlet process mixture models to define the cluster structure. We update the variable selection index using a Metropolis algorithm and obtain inference on the cluster structure via a split-merge Markov chain Monte Carlo technique. We explore the performance of the methodology on simulated data and illustrate an application with a DNA microarray study.

On least-squares regression with censored data

Abstract: SUMMARY The semiparametric accelerated failure time model relates the logarithm of the failure time linearly to the covariates while leaving the error distribution unspecified. The present paper describes simple and reliable inference procedures based on the least-squares principle for this model with right-censored data. The proposed estimator of the vector valued regression parameter is an iterative solution to the Buckley-James estimating equation with a preliminary consistent estimator as the starting value. The estimator is shown to be consistent and asymptotically normal. A novel resampling procedure is developed for the estimation of the limiting covariance matrix. Extensions to marginal models for multivariate failure time data are considered. The performance of the new inference procedures is assessed through simulation studies. Illustrations with medical studies are provided.

A Construction Method for Orthogonal Latin Hypercube Designs

Abstract: SUMMARY The Latin hypercube design is a popular choice of experimental design when computer simulation is used to study a physical process. These designs guarantee uniform samples for the marginal distribution of each single input. A number of methods have been pro posed for extending the uniform sampling to higher dimensions. We show how to construct Latin hypercube designs in which all main effects are orthogonal. Our method can also be used to construct Latin hypercube designs with low correlation of first-order and second-order terms. Our method generates orthogonal Latin hypercube designs that can include many more factors than those proposed by Ye (1998).

Multivariate logistic models

Abstract: SUMMARY The multivariate logistic transform is a reparameterisation of cell probabilities in terms of marginal logistic contrasts. It is known that an arbitrary set of logistic contrasts may not correspond to a valid joint distribution. In this paper we present an efficient algorithm for detecting whether or not the inverse transform exists, and for computing it if it does.

Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling

Abstract: We consider parameter-driven models of time series of counts, where the observations are assumed to arise from a Poisson distribution with a mean changing over time according to a latent process. Estimation of these models is carried out within a Bayesian framework using data augmentation and Markov chain Monte Carlo methods. We suggest a new auxiliary mixture sampler, which possesses a Gibbsian transition kernel, where we draw from full conditional distributions belonging to standard distribution families only. Emphasis lies on application to state space modelling of time series of counts, but we show that auxiliary mixture sampling may be applied to a wider range of parameter-driven models, including random-effects models and panel data models based on the Poisson distribution.

Range of correlation matrices for dependent Bernoulli random variables

Abstract: SUMMARY We say that a pair (p, R) is compatible if there exists a multivariate binary distribution with mean vector p and correlation matrix R. In this paper we study necessary and sufficient conditions for compatibility for structured and unstructured correlation matrices. We give examples of correlation matrices that are incompatible with any p. Using our results we show that the parametric binary models of Emrich & Piedmonte (1991) and Qaqish (2003) allow a good range of correlations between the binary variables. We also obtain necessary and sufficient conditions for a matrix of odds ratios to be compatible with a given p. Our findings support the popular belief that the odds ratios are less constrained and more flexible than the correlations.

Adaptive and nonadaptive group sequential tests

Abstract: SUMMARY Methods have been proposed for redesigning a clinical trial at an interim stage in order to increase power. In order to preserve the type I error rate, methods for unplanned design-change have to be defined in terms of nonsufficient statistics, and this calls into question their efficiency and the credibility of conclusions reached. We evaluate schemes for adaptive redesign, extending the theoretical arguments for use of sufficient statistics of Tsiatis & Mehta (2003) and assessing the possible benefits of preplanned adaptive designs by numerical computation of optimal tests; these optimal adaptive designs are concrete examples of optimal sequentially planned sequential tests proposed by Schmitz (1993). We conclude that the flexibility of unplanned adaptive designs comes at a price and we recommend that the appropriate power for a study should be determined as thoroughly as possible at the outset. Then, standard error-spending tests, possibly with unevenly spaced analyses, provide efficient designs, but it is still possible to fall back on flexible methods for redesign should study objective change unexpectedly once the trial is under way.

On Bartlett correction of empirical likelihood in the presence of nuisance parameters

Abstract: SUMMARY Lazar & Mykland (1999) showed that an empirical likelihood defined by two estimating equations with a nuisance parameter need not be Bartlett-correctable. This paper shows that Bartlett correction of empirical likelihood in the presence of a nuisance parameter depends critically on the way the nuisance parameter is removed when formulating the likelihood for the parameter of interest. We establish in the broad framework of estimating functions that the empirical likelihood is still Bartlett-correctable if the nuisance parameter is profiled out given the value of the parameter of interest.

Bayesian alignment using hierarchical models, with applications in protein bioinformatics

Abstract: SUMMARY An important problem in shape analysis is to match configurations of points in space after filtering out some geometrical transformation. In this paper we introduce hierarchical models for such tasks, in which the points in the configurations are either unlabelled or have at most a partial labelling constraining the matching, and in which some points may only appear in one of the configurations. We derive procedures for simultaneous inference about the matching and the transformation, using a Bayesian approach. Our hierarchical model is based on a Poisson process for hidden true point locations; this leads to consider able mathematical simplification and efficiency of implementation of EM and Markov chain Monte Carlo algorithms. We find a novel use for classical distributions from directional statistics in a conditionally conjugate specification for the case where the geometrical transformation includes an unknown rotation. Throughout, we focus on the case of affine or rigid motion transformations. Under a broad parametric family of loss functions, an optimal Bayesian point estimate of the matching matrix can be constructed that depends only on a single parameter of the family. Our methods are illustrated by two applications from bioinformatics. The first problem is of matching protein gels in two dimensions, and the second consists of aligning active sites of proteins in three dimensions. In the latter case, we also use information related to the grouping of the amino acids, as an example of a more general capability of our methodology to include partial labelling information. We discuss some open problems and suggest directions for future work.

Modelling of covariance structures in generalised estimating equations for longitudinal data

Abstract: SUMMARY When used for modelling longitudinal data generalised estimating equations specify a working structure for the within-subject covariance matrices, aiming to produce efficient parameter estimators. However, misspecification of the working covariance structure may lead to a large loss of efficiency of the estimators of the mean parameters. In this paper we propose an approach for joint modelling of the mean and covariance structures of longitudinal data within the framework of generalised estimating equations. The resulting estimators for the mean and covariance parameters are shown to be consistent and asymptotically Normally distributed. Real data analysis and simulation studies show that the proposed approach yields efficient estimators for both the mean and covariance parameters.

Applying the Horvitz-Thompson criterion in complex designs: A computer-intensive perspective for estimating inclusion probabilities

Abstract: A modification of the Horvitz-Thompson estimator is proposed for complex sampling designs. The inclusion probabilities are estimated by means of independent replications of the sampling scheme. The properties of the resulting estimator are derived. Guidelines for choosing the appropriate number of replications are given and some applications are considered.

Analysing panel count data with informative observation times

Abstract: In this paper, we study panel count data with informative observation times. We assume nonparametric and semiparametric proportional rate models for the underlying event process, where the form of the baseline rate function is left unspecified and a subject-specific frailty variable inflates or deflates the rate function multiplicatively. The proposed models allow the event processes and observation times to be correlated through their connections with the unobserved frailty; moreover, the distributions of both the frailty variable and observation times are considered as nuisance parameters. The baseline rate function and the regression parameters are estimated by maximising a conditional likelihood function of observed event counts and solving estimation equations. Large-sample properties of the proposed estimators are studied. Numerical studies demonstrate that the proposed estimation procedures perform well for moderate sample sizes. An application to a bladder tumour study is presented.

Semiparametric efficient estimation of survival distributions in two-stage randomisation designs in clinical trials with censored data

Abstract: SUMMARY Two-stage randomisation designs are useful in the evaluation of combination therapies where patients are initially randomised to an induction therapy and then, depending upon their response and consent, are randomised to a maintenance therapy. In this paper we derive the best regular asymptotically linear estimator for the survival distribution and related quantities of treatment regimes. We propose an estimator which is easily computable and is more efficient than existing estimators. Large-sample properties of the proposed estimator are derived and comparisons with other estimators are made using simulation.

A Haar-Fisz technique for locally stationary volatility estimation

Abstract: SUMMARY We consider a locally stationary model for financial log-returns whereby the returns are independent and the volatility is a piecewise-constant function with jumps of an unknown number and locations, defined on a compact interval to enable a meaningful estimation theory. We demonstrate that the model explains well the common characteristics of log returns. We propose a new wavelet thresholding algorithm for volatility estimation in this model, in which Haar wavelets are combined with the variance-stabilising Fisz transform. The resulting volatility estimator is mean-square consistent with a near-parametric rate, does not require any pre-estimates, is rapidly computable and is easily implemented. We also discuss important variations on the choice of estimation parameters. We show that our approach both gives a very good fit to selected currency exchange datasets, and achieves accurate long- and short-term volatility forecasts in comparison to the GARCH(1, 1) and moving window techniques.

Semiparametric transformation models for the case-cohort study

Abstract: A general class of semiparametric transformation models is studied for analysing survival data from the case-cohort design, which was introduced by Prentice (1986). Weighted estimating equations are proposed for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance-covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to a case-cohort dataset from the Atherosclerosis Risk in Communities study is also given to illustrate the methodology.

Spatially adaptive smoothing splines

Abstract: SUMMARY We use a reproducing kernel Hilbert space representation to derive the smoothing spline solution when the smoothness penalty is a function A(t) of the design space t, thereby allowing the model to adapt to various degrees of smoothness in the structure of the data. We propose a convenient form for the smoothness penalty function and discuss computational algorithms for automatic curve fitting using a generalised crossvalidation measure. of order m, where m is typically taken to be two. The smoothing spline solution uses a global smoothing parameter il which implies that the true underlying mean process has a constant degree of smoothness. In this paper we suggest a more general, 'spatially adaptive', framework that accommodates varying degrees of roughness by seeking solutions where the smoothness penalty A(t) is now a function of the design t. We derive the solution within a reproducing kernel Hilbert space (Gu, 2002, Ch. 2; Wahba, 1990, Ch. 1). When applying the method to data we propose a piecewise-constant model for the smoothing function A(t). This provides a convenient computational framework with closed-form solutions for the corresponding reproducing kernels of the Hilbert space. These kernels show similarities to regression splines in that the use of a piecewise-constant i~(t) induces breaks in the mth, ... ., (2m -l1)th derivatives at the changepoints in L(t).

Estimating survival under a dependent truncation

Abstract: SUMMARY The product-limit estimator calculated from data subject to random left-truncation relies on the testable assumption of quasi-independence between the failure time and the truncation time. In this paper, we propose a model for a truncated sample of pairs (Xi, Y) satisfying 1 > Xi. A possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas. The model also features a distribution function Fx(.) and a survival distribution Sy(.) associated with the marginal behaviours of X and Y in the observable region Y> X. Semiparametric estimators for these two functions are proposed; they do not make any parametric assumption about either Fx(.) or Sy(.). We derive an estimator for the copula parameter cX based on the conditional Kendall's tau. We generalise the copula-graphic estimators of Zheng & Klein (1995) to truncated variables. The asymptotic distributions of all these estimators are then investigated. The methods are illustrated with a real dataset on HIV infection by transfusion and by simulations.

Efficient semiparametric estimator for heteroscedastic partially linear models

Abstract: SUMMARY We study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function. An interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated. To avoid this, we propose a family of consistent estimators and investigate their asymptotic properties. We show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family. Building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing, we conjecture that a weighted partial spline estimator could also be semiparametric efficient. Properties of the proposed estimators are presented through theoretical illustration and numerical simulations.

Multi-level modelling under informative sampling

Abstract: We consider a model-dependent approach for multi-level modelling that accounts for informative probability sampling of first- and lower-level population units. The proposed approach consists of first extracting the hierarchical model holding for the sample data given the selected sample, as a function of the corresponding population model and the first- and lower-level sample selection probabilities, and then fitting the resulting sample model using Bayesian methods. An important implication of the use of the model holding for the sample is that the sample selection probabilities feature in the analysis as additional data that possibly strengthen the estimators. A simulation experiment is carried out in order to study the performance of this approach and compare it to the use of ‘design-based’ methods. The simulation study indicates that both approaches perform in general equally well in terms of point estimation, but the model-dependent approach yields confidence/credibility intervals with better coverage properties. Another simulation study assesses the impact of misspecification of the models assumed for the sample selection probabilities. The use of maximum likelihood estimation is also considered.

Linear life expectancy regression with censored data

Abstract: SUMMARY In the statistical literature, life expectancy is usually characterised by the mean residual life function. Regression models are thus needed to study the association between the mean residual life functions and their covariates. In this paper, we consider a linear mean residual life model and develop inference procedures in the presence of potential censoring. The new model and inference procedures are applied to the Stanford heart transplant data. Semiparametric efficiency calculations and information bounds are also considered.

Nonparametric k-sample tests with panel count data

Abstract: We study the nonparametric k-sample test problem with panel count data. The asymptotic normality of a smooth functional of the nonparametric maximum pseudolikelihood estimator (Wellner & Zhang, 2000) is established under some mild conditions. We construct a class of easy-to-implement nonparametric tests for comparing mean functions of k populations based on this asymptotic normality. We conduct various simulations to validate and compare the tests. The simulations show that the tests perform quite well and generally have good power to detect differences among the mean functions. The method is illustrated with a real-life example.

Efficient Bayes factor estimation from the reversible jump output

Abstract: SUMMARY We propose a class of estimators of the Bayes factor which is based on an extension of the bridge sampling identity of Meng & Wong (1996) and makes use of the output of the reversible jump algorithm of Green ( 1995). Within this class we give the optimal estimator and also a suboptimal one which may be simply computed on the basis of the acceptance probabilities used within the reversible jump algorithm for jumping between models. The proposed estimators are very easily computed and lead to a substantial gain of efficiency in estimating the Bayes factor over the standard estimator based on the reversible jump output. This is illustrated through a series of Monte Carlo simulations involving a linear and a logistic regression model.