Biometrika - current issue

Hierarchical testing of variable importance

Meinshausen, N. — 2008-05-24

A frequently encountered challenge in high-dimensional regression is the detection of relevant variables. Variable selection suffers from instability and the power to detect relevant variables is typically low if predictor variables are highly correlated. When taking the multiplicity of the testing problem into account, the power diminishes even further. To gain power and insight, it can be advantageous to look for influence not at the level of individual variables but rather at the level of clusters of highly correlated variables. We propose a hierarchical approach. Variable importance is first tested at the coarsest level, corresponding to the global null hypothesis. The method then tries to attribute any effect to smaller subclusters or even individual variables. The smallest possible clusters, which still exhibit a significant influence on the response variable, are retained. It is shown that the proposed testing procedure controls the familywise error rate at a prespecified level, simultaneously over all resolution levels. The method has power comparable to the Bonferroni–Holm procedure on the level of individual variables and dramatically larger power for coarser resolution levels. The best resolution level is selected adaptively.

On weighted Hochberg procedures

Tamhane, A. C., Liu, L. — 2008-05-24

We consider different ways of constructing weighted Hochberg-type step-up multiple test procedures including closed procedures based on weighted Simes tests and their conservative step-up short-cuts, and step-up counterparts of two weighted Holm procedures. It is shown that the step-up counterparts have some serious pitfalls such as lack of familywise error rate control and lack of monotonicity in rejection decisions in terms of p-values. Therefore an exact closed procedure appears to be the best alternative, its only drawback being lack of simple stepwise structure. A conservative step-up short-cut to the closed procedure may be used instead, but with accompanying loss of power. Simulations are used to study the familywise error rate and power properties of the competing procedures for independent and correlated p-values. Although many of the results of this paper are negative, they are useful in highlighting the need for caution when procedures with similar pitfalls may be used.

A family of Bayes multiple testing procedures

Cohen, A., Sackrowitz, H. B., Xu, M., Buyske, S. — 2008-05-24

Under the model of independent test statistics, we propose a two-parameter family of Bayes multiple testing procedures. The two parameters can be viewed as tuning parameters. Using the Benjamini–Hochberg step-up procedure for controlling false discovery rate as a baseline for conservativeness, we choose the tuning parameters to compromise between the operating characteristics of that procedure and a less conservative procedure that focuses on alternatives that a priori might be considered likely or meaningful. The Bayes procedures do not have the theoretical and practical shortcomings of the popular stepwise procedures. In terms of the number of mistakes, simulations for two examples indicate that over a large segment of the parameter space, the Bayes procedure is preferable to the step-up procedure. Another desirable feature of the procedures is that they are computationally feasible for any number of hypotheses.

Kernel stick-breaking processes

Dunson, D. B., Park, J.-H. — 2008-05-24

We propose a class of kernel stick-breaking processes for uncountable collections of dependent random probability measures. The process is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and beta-distributed random weights are assigned to each location. Predictor-dependent random probability measures are then constructed by mixing over the locations, with stick-breaking probabilities expressed as a kernel multiplied by the beta weights. Some theoretical properties of the process are described, including a covariate-dependent prediction rule. A retrospective Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using a simulated example and an epidemiological application.

Objective Bayesian analysis for the Student-t regression model

Fonseca, T. C. O., Ferreira, M. A. R., Migon, H. S. — 2008-05-24

We develop a Bayesian analysis based on two different Jeffreys priors for the Student-t regression model with unknown degrees of freedom. It is typically difficult to estimate the number of degrees of freedom: improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. We show that Bayesian analysis with either of the two considered Jeffreys priors provides a proper posterior distribution. Finally, we show that Bayesian estimators based on Jeffreys analysis compare favourably to other Bayesian estimators based on priors previously proposed in the literature.

Multi-parameter automodels and their applications

Hardouin, C., Yao, J.-F. — 2008-05-24

Motivated by the modelling of non-Gaussian data or positively correlated data on a lattice, extensions of Besag's automodels to exponential families with multi-dimensional parameters have been proposed recently. We provide a multiple-parameter analogue of Besag's one-dimensional result that gives the necessary form of the exponential families for the Markov random field's conditional distributions. We propose estimation of parameters by maximum pseudolikelihood and give a proof of the consistency of the estimators for the multi-parameter automodel. The methodology is illustrated with examples, in particular the building of a cooperative system with beta conditional distributions. We also indicate future applications of these models to the analysis of mixed-state spatial data.

Estimating functions for inhomogeneous spatial point processes with incomplete covariate data

Waagepetersen, R. — 2008-05-24

The R package spatstat provides a very flexible and useful framework for analysing spatial point patterns. A fundamental feature is a procedure for fitting spatial point process models depending on covariates. However, in practice one often faces incomplete observation of the covariates and this leads to parameter estimation error which is difficult to quantify. In this paper, we introduce a Monte Carlo version of the estimating function used in spatstat for fitting inhomogeneous Poisson processes and certain inhomogeneous cluster processes. For this modified estimating function, it is feasible to obtain the asymptotic distribution of the parameter estimators in the case of incomplete covariate information. This allows a study of the loss of efficiency due to the missing covariate data.

Modelling multiple time series via common factors

Pan, J., Yao, Q. — 2008-05-24

We propose a new method for estimating common factors of multiple time series. One distinctive feature of the new approach is that it is applicable to some nonstationary time series. The unobservable, nonstationary factors are identified by expanding the white noise space step by step, thereby solving a high-dimensional optimization problem by several low-dimensional sub-problems. Asymptotic properties of the estimation are investigated. The proposed methodology is illustrated with both simulated and real datasets.

Simultaneous confidence bands in spectral density estimation

Neumann, M. H., Paparoditis, E. — 2008-05-24

We propose a method for the construction of simultaneous confidence bands for a smoothed version of the spectral density of a Gaussian process based on nonparametric kernel estimators obtained by smoothing the periodogram. A studentized statistic is used to determine the width of the band at each frequency and a frequency-domain bootstrap approach is employed to estimate the distribution of the supremum of this statistic over all frequencies. We prove by means of strong approximations that the bootstrap estimates consistently the distribution of the supremum deviation of interest and, consequently, that the proposed confidence bands achieve asymptotically the desired simultaneous coverage probability. The behaviour of our method in finite-sample situations is investigated by simulations and a real-life data example demonstrates its applicability in time series analysis.

Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity

Li, G., Li, W. K. — 2008-05-24

We consider a unified least absolute deviation estimator for stationary and nonstationary fractionally integrated autoregressive moving average models with conditional heteroscedasticity. Its asymptotic normality is established when the second moments of errors and innovations are finite. Several other alternative estimators are also discussed and are shown to be less efficient and less robust than the proposed approach. A diagnostic tool, consisting of two portmanteau tests, is designed to check whether or not the estimated models are adequate. The simulation experiments give further support to our model and the results for the absolute returns of the Dow Jones Industrial Average Index daily closing price demonstrate their usefulness in modelling time series exhibiting the features of long memory, conditional heteroscedasticity and heavy tails.

On the asymptotics of penalized splines

Li, Y., Ruppert, D. — 2008-05-24

We study the asymptotic behaviour of penalized spline estimators in the univariate case. We use B-splines and a penalty is placed on mth-order differences of the coefficients. The number of knots is assumed to converge to infinity as the sample size increases. We show that penalized splines behave similarly to Nadaraya--Watson kernel estimators with ‘equivalent’ kernels depending upon m. The equivalent kernels we obtain for penalized splines are the same as those found by Silverman for smoothing splines. The asymptotic distribution of the penalized spline estimator is Gaussian and we give simple expressions for the asymptotic mean and variance. Provided that it is fast enough, the rate at which the number of knots converges to infinity does not affect the asymptotic distribution. The optimal rate of convergence of the penalty parameter is given. Penalized splines are not design-adaptive.

Nonparametric variance estimation in the analysis of microarray data: a measurement error approach

Carroll, R. J., Wang, Y. — 2008-05-24

We investigate the effects of measurement error on the estimation of nonparametric variance functions. We show that either ignoring measurement error or direct application of the simulation extrapolation, SIMEX, method leads to inconsistent estimators. Nevertheless, the direct SIMEX method can reduce bias relative to a naive estimator. We further propose a permutation SIMEX method that leads to consistent estimators in theory. The performance of both the SIMEX methods depends on approximations to the exact extrapolants. Simulations show that both the SIMEX methods perform better than ignoring measurement error. The methodology is illustrated using microarray data from colon cancer patients.

Model diagnosis for parametric regression in high-dimensional spaces

Stute, W., Xu, W. L., Zhu, L. X. — 2008-05-24

We study tools for checking the validity of a parametric regression model. When the dimension of the regressors is large, many of the existing tests face the curse of dimensionality or require some ordering of the data. Our tests are based on the residual empirical process marked by proper functions of the regressors. They are able to detect local alternatives converging to the null at parametric rates. Parametric and nonparametric alternatives are considered. In the latter case, through a proper principal component decomposition, we are able to derive smooth directional tests which are asymptotically distribution-free under the null model. The new tests take into account precisely the ‘geometry of the model’. A simulation study is carried through and an application to a real dataset is illustrated.

Determining the dimension of the central subspace and central mean subspace

Zeng, P. — 2008-05-24

The central subspace and central mean subspace are two important targets of sufficient dimension reduction. We propose a weighted chi-squared test to determine their dimensions based on matrices whose column spaces are exactly equal to the central subspace or the central mean subspace. The asymptotic distribution of the test statistic is obtained. Simulation examples are used to demonstrate the performance of this test.

The prognostic analogue of the propensity score

Hansen, B. B. — 2008-05-24

The propensity score collapses the covariates of an observational study into a single measure summarizing their joint association with treatment conditions; prognostic scores summarize covariates' association with potential responses. As with propensity scores, stratification on prognostic scores brings to uncontrolled studies a concrete and desirable form of balance, a balance that is more familiar as an objective of experimental control. Like propensity scores, prognostic scores can reduce the dimension of the covariate, yet causal inferences conditional on them are as valid as are inferences conditional only on the unreduced covariate. As a method of adjustment unto itself, prognostic scoring has limitations not shared with propensity scoring, but it holds promise as a complement to the propensity score, particularly in certain designs for which unassisted propensity adjustment is difficult or infeasible.

Diagnostic measures for empirical likelihood of general estimating equations

Zhu, H., Ibrahim, J. G., Tang, N., Zhang, H. — 2008-05-24

We develop diagnostic measures for assessing the influence of individual observations when using empirical likelihood with general estimating equations, and we use these measures to construct goodness-of-fit statistics for testing possible misspecification in the estimating equations. Our diagnostics include case-deletion measures, local influence measures and pseudo-residuals. Our goodness-of-fit statistics include the sum of local influence measures and the processes of pseudo-residuals. Simulation studies are conducted to evaluate our methods, and real datasets are analyzed to illustrate the use of our diagnostic measures and goodness-of-fit statistics.

A note on deletion diagnostics for estimating equations

Preisser, J. S., Qaqish, B. F., Perin, J. — 2008-05-24

We describe an algorithm based upon the Sherman–Morrison–Woodbury formula for the inversion of matrices with special structure that occur in formulae for deletion diagnostics. Substantial computational savings relative to a method based upon Cholesky's decomposition are illustrated. The result has broad application to regression diagnostics for clustered data.

A new class of average moment matching priors

Ganesh, N., Lahiri, P. — 2008-05-24

We derive a new class of priors for the variance component in the Fay–Herriot model, a mixed regression model widely used in small area estimation. This class includes the well-known uniform or superharmonic prior. Through simulation we illustrate the use of our class of priors.